Ionel Dinu shared a new video of a wire loop with the electrigen ("air" analog) and aether ("water" analog) flow dynamics to this/these interface(s) around it where he uses computational fluid dynamics to model his aether theory:

(Ionel had published a previous similar video that was not as clear/informational and seemingly deleted it, so here's an archive for anyone interested.)

It appears Ionel has also published a new paper (added to the end of the previously shared list):

The comments have some interesting and great remarks:

It may possible that we could actually take the computational fluid dynamics (CFD) approach and theory that Ionel Dinu has pulled together and possibly integrate it with Joseph Cater's work as well... More thought and study would be required on this topic as well as a re-read of both corpora of work.

In one of the comments Ionel requested that someone try placing an electromagnet onto a ferrocell and Lori Gardi (Fractal Woman) had already done this: Fractal Woman - YouTube

Ionel's video on the topic of electrigen was very interesting and the comments on "water" and "air" were of special interest. It makes sense to have "multiple" ethers, so to speak. Even Paul LaViolette's proposition posits that this is a necessity in his reaction-diffusion ether model aka Subquantum Kinetics (eg, see the Belousov-Zhabotinskii reaction).

It seems as though there may be a possibility that his electrigen (aka the "air") moving the aether ("water") is perhaps the only thing we're really familiar with that evokes motions or change in the aether. It appears to be a or the motive force.

I also cannot help but wonder if this is also "spirit" in some way or another - or is there another "will" force that is the spirit that is yet unnamed and a 3rd "aether"?

Sorry this is a somewhat nebulous and not-wellformed thought. I wanted to mostly document this so that it doesn't slip away and since it has been nagging at me for quite a long time.

The magnetic field (or, as I prefer to say, the magnetic current) of a DC-carrying wire is helical shaped like in the image below:

Imagine that you are turning this helix to the right holding it on the near end (where the red arrow is). In this way one can make a simulation of the magnetic helical motion in a DC-carrying wire.
When you shape the wire in a loop, then you get this:

That's why I say that the magnetic field of an electromagnet (but also of a permanent magnet) is twisted, something similar to a stranded wire:

The black part should represent the (electro)magnet, whereas the stranded wire should represent the invisible magnetic field.
There are many experimental indications and proofs that this is true.

I cannot stress strongly enough of how huge importance is to understand that the magnetic field of a current-carrying wire is helical shaped.

So @Mitko_Gorgiev what you seem to be saying is that the magnetic/"m"(/ aka aether) flow (which is the outer "water" or liquid as it were in the animation) is also flowing helically around and may not be fully illustrated in the animation, correct?

What are your thoughts on the "electrigen"?... and what visual/mental modeling improvements might you suggest for that?

Dear Soretna,
I have to write a longer reply to answer your question. So, please give me some time.
Regarding the first passage of your reply, I can say this: the magnetic flow (just as the electric flow) is not around the wire(s), but in the wire(s).
However, its effect is present also around the wire(s).
The animation in the video follows the tremendous misconception of the contemporary science, which is this:

I'm sorry for the additional trouble - thank you for offering @Mitko_Gorgiev. I look forward to your details.

I have spent a good bit of time with Ionel Dinu's papers before discovering your work, so I have to admit a greater familiarity and comfort level in terms of understanding and so forth.

Again, I hate to cause you any additional burden or frustration and very much appreciate your endeavoring to explain things from your perspective / theory!

I am, however, not married to any singular theory. I am attempting to understand every one that I encounter so as to be able to come to the truth or synthesize a superior functional mental model if possible that I can attach truth to and that which may provide proper context for expansion beyond.

No problem at all.
I will write not only for you, but for many others who will read the posts.
Regarding the things said in your last passage, you should ask yourself, which theory fits in the experiments the best, that is, without any contradictions. The experiments are all that matters.
The problem nowadays is that most of the people don't make experiments, but they only want to philosophize.
I have presented experiments where electricity is flowing in an open circuit. With those experiments I can ruin all the existing theories at once. But the people don't have the courage even to replicate them, because their beliefs will be ruined. So, they will only say that it is fake and go on with their empty beliefs.

Thanks so much. Indeed you are correct and this is what I am already doing insomuch as my own knowledge and awareness are aware of such things. Obviously one may not (perhaps cannot) know all of the experiments available. Such a compilation that succinctly states such and offers a compendium for such reference would be invaluable... makes me wonder about making such a thing in a digital searchable format. Hm.

@Mitko_Gorgiev , @Soretna , thanks so much for bringing up this topic and your comments on it...I was musing about this subject more from the perspective of geometry , which I have always believed can provide astonishing insights that might otherwise elude us .

So , here is a research work that you must look at...you see , the helix form is deeply connected to a toroidal shape & that has profound significance in nature - even DNA is a double Helix structure ! While on the face of it , that by itself has nothing to with electromagnetism , but when you see all the shapes listed in this excellent research , you will start getting ideas & inspiration , sparks of intuition , which I am sure would help us model this even better and further unravel this mystery - possibly creating a new paradigm for science !!

There is so much depth and invaluable content in the enclosed article that I had to split it up into 2 parts , just so that mindreach platform could allow me to mail this across to you ! So here goes Part 1 , which will be continued in my immediate next post as Part 2 :

The purpose of this document is threefold. Firstly it constitutes a further development of the possibilities of 3D visualization of the 5 nonagons previously outlined with respect to what was termed a Concordian Mandala -- in contrast with the Discordian Mandala in 2D (Three-dimensional Configuration of Nonagon Mandala, 2016). Those experiments made it apparent that a key to this development lay in helicoidal (sinusoidal) windings on a torus rather than a simple ring structure, especially in a quest for parameters which might be consistent with the pattern of a set of Borromean rings.

The second purpose is to render comprehensible for others the approach to the dynamics of 3D visualization of the helicoidal structure using readily accessible software through with the relevant mathematical formula could enable the configuration. In this sense it is intended as a guide reflecting the progressive learnings of anyone with only a modest degree of competence in the mathematics or the visualization software. The focus is to enable dissemination of such models over the web via standard web browsers. Given the time taken (by the author) to discover how to render what is indicated below, the intention is to reduce significantly the obstacles to others of exploring such possibilities further.

Contextually the theme is of some relevance in a period when the market for virtual reality devices is about to expand significantly, when a 2016 Nobel Prize in Physics was awarded for insights into the mathematics of vortices, and when the US presidential campaign of 2016 is demonstrating the fundamental inability of democracies to transcend the primitive dynamics to which many would appear to subscribe -- for lack of any credible alternative.

Rather than proposing any new "model", one purpose of the following exploration is to enable model exploration of a new kind -- with a visual dimension enhancing comprehension in new ways, somewhat reminiscent of the explosion of interest in online gaming. The following could be understood as creating a learning pathway or bridge between the "trivial" for the few and the "obscure" for the many.

Potential merit of a toroidal helical configuration

Preliminary indications are relevant regarding the appropriateness of the torus-helical combination for holding cognitive dynamics and changing perspectives. Indications include widespread understanding and development of the following perspectives:

cycles: There is widespread interest in cycles, whether business cycles, economic cycles (longer and shorter term), weather cycles, social and historical cycles. With respect to change, there is particular interest in the adaptive cycle and the challenges of its navigation

continuity and recurrence: There is interest of a different style in "return", especially recognition of "eternal return"

multiple cycles: Clearly there is a case for exploring multiple cycles, notably those in parallel, as well as any form of cyclic displacement

phasing: Cycles highlight interest in the recognizable phases they embody, namely phase transitions within a cycle

technological significance: Fundamental to the industrial revolution was the transition to electromechanical and radio technologies:

rope design: clearly helical winding has long been essential to reinforcing the strength of ropes

springs: typically helical windings in springs are in linear rather than toroidal form

dynamos: helical coils, typically requiring toroidal windings, are fundamental to the generation of electricity

motors: helical coils, typically in toroidal form, are fundamental to the transfer of electricity into movement

aerials: reception and transmission of radio signals may well require aerials of helical and/or toroidal form

biological significance: This is most strikingly evident in the relatively recent discovery of the coiled coil of the DNA molecule so vital to genetic replication (as discussed below)

aesthetic design: There is a long tradition, and continuing exploration, of circular designs for rings, bracelets and necklaces -- notably involving a degree of interweaving of threads around the circle (as with knot rings, head rings)

As noted in concluding the earlier experiments, there appeared to be value in exploring configurations suggested by the following. However it also became clear that there was extensive literature on helical coils (and springs) but relatively limited references to their toroidal form -- especially with respect to their rendering using 3D applications.

Illustrative depictions of a circular helix
Rope quoit Double-spun helix Circular helix Torus knot

Following from the earlier experiments with 5 nonagons, the approach envisaged was to have 5 helical windings around a common torus -- each winding to be understood as a nonagon. However, rather than the nonagons being of polygonal (or polyhedral) form, the concern was how to combine the 5-fold and 9-fold patterns into an aesthetically agreeable visualization, for reasons articulated in the earlier paper (Meaningful configuration engendered only by tacit aesthetic entanglement, 2016).

As noted with regard to the previous experiments, a degree of guidance was kindly provided by Sergey Bederov, Senior Developer of Cortona3D. The latter is a web plugin which enables renderings in web browsers of 3D models according to the legacy virtual reality modelling standard (VRML). The models were however developed for these experiments using the X3D Edit application (namely according to X3D, a more recent standard), and exported into VRML. Bederov provided models (presented in the previous paper) to show that the Discordian 2D representation could not be presented in 3D in a manner consistent with the Borromean condition. However he provided an alternative which evoked the possibility of a sinusoidal form of nonagon -- the focus of this document.

To that end Bederov provided a formula through which a toroidal knot could be constructed, consistent with the image on the right above. As is perhaps more evident in the image below, this seemingly involves 2 windings which are however continuous, not separate. It is therefore merely indicative of the need for a distinct formula to pursue the configuration of nonagons as envisaged here.

The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of rotational symmetry. If p and q are not relatively prime, then we have a torus link with more than one component. where r = cos(qφ) + 2 and r=cos(qφ)+2 and 0 < φ < 2π and 0 < φ < 2 π ~

Visualization of a toroidal knot (using 201 rows of xyz coordinates)
Formula x=rCos(pφ) y=rSin(pφ) z= -Sin(pφ)
Spreadsheet (A1=0)
(namely 12/201= 0.062832) A2=A1+0.062832 B1=SIN(A1)(1+COS(A14.5)0.3) C1=COS(A1)(1+COS(A1*4.5)0.3) D1=SIN(A14.5)*0.3

Continuous toroidal knot with 9 windings
(Interactive variants: X3D or VRML)

Deriving coordinates for a toroidal helix (helical toroid)

Although trivial from a mathematical perspective, a concern (as indicated above) was how to get from a formula to a 3D rendering through appropriate software, using a set of coordinates which are aesthetically agreeable -- however this might be understood in this context. Some of the experiments undertaken to converge on the requisite parameters for the formula are presented below.

The key parameters are for a helix of n winds wrapped around a torus of major radius R and minor radius r -- with t based on the number of sets of xyz coordinates. The quest was for an optimal balance between these four parameters -- once the implications of varying one with respect to the other were (finally) understood. The process was constrained by only modest competence in spreadsheet usage, the geometry and the 3D editing software. The formula for the coordinates in 3D are as follows:

Visualization of a toroidal helix (using 201 rows of xyz coordinates, for 9 winds; R=6, r=3)
Column A
(A1=0) Column B
(x coordinate) Column C
(y coordinate) Column D
(z coordinate)
Formula x=(R+rcos(nt))cos(t) y=(R+rcos(nt))sin(t) z= r(sin(nt))
Spreadsheet
(namely 2π/201= 0.062832) A2=A1+0.062832 B1=(6+COS(A29))(COS(A2)) C1=(6+COS(A29))(SIN(A2)) D1=3SIN(A29)
or 2π/100=0.0628318? (for 72) A2=A1+0.0628318 B1=(6+3COS(A272))(COS(A2)) C1=(6+3COS(A272))(SIN(A2)) D1=3SIN(A272)

Experimental "extremes" in the quest for desirable parameters

The decision finally taken was the following, in the light of the focus on 9 and 5 in this exercise:

given the quest for an aesthetic solution, it appeared appropriate to make use of the golden ratio (symbolized by Φ) in seeking an appropriate relationship between R (the major radius) and r (the minor radius) of the torus. This suggested that R/r should be 1.61803. With R (arbitrarily) specified as 6, this gave r=6/1.61803 = 3.708.

given the quest for a sinusoidal nonagon, alternatives examined were r = 3.708/3 = 1.2361, or r = 3.708/9 = 0.412.

the last was chosen. Use of Φ (1.61803) on R=6 then r=6/1.61803 = 3.708 or 3.708/3= 1.23607 or 3.708/9=0.412

the number of windings was defined by the experiment as 9, consistent with the focus on nonagons

the number of sets of xyz coordinates appropriate to visualization of a coil was initially set arbitrarily at 201 (following the Bederov model). The more such sets, the smoother the visual rendering. Further reflection suggested that the number of segments in a given coil could be related to the 9-fold emphasis. Consideration was given to having 9x9 (namely 34), 3x27 (35), and 2x81 (2x92). The last was finally chosen, namely 162, since it gave 18 segments in each sinusoidal loop (namely 2x9 in each half loop). The number of sets should be a multiple of the number of windings; a multiple of 4 gives a 4-sided form to each winding, 5 a pentagonal, etc -- the higher the multiple the smoother -- a point of consideration where the number of windings is much higher -- then possibly useful to reduce the number of digits after the decimal in each coordinate

The choices above enabled 162 sets of xyz coordinates to be constructed through spreadsheet formulae as follows, where the number of sets of coordinates is related to 2π (by 2π/162 =0.038785. [This was incorrectly stated as 0.07757 in an earlier version, which gave rise to a helix of linear elements, possibly useful for some purposes]. The formula for each column were then distributed down to A162

Column A
(A1=0) Column B
(x coordinate) Column C
(y coordinate) Column D
(z coordinate)
(R+rcos(nt))cos(t) (R+cos(nt))sin(t) rsin(nt)
A1+0.038785 (R+rCOS(A19))(COS(A1)) (R+rCOS(A19))(SIN(A1)) rSIN(A1*9)

The further decision required was the diameter of the sinusoidal winding around an invisible torus (namely one which would not be visually rendered). This was achieved within the extrusion option of the X3D-Edit application. The choices there were:

cross-section of the sinusoidal winding: given its relatively small diameter this could be defined by a polygon of a limited number of sides. 5 was chosen

scale of cross section: this was scaled down to 30%

following the previous experiment, 5 such sinusoidal coils were extruded and coloured distinctively

Convergence on desired configuration (with interactive variants)
3 toroidal coils
(X3D or VRML) Single toroidal coil
(X3D or VRML) 5 toroidal coils
(X3D or VRML)

Dynamic possibilities of 3D visualizations of a helical toroid with multiple windings

The 3D rendering of a torus wound with 5 sinusoidal nonagons (indicated above) offers possibilities for exploring dynamics which could be added to it. This implies choices relating to the position and movement of the coils relative to one another.

One approach was to move each coil outwards, in the plane of the torus, in a pentagonal pattern. Two possibilities were explored:

moving them outward to the sides of the pentagon, starting a rotation out of the plane from 50 percent of the displacement. Each coil then rotates around one side of the pentagon. Screen shots of the sequence are shown immediately below

moving them outward to the diagonals of the pentagon, starting a rotation out of the plane from the limit of that displacement. Screen shots of the sequence are shown in a second set of images below

Animations of possible dynamics of toroidal 5 coil configuration
(Variants: interactive X3D or VRML; videos: MP4; MOV)
"Top views"

"Side views"

The approach taken followed from the possibility that the sinusoidal coiling might create "windows" during the cycle of rotation which would correspond to the Borromean non-touching condition -- if the coils were appropriately out of phase. Other choices of parameters (deeper or thinner coils, larger diameter torus, etc) might offer better possibilities or larger "windows".

Movement of spheres along the pathways of a toroidal helix

As part of both exercises, the possibility of moving a sphere along the pathway defined by each coil was explored. The coordinates of the "spine" of the extruded pathway were used. The result is evident from the images above (and especially the animations). It is then also of interest to switch the visualization into wireframe mode. One purpose of this exercise is partially enabled by consideration of the spheres as a focus of attention, whether individually or collectively. This relates to concerns with eye-scanning of an image.

In the first approach, the speed of movement of the spheres was timed to match that of the rotational cycle of the animation (namely using the same "clock"). Clearly the exercise can be taken further by increasing the speed (namely using a second "clock"). Of particular interest is using a much greater speed, to the degree that the location of a sphere cannot then be determined at any given time. Visually the pathway of each coil is then defined primarily by the movement of the spheres rather than by the "tunnel" constituted by each coil -- especially in wireframe mode. The coils could in fact be rendered transparent, if only for a portion of the cycle (obviating the need to use wireframe mode).

One purpose of this approach follows from the possibility that a "Borromean condition" of a kind would be defined by the movement of the spheres rather than by that of the coiled pathways themselves. In other words, rather than expecting a condition in which the coils would not touch during their cycle, the focus would then be on whether the spheres collided during the course of their movement along the coiled pathways.

Using the spheres to define the cycle is consistent with the sinusoidal pathway of the coils, effectively then even more suggestive of a sine wave with its implications.

Potential cognitive implications of toroidal helical movement

As stressed above, one purpose of this exploration is to determine whether the visualized dynamics in 3D are suggestive of fruitful ways of mapping complexity in 3D -- complexity which it is possibly difficult, if not impossible, to visualize in 2D. Some possibilities are discussed separately (Cognitive implication of toroidal forms and dynamics, 2011)

The issue here is whether the visual effects offer integrative insights enabling cognitive content to be associated comprehensibly with features of the configuration. As a "holy grail" for this exercise, the assumption is made that the Borromean condition -- if it can be engendered by a particular set of parameters -- may possibly offer particular insights into viability and sustainability.

Related metaphors: So framed it could then be said (for example) that the configuration is variously illustrative (or reminiscent) of situations in which the "balls" necessarily do not touch and could be understood as engendering a form of "Borromean condition". There is a shift in perspective to pathways defined dynamically -- pathways which are only "occupied" infrequently at any particular location. Examples include:

a chemical molecule characterized by electron orbitals. This is especially suggestive because of the many possible cognitive implications of the wave-particle duality. Some have been discussed separately (as noted below)

juggling a set of balls, as remarkably discussed by Burkard Polster (The Mathematics of Juggling, 2006) which fruitfully integrates perspectives regarding polyhedra (and their great circles), Hamiltonian cycles and distinctive complex juggling patterns (with an annex of stereograms of more complex Hamiltonian cycles). The author extends his argument to include the seemingly disparate domains of bell ringing and knot theory (including braid theory). Understood more generally these all reflect forms of dynamic interweaving -- all of which have long been appreciated as comprehensible in practice, notably through their aesthetics. (Some juggling demonstrations, 1998; Juggling demo, 1998; 46 Juggling animations, Wikimedia; X3D Robotic hand juggling primitives, 2014)

Cognitive fusion: It is appropriate to note that hopes for new sources of energy are notably framed by nuclear fusion. This currently requires a toroidal reactor through which plasma is held away from the walls of that container by magnetic rings. These control the instabilities of the snake-like dynamics of the circulating plasma.

In this case the "plasma, could be understood as intimately related to a "confusion" of what is conventionally conceptualized in terms of attention, confidence and belief.

Category juggling reframed through visualization dynamics

Insights of relevance to one purpose of this exploration are perhaps most appropriately framed by metaphor, given the manner in which it interrelates practical skills, distinction of patterns, aesthetics, coordination/collaboration, a challenge to comprehension, a degree of identification with the process, and mathematical insights into complex patterns.

Psychosocial relevance of juggling as a metaphor: There are numerous references to the insights to be gained from juggling as a metaphor, as noted by Arthur Chandler (Life Juggling, Juggler's World, 42, 1996, 4):

I've seen the term "juggling" in a number of context... Newspaper articles are headed "Juggling Family and Career," and friends speak of "juggling too many commitments." The public at large seems to be adopting juggling as a figure of speech for trying to keep parts of life in sync with each other. Juggling, as a 1990s metaphor, comes to stand for the attempt to attain a state of dynamic equilibrium in which several ongoing commitments are kept in balance through constant effort.... "Juggling" in 1991 has become the metaphor for life's major hassles - not just the little annoyances of waiting in line at the checkout stand, or even getting audited by the IRS. "Life Juggling" is a defensive activity.

I have encountered three metaphors for what most people call the 'work-life balance' issue. These are: juggling, keeping multiple plates spinning on sticks, and surfing. Each has its strengths and flaws. All share in common the problems that arise from calling the whole thing a 'balance' problem in the first place, but the 'balance' point of view has some merits.

The juggling act, for those of us with career and family balls in play, doesn't work if either crashes to the ground...
But if we want to juggle different kinds of things in our lives, the awareness of how those slices of our experience are different from each other is part of the fun: the feel of the delicate crystal, the fresh firm apple, or the taped grip of the flaming club as we confidently catch it and then send it back up.

Margaret W. Ferguson: Juggling the Categories of Race, Class and Gender (In: A. R. Jones and Betty S. Travitsky, Women in the Renaissance: An Interdisciplinary Forum, 1991)

Juggling priorities: Common to use of juggling as a metaphor, as indicated by the example above, is the experiential sense in which priorities are juggled. The priorities could be understood as strategic initiatives, preoccupations with problems, or cultivation of values. The metaphor is appropriate when the number in each case (or together) becomes a challenge to handling or coping. Greater skill is required as the number increases.

It is interesting that a very common device for handling such complexity is through some form of scheduling, typically requiring a 2D spreadsheet. This is of interest in that that tool was used to configure the visualizations described here. Given the timing issues involved, it is appropriate to ask what patterns might need to be designed in 3D and 4D, or more, as speculatively considered elsewhere (Spherical Accounting: using geometry to embody developmental integrity, 2004).

In general, and with respect to any form of governance, it is of particular interest to note recognition of juggling strategies/policies, obligations, concepts/categories, or factors:

"Dropping the ball": In addition to any sense of "keeping the ball in play", of particular interest is the experience framed by "dropping the ball" -- necessarily a common experience in engaging with more complex challenges of governing:

Such failure may result from failure to "catch the ball" or from failing to ensure that it avoids colliding with another ball in play -- engendering incompatibility or a "clash".

So framed, the existence of "windows" of opportunity becomes of great interest -- exemplified by the launch windows through which spacecraft are launched, such as to avoid the thousands of objects of orbiting space debris. How might "cognitive launch window" be recognized -- or those relating to any strategy?

Quantitative challenge: how many patterns, balls and partners? As noted above, Wikimedia provides access to 46 patterns (as gif animations) -- distinguishing the number of "balls" and jugglers. This is clarified in a separate table.

In his study Burkard Polster explores the question of how many ways there are to juggle -- as being the question most frequently asked of jugglers. Unconstrained, his response is infinite, but with a preliminary answer to the effect that:

However, it still makes sense to ask for the number of juggling sequences that are distinguished in some way. The three most natural parameters used to define distinguished classes of juggling are:

the number of balls used to juggle a juggling sequence

the period of a juggling sequence

the maximum height of a throw in a juggling sequence

If we only fix the number of balls, or the period, or a maximum throw height, the resulting class of juggling sequences will still be infinite, except for some trivial exceptions. Fixing the period p and a maximum throw height h yields a finite class of juggling sequences. Clearly, there are no more than (h+1)p such sequences. (p. 37)

He also offers a more complex indication in the following terms:

Numbers of juggling sequences

(Burkard Polster, The Mathematics of Juggling, 2006, p. 40)
See explanation of Möbius function

Of relevance here is the apparent absence of consideration of constraint on the number of balls which can be effectively juggled, notably as these might relate to the number of participants between which they are passed. Some constraints are evident from the details listed by Wikipedia with respect to juggling world records. It is curious that the number for an individual is consistent with the psychological constraint famously highlighted by George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956). How this constraint might relate to a limited group of individuals (or a group of limited individuals) is another matter.

Dialogue as juggling points, topics and themes: The vocabulary of discourse establishes the relevance of a juggling perspective: "making a point", "taking a point", "ball in your court", "over to you", "missing the point", and "dropping the ball".

In the light of the analysis of juggling patterns, this highlights the question as to how many patterns of discourse there might be, whether 2-person or multi-person -- given the cognitive and coordination constraints -- and usefully illustrated by the 46 ball-passing animations offered by Wikipedia?. Are there dialogue records to be recognized by analogy with those of juggling noted above?

As noted by Polster in a discussion of enumerating and creating new interesting patterns:

Using algorithms that are based on results in this book, computers have been programmed to enumerate all juggling sequences satisfying any conceivable set of constraints. Many new interesting juggling sequences have been found in this way. Since we now know "all" possible juggling sequences, what remains to be done is to identify those that, in themselves, are interesting from either a juggler's or a mathematician's point of view (pp. 137-138)

With respect patterns of dialogue, Polster's subsequent comment is especially valuable: Also, if you want to find out how you can smoothly move from one pattern to the next, tools such as state graphs are very helpful . One accessible summary is provided by Harri Varpanen (Toss and Spin Juggling State Graphs, 12 May 2014). Use of "spin" in that title might offer particular insights to a world in which dialogue is increasingly characterized by "spin".

Problematic experience of "being juggled"? Clearly a quite different perspective is offered if there is a sense of "being juggled", whether by obligations, employers or other agencies. This is typically recognized in a sense of "manipulation", which would indeed be the perspective of the juggler -- possibly recognized as a puppet master.

The experience of structural violence could be usefully explored as the sense of "being juggled".

Requisite controversy engendered and encoded by a counter-coil pattern?

It has long been claimed that There is nothing new under the Sun. Curiously however the creativity of a knowledge-based civilization is characterized by multiple initiatives to "rediscover the wheel", accompanied by desperate efforts to patent the originality of the discovery, to franchise its use, and to associate it possessively with the name of the discoverer. Not the slightest humility is associated with naming features of the universe (stars, mountains, rovers, species) after those who claim to have discovered them.

Controversy: The Rodin coil offers one remarkable case study of the nature of the controversy engendered by unconventional patterns and their discoverers. The "thought police" have adjudicated definitively on the pattern from a conventional perspective -- to the point of describing their presentation (anonymously) as the best example of the worst of the TED talks (The Ugly Side of TED talks, Physicis Central, 1 June 2012).

The process is characteristic of most innovation calling into question the patterns of traditional conventional ("mainstream") thinking, as has also been even more remarkably demonstrated by the extensive controversy about the arguments of Ruper Sheldrake (The Science Delusion, 2012) [The debate about Rupert Sheldrake's talk, TED; Rupert Sheldrake, The TED Controversy]. These acquired fame through the editor of an eminent science journal framing their original presentation as "a book fit for burning". Groups, claiming the role of gatekeepers, effectively define themselves as assemblies of the righteous -- indistinguishable as such from those they disparage.

As illustrated by the 2016 US presidential campaign, there is currently virtually no perspective from which assertion and counter-assertion can be fruitfully explored, as a means of transcending the mutual demonisation characteristic of the least civilized cultures of the past. The problematic dynamic is of course also characteristic of relations between (and within): political ideologies, disciplines, religions, cultures, and languages, as remarkably clarified by Nicholas Rescher:

For centuries, most philosophers who have reflected on the matter have been intimidated by the strife of systems. But the time has come to put this behind us -- not the strife, that is, which is ineliminable, but the felt need to somehow end it rather than simply accept it and take it in stride (The Strife of Systems: an essay on the grounds and implications of philosophical diversity, 1985)

The knee-jerk reaction associated with such "strife" (justified with greater or lesser sophistication) is to frame the abnormal as debased (gross), pathological (mad), evil, heroic, inspired (aesthetically or spiritually), or a form of genius. Individually, or in combination, they may all be experienced as constituting a (radical) threat.

The challenge is to frame a space (with an inherent dynamic) in which the dynamic between contrasting views of right and wrong can coexist as competing senses of self-appreciation. Despite the qualities for which it is otherwise much appreciated, it is evident that the TED process has neither the capacity nor the motivation to do so. There is clearly a problematic paradox to being positioned at any "leading edge" (Seeking the "Cutting Edge" of Sustainable Community, 1997).

More interesting however is the sense in which the media lynch-mob piling righteously onto Donald Trump somehow engenders a fulfilling charge in the participants -- as the victim is torn apart. This is only too evident in any form of feeding frenzy and the dynamics of crowd psychology. Irrespective of the denigration of Rodin's "vortex mathematics", self-reflexive exploration of a counter-coil might offer clues to a tragic feature of a supposedly mature knowledge-based civilization indulging in uncritical liking ("likes") or disliking ("dislikes").

Critical distinction: Given the articulation by Marko Rodin, the concern here is to dissociate the geometry of the pattern from the claims as to their significance by him (and variously promoted by his supporters) -- and variously denigrated by his critics. The pattern is interesting in its own right, notably in relation to the argument presented above.

Whilst the pattern may indeed invite "buy in" from speculative enthusiasts, such engagement is quite another matter. However it is also a matter of interest how a complex pattern may both function as a strange attractor and evoke controversy and opposition -- especially if the discoverer identifies possessively with it. The Rodin coil is potentially of further interest in that the counter-coiling it embodies could be considered as modelling to some degree the complex controversies of governance, religions, and belief systems in general -- controversies which may well lead to violence of one form or another.

Description: A number of descriptions of the Rodin coil are available (Marko Rodin, The Rodin Solution Project, Rodin Aerodynamics, 2001; Russ Blake, Analysis of the Rodin Coil and it's Applications, SERI-Worldwide; Marko Rodin and Greg Volk, The Rodin Number Map and Rodin Coil, Proceedings of the NPA, 2010). Numerous images of variants of the coil are available on websites, notably those preoccupied with the controversial possibility of "free energy".

Rodin toroidal counter-coiling

Experimental counter-coiling of nuclear fusion reactor

Marko Rodin has discovered a series of regularities in the decimal number system. From these he derives his "Vortex mathematics". Vortex mathematics explain all mathematical operations, the genes, and non-decaying energy. These regularities are inherent to the system because of its base. You can find similar regularities in systems other than the decimal...

The Rodin Coil consists of a pair of wires wrapped around a doughnut-shaped core in a star pattern. Rodin claims this particular design, deduced from his number theory, yields different electromagnetic properties than any other coil -- enabling it to create perpetual energy, and thus breaking fundamental laws of physics. Rodin admits not being able to build those coils himself -- as he is "not an electrical engineer" [see video]. In absence of any scientific proof, it remains therefore highly questionable, to say the least, whether the Rodin coil has any special physical properties.

According to Rodin, the coil represents the underpinning geometry of the universe, which draws its non-decaying energy from the vortex, the zero.

Towards a 3D visualization of toroidal counter-coiling dynamics

Adapting the method described above, the question is how to introduce a "counter-coil" into the visualization (especially given the distinctive cognitive implications of use in any mandala-style mapping). Approaches explored included:

pseudo-counter-coiling:

reversing the pathway along which the helix is extruded for visualization by rendering negative x and z coordinates in all the sets. This suffices to provide a spine along which a sphere can be moved in the reverse direction (from that on a related helix) -- provided the order of the sets of coordinates is itself reversed in the spreadsheet. Although the direction of movement is reversed, the transformed helix is merely out of phase with the untransformed variant, as indicated by the central image below. It is not the required mirror variant

reversing the direction of rotation relative to the original, again implies an opposing movement, but again there remains a similarity between the two. Such reversal of direction (even if spheres on them move in opposite directions) does not achieve the required mirror condition.

reversing the sequence of component helices, namely by reversing the pattern of colours

combining any of the above

creating a mirror version, achieved by rendering negative one coordinate (x, y or z) in all the sets in order to ensure the distinction required between a right-handed helix and a left-handed helix (which is not achieved by the previous approach).

Results are indicated by the images below. Note the contrast between right and left-handed windings.

Pseudo-counter-coiling Original coil
around an invisible torus
(X3D or VRML) Pseudo-counter-coil addition
(helixes out of phase, but spheres travel
in opposite directions)
(X3D or VRML) Addition of pseudo-counter-coil
to 5-coil pattern
(black addition of different phase only)
(X3D or VRML)

Given the time taken (by the author) to discover the seemingly trivial technique of how to provide a 3D rendering of the right-handed variant of the helix, and the confusion engendered by models characterized by pseudo-counter-coiling, and counter-rotation, it is useful to provide images of the variants. Since one purpose of the whole exercise is to trigger imaginative reflection on ordering complexity, a variety of renderings is presented. Note that the quality of the images and animations can be easily improved (typically by modification of the colours, changing the crease-angle of the extruded nonagons, or changing the speed of rotation and/or counter-rotation). Such changes can be made in a text editor.

Clarification of mirror-image counter-coiling
Counter-rotation; not counter-coiling
(X3D or VRML) Helix mirror images Counter-rotation; not counter-coiling
(flipped image)
left-handed right-handed

Counter-rotation; not counter-coiling
(X3D or VRML) Counter-rotation; not counter-coiling
(variant) Counter-rotation with counter-coiling
(X3D or VRML)

Counter-rotation; not counter-coiling
(X3D or VRML) Counter-rotation; not counter-coiling
(X3D or VRML)? 5 only Counter-rotation with counter-coiling
(X3D or VRML)

Having achieved the ability variously to position a counter-rotating mirror helix in relation to its complement, the focus then shifted to the possibility of reverting to the experiment with nesting the counter-rotating helix within its complement using phi-based proportions for the outer helix.

Examples of nesting peudo-counter-coiling within coiling
(X3D or VRML) (X3D or VRML) (X3D or VRML)

Transforming vehicles of identity between global and toroidal forms

Globality: On a spherical world, intensely preoccupied with "globalization", it might be asked how a torus could be understood to be of any relevance. It is indeed already a challenge to comprehend the full significance of globality -- despite the irony of the widespread focus on the balls of the variety of ball games. Supposedly these provide a degree of complex experiential insight into globality. Much of geometry can also be understood as a quest for more precisely articulated understanding of it (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009).

That exploration gave focus to the question of what geometry serves as a vehicle of personal or collective identity, whether it be points, lines, planar surfaces or the like, as explored in more detail (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009). Clearly there is a sense in which people identify with a "point" when making one -- or with a "line" when pursuing a line of argument. The identification with a surface is evident in the case of land ownership (with regard to which there is so much conflict), or with the cubic volumes of a dwelling or place of work. More generally is it somehow with a "sphere" that the wholeness of personal identity is associated -- and hence a degree of resonance with globality?

Toroidal vehicles: One indication of the relevance of a torus is that the planetary globe travels through (and defines) a toroidal tunnel around a sun vital to life on that globe. The torus thus becomes of significance from a temporal perspective -- an orbit ensuring the systemically healthy dynamic of the seasons. That the solar system is moving as a whole, with the planetary orbit then tracing a helix, is a different matter.

The interplay between that orbit and the central sun has been a theme of reflection, symbolism and metaphor since the beginning of civilization. A striking example is provided by the symbol of the Ouroboros and its relation to the sense of eternal return. Of potential relevance however is any shift from preoccupation with globality and its distorting representation on planar projections, to a perspective which encompasses the toroidal orbit and the mysterious role of the sun.

Requisite toroidal complexity: A doughnut could however be recognized to be excessively simplistic in a context calling for requisite complexity in cybernetic terms. Hence the argument above in terms of a complex of toroidal helices in dynamic relation to one another. Can meaning be feasibly associated with such complexity and mapped fruitfully onto it to provide a vehicle for identity? Clearly visualization offers a means of rendering comprehensible complexity which is otherwise articulated primarily in mathematical equations for the few.

As noted above, one focus to the argument can be expressed in terms of the requisite toroidal design of a nuclear fusion reactor -- from which it is claimed for public relation purposes that that the "energy of the sun" can be readily and efficiently obtained. Does this suggest the case for exploring the design of an analogous "reactor" of relevance to psychosocial energy -- whether individually or collectively -- as separately argued (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006)? Given the reference to the Ouroboros, it is delightfully ironic that a particular preoccupation in the case of the nuclear reactor is framed as containing the "snake-like" dynamics of the circulating plasma.

Topology: Reference to the investment in such complex technology based on the torus suggests the merit of further exploration of the interplay between torus and sphere -- succinctly clarified by the following animations.

Animations of toroidal complexification
Torus-to-Sphere transformation Trefoil knot Clifford torus
Animations reproduced from Wikipedia

Mathematics, especially topology, has a wide array of insights of potential relevance in this regard -- further to the pointers highlighted above. What might these suggest with respect to optional vehicles for individual or collective identity?

Especially intriguing is the sense of a cognitive challenge somewhat analogous to achieving orbit of the globe -- both conceptually (through an understanding of globality and spherical geometry) and through the technology required. As argued, "technomimicry" may well offer a pathway for creative innovation.

Appropriateness of a hypersphere as a vehicle for identity? As noted above, every reason is offered to constrain any sense of identity to the simplest geometric forms in 2D or 3D. Collective may embody these in heraldic signs, symbols on flags, or statues. In addition to the insights of mystics, this constraint has been variously challenged by mathematicians (Ian Stewart, Flatterland, 2001; Dionys Burger, Sphereland: a fantasy about curved spaces and an expanding universe, 1965). Especially relevant is the question if Ron Atkin (Multidimensional Man: can man live in three dimensions? 1982).

The conceptual adventures of mathematicians and astrophysicists are an invitation to associate personal and collective identity with ever more complex topology. The hypersphere (especially the 3-sphere) is one example of the space within which a Concordian Mandala might be more appropriately located to enable any such sense of identity. The possibility has notably been highlighted by Mark A. Peterson, pointing out that language in Dante's Divine Comedy suggesting that he visualized his universe in the same way (Dante and the 3-Sphere, American Journal of Physics, 47, 1979; S. Lipscomb, Art Meets Mathematics in the Fourth Dimension, Springer, 2014, chapter 2; and extensive discussion thereof, Dante and the 3-Sphere, Science and Philosophy Chat Forums). Perhaps appropriately, the 3-sphere is also known as a glome -- a term employed for the fictitious kingdom of the novel of C. S. Lewis (Till We Have Faces, 1956).

Despite some articulation of perceptions for those living in such geometry, the principal inadequacy of the insights of mathematicians -- is that they naturally tend to avoid any exploration of the embodiment of identity therein or by such spaces. This is the challenge articulated by George Lakoff and Mark Johnson (Philosophy in the Flesh: the embodied mind and its challenge to western thought, 1999) and in the subsequent argument (George Lakoff and Rafael Nuñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

It is in this sense that any study of the organizing function of the brain is especially valuable, as with that of Arturo Tozzi and James F. Peter (Brain Activity on a Hypersphere). The authors note that:

Folks , you must also see what follows from the above , because it contains some mind boggling 3D visualisations - Part 2 (continued from my previous Post) :-

Current advances in neurosciences deal with the functional architecture of the central nervous system, paving the way for "holistic" theories that improve our understanding of brain activity. From the far-flung branch of topology, a strong concept comes into play in the understanding of brain signals, namely continuous mapping of the signals onto a "hypersphere": a 4D space equipped with a donut -- like shape undetectable by observers living in a 3D world. Here we show that the brain connectome may be regarded as a functional hypersphere.... We anticipate that this introduction to the brain hypersphere is a starting point for further evaluation of a nervous fourth spatial dimension, where mental operations take place both in physiological and pathological conditions. The suggestion here is that the brain is embedded in a hypersphere, which helps solve long-standing mysteries concerning our psychological activities such as mind-wandering and memory retrieval or the ability to connect past, present and future events.

Of particular relevance to the above argument, the authors offer the following images illustrating the structure of a hypersphere (or glome). They note that the shape of the glome is ever changing, depending on the number of circles taken into account (in the left hand image) and their trajectories (see video by Niles Johnson, A visualization of the Hopf fibration). The other images illustrates depiction of a hypersphere as two spheres glued together along their spherical boundary, giving rise to a Clifford torus (see animation above, showing a stereographic projection of a Clifford torus performing a simple rotation through the xz plane). The image on the left is a suggestive illustration of how periodicity might be cognitive embodied, whether in the case of the organization of music or as the elements of the periodic table.

Alternative representations of a hypersphere
Arrangement of circles in 4D Clifford torus 2 Spheres glued together

Tozzi and Peter also make use of a single image (one phase from Johnson's video, also figuring in the Wikipedia commentary on Hopf fibration), presented in the following animation using changing colour values as a suggestion of the dynamics within which brain organization might be associated. Use of a similar technique can be used with respect to the 6-dimensional Calabi-Yau manifold of significance to superstring theory, with which the extra dimensions of spacetime are conjectured to be associated, as well as mirror symmetry. Given its relevance to the branes of astrophysics, such speculation has been explored in terms of hypothetical correspondence between global brane and global brain (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? 2011).

The typical approach of mathematics is also somewhat misleading in that it appears to emphasize static organization of structures in 4D (effectively snapshots) when the dynamics may be especially significant to embodiment of identity therein. It is in this sense that explorations of the organization of music by the brain is especially valuable (Dmitri Tymoczko, The Geometry of Musical Chords. Science, 313, 5783, 7 July 2006, pp. 72-74; A Geometry of Music: harmony and counterpoint in the extended common practice, 2011). This is discussed separately (Engaging creatively with hyperreality through music, 2016).

Cognitive twist via a hole? The conceptual challenge can however be usefully understood as involving a degree of paradox and a recognition of what is not immediately obvious. This can be framed as a "cognitive twist" (Configuring a focus for awareness through a cognitive twist, 2015). This is potentially to be understood in terms of self-reflexivity and higher orders of cybernetic feedback, as argued by Maurice Yolles and Gerhard Fink (Generic Agency Theory, Cybernetic Orders and New Paradigms, 2014). The twist can be partially recognized through images such as those of M. C. Escher. It can be explored through the topology of knots (Jacques Lacan, R. D. Laing), through the symbolism of Celtic knots, with the associated poetic constraints of cynghanedd, and mathematical representation (Jessica Connor and Nick Ward, Celtic Knot Theory, University of Edinburgh, 2012).

In its simplest form, it is in this sense that the torus knot is potentially of great significance as a vehicle for identity -- especially through the mystery of the central hole with which it is associated. The neglected significance of a "hole" has been remarkably discussed by Roberto Casati and Achille C. Varzi (Holes and Other Superficialities, 1994) -- with respect to the borderlines of metaphysics, everyday geometry, and the theory of perception (as they summarize in the entry on holes in the Stanford Encyclopedia of Philosophy).

The authors seek to answer two basic questions: Do holes really exist? And if so, what are they? Philosophers would typically like to expel holes from their ontological inventory. Arguing in favour of the "existence" of such absences as full-fledged cognitive entities, the authors examine the ontology of holes, their geometry, their part-whole relations, their identity, their causal role, and the ways they are perceived. In cylindrical form holes are centred on an implied axis, with which a "point" can only be associated dynamically -- or as a succession of "points".

So framed, what is the strange attraction of the hole which is such a powerful catalyst for intercourse? This includes recognition of a "hole in an argument" typically framed pejoratively as a fallacy.

The significance can be partly explored through the arguments regarding the curious potential of "nothing" and the "missing", as presented by Terrence Deacon (Incomplete Nature: how mind emerged from matter, 2011), under the heading Nothing Matters:

When Western scholars finally understood how operations involving zero could be woven into the fabric of mathematics, they gained access to unprecedented and powerful new tools for modeling the structure and dynamics of the physical world. By analogy, developing a scientific methodology that enables us to incorporate a fundamental role for the possibilities not actualized -- constraints -- in explaining physical events could provide a powerful new tool for precisely analyzing a part of the world that has previously been shrouded in paradox and mystery. The mathematical revolution that followed an understanding of the null quantity in this way may presage a similarly radical expansion of the sciences that are most intimately associated with human existence. (pp. 540-541)

Intercourse: More speculatively, what is the significance of the hole (goal, or basket) of many ball games -- the strange attractor which is such a primordial preoccupation as a symbolic indicator (and means) of scoring competitively? Perhaps provocatively, of necessity, how might this be understood as an unconscious surrogate for sexual intercourse -- especially given metaphorical borrowing of "scoring"?

Consideration of any such "hole" could be "framed" by the following configuration, understood to an appropriate degree through their metaphorical and symbolic associations. This graphics package draws objects in the hyperbolic plane using the Poincaré model. Comprehension of the central significance could be further associated with the through mathematical consideration of the Poincaré disk and the tessellation of the hyperbolic plane with which it is so closely associated. as discussed separately in relation to the imaginative images of Escher (Global communication patterns in a hyperbolic space of negative curvature, 2016).

Renderings of the hyperbolic plane
W. Goldman, Ultraideal Triangles, 2004 Escher

Inversion: Of particular interest is the cognitive relationship between "inside and outside" or "without and within", a theme of great interest to various authors, including Buckminster Fuller and Joseph Campbell (The Inner Reaches of Outer Space: metaphor as myth and as religion, 1986).

Interlocking tori: There would appear to be a powerful electrical metaphor, curiously unexplored, for the process of intercourse and the energy it engenders. That some form of energy is engendered is of course a theme of frames of references deemed marginal from a mainstream perspective unable to integrate that perspective into politically correct discourse. Examples include the insights associated with the tantric sex of Hinduism and western consideration of personal development (Shiv Joshi, Electric Intimacy, Complete Wellbeing, 19 Feb 2009).

Curiously the relationship is evident through use of "coupling" both in an electrical and sexual context. In the former, coupling is an interaction between two electrical components by electromagnetic induction, electrostatic charge, or optical link. The overlap in terminology goes much further, notably through explicit reference to male and female connectors (Electrical Systems as a Guiding Metaphor for Stages of Group Dialogue, 2001; Modulating cognitive transformations: electrical metaphors and semiconduction, 2012). The common solenoid, and the visual depiction of its energy fields, could be considered more than suggestive -- given any movement in relation to it which results in electromagnetic induction.

Screen shots of a dynamic virtual reality model of intertwined tori
(click on each variant to access and manipulate in 3D;
in the free Cortona VRML viewer, right click for preferences to switch from/to the "wireframe" presentation) Red torus has a vortex (smoke ring) dynamic in the model
Blue torus has a wheel-like dynamic in the model
VRML animations by Bob Burkhardt.

Rather than assume that the two tori are simply static representations of sets of categories, there is a strong case for taking account of the dynamics associated with many occurrences of a torus in natural phenomena -- from smoke rings, through plasma containment in fusion reactors, to the environment of black holes. Anything defining the surface of a torus may then have two dynamics: (a) around the tube of the torus, and (b) along the tube of the torus.

Give that such dynamics may occur in both tori when interlocked, interlocking is most smoothly achieved when the direction of the dynamics in each is such as to be mutually reinforcing. In effect the dynamics around the tube of one reinforces the dynamics along the tube of the other. If the tori are understood to be serrated (as in the aesthetic representation in the original paper), then the serrations effectively function like the teeth of gears at the interface between the two tori.

Further insights may be obtained by manipulation of the more elegantly compact X3D variant of the model (kindly developed by Sergey Bederov), or the equivalent VRML model (accessible via the Cortona3D plugin).

The cognitive content of the two ("doubly curled") matrices, potentially to be interrelated in this way, can then be recognized in terms of a dynamic situation. These are the "preferred modes of knowing" (potentially emergent within any psychosocial system) interact with the "sub-systems" -- variously perceived as important (to understanding and ensuring the operation of any psychosocial system) -- through the dynamics of the changing contact at the interface of the two tori.

When, as noted above, the dimensions of the torus are different, their relationship may reflect cognitive challenges corresponding to those the Kama Sutra delights in articulating through euphemism -- regarding the relationship between partners of different dimension: males (as rabbit, bull or horse) and females (as doe, cow or she-elephant) [more].

Dynamic configuration of holes: So framed, the vortex dynamics (notably any vortex ring) of engagement with holes as attractors is very suggestive of the cognitive engagement with any form of intercourse (as an "unknown). The self-reflexive engagement is minimally framed by the trefoil knot (depicted below). Such considerations raise the question as to whether the integrative challenge of many configurations merits exploration as a configuration of holes (Holiness framed by a triangulated configuration of holes, 2014).

Representational challenge of the periodic table of chemical elements

In the quest of a pattern of relevance to the Concordian Mandala of this argument, the thinking over a century with regard to the ordering of chemical elements merits the most careful attention. As a periodic table it should, in principle, imply a form of order of relevance to any understanding of global concord -- especially in the light of the arguments of George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). In that light, one potentially fruitful approach is the compilation of Denis H. Rouvray and R. Bruce King (The Mathematics of the Periodic Table, 2005).

The Periodic Table with a new double numerical structure, presented here is attempt to find table form which will in some new way represent the periodicity and symmetry of the Elements, with the Periodic System as base. Also this tetrahedral laminar table structure maybe will became a base for developing a new shell structure of atomic nucleus. This new rearrangement of the chemical element is based on mathematical formula which result is simple, length of the periods:

With respect to the toroidal focus of the above argument, of particular interest is that of Rafael Poza (Elements and the Magnetosphere, 2008).

The concern here is to relate the conventional challenges of depiction with the cognitive challenges of integrative comprehension -- those encouraging the exploration of a Concordian Mandala. A useful linking argument is provided by the approach of Edward Haskell which is centred on the role of a coaction cardioid as depicted below.

As a cardioid, Haskell's coaction compass featured in a separate discussion of the mathematical derivation of the "heart pattern" potentially characteristic of the widespread cognitive acceptability of playing cards (Radical Localization in a Global Systemic Context Distinguishing normality using playing card suits as a pattern language, 2015). The distinctions between a "cardioid" and the conventions of "heart" depiction were highlighted. That discussion also included interactive demonstration of 3D heart shapes based on different equations for cardioid renderings (Equations for Valentines, Wolfram Demonstrations Project). The argument can be taken further in this context by considering the pattern implied by the following as presented there.

Defining the heart pattern using the golden ratio (phi)
Heart pattern framed by 4 circles
(phi is the ratio of separation of centres of the smaller circles to that separating the larger) Heart pattern on the left overlayed
(highlighting the relation to the form of the Gaussian distribution curve above)

2D animations : These suggest he possibility of 2D animations of the (circular) dynamics around which implicitly frame a heart pattern, as shown below -- and potentially suggestive of a cognitive dynamic analogous to that between the four "chambers of the heart".

Various 2D animations of dynamics defining the 4 conditions of the heart-pattern
using juxtapositioned cross-sections of two 3D tori

3D animations: The image on the left suggests the possibility of a 3D rendering using the juxtaposition of two tori with their major radius related by phi.

The resulting heart pattern differs significantly from the conventional cardioid but it does recall the domes of many mosques -- with the much appreciated cognitive implications these may elicit. Curiously only a small proportion of constructed mosques closely approximate that form (more especially those in Malaysia, but including the Taj Mahal), whereas many Islamic greeting cards do so to a far higher degree in their idealized representations of a mosque (for Ramadan and Eid). The domes of most constructed mosques bear a closer resemblance to the conventional cardioid -- ornamented with a minimally tapering spire.

In anticipation of more appropriate animation possibilities to highlight the heart pattern in 3D, the following justaposition of horn tori frames such a pattern to some degree, given the difficulty of ensuring contiguity. The movement into (or out of) the central vortex is consistent with the animations in the 2D cross-sections above -- meshing correctly between the upper and lower tori.

Animation of two horn tori of major radius in proportion of phi
(2 of the 4 variants presented in the 2D animations above)
implied upright 3D heart pattern implied inverted 3D heart pattern

The other two variations noted in the 2D animations are inversions of those above. The clockwise and counter-clockwise rotational dynamic (not indicated in the 2D animations) suggest that there are 8 distinctive patterns that could be considered.

A much clearer understanding of the heart pattern embedded within, or framed by, the two contiguous horn tori is evident from the animations below, although these lack the longitudinal and latitudinal rotations of those above -- which could be added. Note that switching between filled and wrireframe renderings is achieved when viewing.

Animations indicative of embedding of heart pattern within contiguous tori
(Variants: X3D: without tori / with tori; VRML: without and with tori)
Upright heart pattern
(filled rendering) Inverted heart pattern
(wireframe rendering)

X3D and VRML models kindly prepared by Sergey Bederov of Cortona3D

Lissajous curves: In relation to the horn torus, Wolfgang Daeumler comments on the presentation of Lissajous curves on the surface, as illustrated by the animation (below). The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. The animation below is somewhat reminiscent of the hypersphere animations suggestive of higher-dimensional brain functioning (as presented above).

Visual renderings of Mandelbrot set: As noted in the earlier discussion, it is widely recognized that the cardioid corresponds closely to the larger "bulb" of the many contrasting visual renderings of the Mandelbrot set fundamental to complexity science -- with all its potential cognitive implications (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005). The variety of renderings presented there can be generated using the interactive XaoS fractal zoomer and morpher. The more comprehensive heart pattern may correspond more closely to the curve which encompasses both the larger "bulb" and the asymptotic extension beyond the smallest "bulb". This possibility is illustrated as one variant of the 2D animations (above).

Comprehension of complex dynamics: It must be stressed that the argument here is preoccupied with how complex dynamics are to to be comprehended and what constraining approximations may be made in framing intuitive recognition of it. The use of the upright heart pattern offers the suggestion that the dynamics associated with the larger torus "beneath it" are in some way implicit or virtual. Use of the inverted heart pattern offers the complementary suggestion that the implicit or virtual dynamics are in some way "above it" (as might be characteristic of the use of that orientation in temple architecture).

Same complex dynamics of higher dimensionality viewed through distinctive "toroidal projections"
concentric nested / lemniscate interlinked / knotted stacked / interlocked

The fractal organization of any Mandelbrot set rendering, and the complex dynamics implied, are necessarily a major challenge to comprehension. The widespread appeal of the heart pattern may constitute a very simplistic recognition of this -- an approximation to a more complex insight, as suggested by the various "projections" above. As may then be imagined, the dynamics of the Concordian Mandala which is the focus of this exercise, cannot be readily mapped through a 2D or 3D rendering.

There is then a case for recognizing the extent to which the central "valley" of the heart pattern could be associated with the entropic focus in Haskell's cardioid (depicted above), with the opposite asymptotic feature associated with his negentropic focus. Framed by two horn tori (as in the left image above), the entropic focus is defined by the smaller and the negentropic focus by the larger. There is the further advantage to this framing in that the smaller torus frames the mystery of a hole (perhaps in its most problematic sense?), whereas the larger frames a larger (more positive?) insight associated with the infinite.

Such a framing offers the possibility of juxtaposing a pattern inspired by 3D rendering of Haskell's "cardioid", understood in cognitive terms, with Poza's toroidal proposal for the periodic table (as presented above).

Symbolic mind maps and learning stories? As a much valued symbol in a society in need of integrative symbols, there is a case for exploring how the heart pattern might be integrated into a more systemically complex learning story with a variety of pathways and dynamics -- a more extensive mind map. As a prelude to a suitable interactive model, the following is such an exercise, using a configuration of 9 heart patterns -- whether configured in a circle or vertically stacked. Any such configuration is best understood as a projection of a complex system.

The stacking is suggestive of a hierarchy of systems, possibly of progressive greater subtlety -- the lowest possibly associated with first order cybernetic processes, the highest with the subtlest envisaged by cybernetics and the wisdom sciences. Positioning one heart pattern midway between the two extremes enables a distinction to be made between the two immediately contiguous (a 3-fold set), between the neighbouring 4 (a 5-fold set), the neighbouring 6 (a 7-fold set), in contrast with the totality (a 9-fold-set). Appropriate polygons are used to reinforce such patterning. The smaller sets could be understood as having constraints on their subtler and/or more fundamental sensitivity.

Of particular interest to fruitful complexification of the systemic story is recognition of the rotational effects. The variants are distinguished in the images below. Each 2-tori variant can be used with others of the same (opposing) directionalities, meshing together when stacked. There are thus two kinds of stack, whether inverted or not. However a further distinction is possible in terms of the rotation around the common axis. Meshing then necessarily requires rotation in the same direction for the whole stack, clockwise or counter-clockwise (as illustrated earlier) -- making a total of 4 upright patterns and 4 inverted patterns.

A further point of interest is the perspective on any stack along the axis. As shown in the central image below, the smallest heart pattern then appears at the centre of what is highly reminiscent of a conventional bullseye target -- for archery, darts, etc -- some of which have 9 rings. Whether as a need hierarchy or otherwise, this can be understood as the focus for the subtlest aspiration. Viewed otherwise the 9 levels of the stack are appropriately reminiscent of the 9-level temples design of some cultures (cf. Nine Story Stupa, Chichen Itza, Tikal) and their association with understandings of 9 levels of consciousness. Such levelling now also features in some online games.

Views of conjoined 2-tori dynamics Circulation through the vortex
(Variant A) View down common axis
(9 rings) Circulation through the vortex
(Variant B)

Framing the heart pattern by two implicit tori suggests that this framing could as well have been understood in terms of the helical windings around an implicit torus as rendered visually (in the earlier argument) -- or possibly between neighbouring tori (as suggested by the Lissajous curves). Hence the insights to be gained from gimbal dynamics (forthcoming paper). Especially intriguing is the sense in which the helical windings -- as a twisted sine wave -- could be considered as the dynamic resolution of the longitudinal and latitudinal rotation dynamics illustrated in the animations above.

Thanks @Mitko_Gorgiev , good to know that you liked Part 1 of my posts above...I just have this gut feel that the Toroid shape is invariably involved in electromagnetic processes , so there must be a much deeper linkage of toroidal geometry (and also that of the helix), with electromagnetism...hope this helps simplify the mathematical modelling for a new paradigm in science...hmm !

Thank you also @sidharthabahadur. I am really impressed by the breadth of your knowledge.
Helical motion is very common in nature. It is present even there where one expects it the least.
The arrow in flight makes also a helical motion. Please watch in this interesting video how an arrow flies (from 7:01 to 7:12 on the timeline)

Amazing @Mitko_Gorgiev ! This "Rabbit Hole" really goes DEEP...!

Given that "computational fluid dynamics" was being discussed in this thread , I have found that water too flows in a helical path (just pour water from a jug & see for yourself) - it's an intrinsic property of water that mainstream science has never bothered to explain ! Nobody has explained why river water always "meanders" , even over a flat plain , instead of neatly flowing in a straight line...well , it is the inherent nature of water that makes it meander !

Anyone can try this simple experiment at home - spill some water from a bucket , onto the flat floor...you will notice that the water "meanders" on it's way to the outlet...you see - water just does NOT like to flow in a straight line :))

Indeed , water loses it's freshness and vitality when it is forcibly transported in a straight line , through pipelines... no wonder , mountain spring water feels so soothing & energising to drink , totally unlike the "stale tasting" tapwater we get...

Actually , it is believed that amongst all physical substances , it is water that most closely resembles the properties of the Ether (even though the ether is ethereal !) , so studying the behaviour of water could give us at least some clues to the nature of the ether .

**The great Austrian naturalist - Viktor Schauberger , had spent his entire life studying water and he made some astonishing discoveries based on that - to such an extent that it caught Hitler's eye during the closing stages of WW II....

***I have also enclosed a mind blowing perspective on how our consciousness itself is a torsion field and that this "twist" or SPIRALITY , is an intrinsic aspect of our Universe itself - as it's built of Vortexes , from the sub-atomic to the Cosmic or Galactic scale !

I noticed today that if I took a glass of water and poured it out slowly (small tilt to the cup), it flowed rather smoothly, but when I increased the tilt the water flowed in a sort of helical manner, this could be observed in the given image in the case of a tap:

What could the possible reason for such a flow be?

Considering Plateau-Rayleigh instability (mentioned in comment), could it be that if speed of water flow be quick enough that the flow doesn't break down to form drops. This is what my limited non mathematical understanding brought me to.

I have misinterpreted the question. This post does not answer the original question of why helical patterns appear in flowing water, which has been already answered here.

Interestingly, my observation of chain-like patterns in flowing water corresponds to the one described in this question.

Unfortunately the above-mentioned question was wrongly marked as a duplicate which prevents me from answering it. I've presented the case before the moderators. (See meta post). Thanks for your attention.

You have made a very interesting observation. In short, this is due to the shape oscillations of the flowing stream of water.

I did the experiment and observed the pattern too. But I'd say that the pattern was not really helical, it was more like the shape of a metal chain. Here you can see the same image from two different angles.

Front view

Side view

When you look carefully at the two images, you would see the flowing water is flattening out in mutually perpendicular directions as it falls down the cup. This works due to a principle similar to the one behind shape oscillations of a freely suspended water droplet. Only difference is that here, you don't have an oscillation in the direction of the flow. So this is actually a 2D case of water drop shape oscillation.

Consider water flowing out from the cup at a uniform rate. We can imagine that the flowing stream of water consists of several cross-sectional films falling down one after another (marked in the picture as green rings). For simplicity, we neglect any interactions between the adjacent films. Let's track the shape of one of the films. When no other force is at play, the film tends to remain in a circular shape.

At the point where water loses contact with the cup (which I shall call 'Base'), the film is stretched out horizontally due to the normal reaction of the cup. When the film leaves the base, surface tension would provide a restoring force for the film. The film would get fully stretched in the perpendicular direction at position A.

This sets up a 2-D oscillation. The film would oscillate between these two states as it falls down.

The oscillations may not be exactly as above but is somewhat similar.

Now imagine all of the films falling one after other down the cup. Note that all the films start out with a horizontal stretch. This would mean that each cross-sectional film would be in the same state of oscillation after falling the same distance from the cup. In other words, initial phase of all the films would be identical.

So in the big picture, as you go down the flow, you are actually observing the different states of the above oscillations with respect to time.

When you plot the different time states of the above depicted 2-D oscillation in along a Z-axis, you get a pattern similar to the one we observe in reality (but less curvy).

Here we have not included any damping forces between adjacent cross-sectional films. If we had, we would have obtained a much smoother shape.

This is a very good answer. The shape of the spout is elongating the shape of the water stream, surface tension is trying to pull it back to round. The tension, and mass of the water, creates an "overshoot", resulting in several observed oscillations as the stream falls. A faster pour will tend to be more chaotic, with greater mass to surface area ratio, reducing chances of the observed pattern.

+1 Impressive answer and effort. @Zheer, the oscillations might be due to the mentioned tension restoring force. Surface tension will try to collect the water into a circular shape. At the onset of the pouring, the water is flattened out, and so tension pulls it towards a circle-shape of its cross-section. Then I guess the idea is that it "overshoots", since there is no significant dampening force involved. So the water will now flatten in the direction of surface tension. In this new flattened shape, a new tension restoring force pulls back from the other dimension, and so it oscillates.

Spiralling towards a new understanding of ‘reality’, consciousness and our role in the universe

Spirality — the condition of being spiral — and ’reality’ are almost interchangeable terms. I made this remark at the beginning of the first chapter of my book, Spirals: The Pattern of Existence (Green Magic, 2006, 2nd edition 2013), my placing of the word ‘reality’ in parenthesis admitting both to the ‘reality’ perceived through our restricted everyday consciousness and to another, deeper reality usually invisible to us.

While spirality manifests itself in the physical world, in nature, in myriad forms, I regard it also as the unseen creative and ordering principle underlying the universe.

Consciousness would seem to be intimately related to torsion energy which, it is proposed, spirals through space-time — a universal ‘implicate order’ (to use the late quantum physicist David Bohm’s term) which underpins our everyday reality but which is also responsible for paranormal phenomena and ‘non-local’ effects observed in quantum physics, such as ‘entanglement’.

If you imagine a sub-atomic particle with zero spin decaying into two other particles which are then separated by great distance, so far apart even that there is no longer any physical force between them, quantum science says that they still retain information about one another, whatever the space between them, in a shared state which is indefinite until measured. If one particle moves in a certain way, so will the other. They are ‘entangled’. Verified experimentally, it supports the idea of a universal inter-connectedness.

Now, as Bernardo Kastrup rightly points out in Meaning in Absurdity (Iff Books, 2012) — although most scientists avoid discussing it at all costs — the concept of entanglement destroys any possible acceptance of a consensus reality, and ushers in a new vision of existence. The results of a study by a group of Polish and Austrian physicists, announced in a paper published in Nature in April, 2007 (‘An Experimental Test of Non-Local Realism’, by Simon Gröblacher et al), showed as untenable the materialistic idea that hidden properties in the quantum substrate caused entanglement; the hypothesis that physical properties in the particles themselves caused it had already been ruled out.

The momentous implication of this is that, as sub-atomic particles are regarded as foundational to nature, entanglement means that the concept of reality, as we normally understand it in everyday terms, must be abandoned. We have to rethink urgently what we mean by ‘reality’. And this, for me, is where torsion field theory, a relatively new branch of physics, comes in. It speculates that consciousness and DNA arise from torsion energy, the quantum twist, or spiralling, of space-time — the macro outcome of the micro-spin of sub-atomic particles.

The pioneering work in torsion field physics was carried out by Albert Einstein and the French mathematician Élie Cartan in the 1920s, resulting in the Einstein-Cartan theory, a classical theory of gravitation in which interest has been revived as theorists have tried to incorporate torsion into quantum theories, or as they explore its cosmological ramifications.

Copernican perspective

I have long believed that sub-atomic particles are better thought of as vanishingly small vortices interacting spirally in space-time rather than as discrete objects whirling about like microcosmic planetary systems, in the way that atoms were once conceived. We have failed to appreciate that vortices are the essence of our experience because of the prevalent Copernican perspective of the universe which tends to flatten things out: for example, the solar system is perceived with the sun at the centre and the planets in orbit around it in more or less the same plane.

But this is wrong. The sun is moving through space so the orbits of the planets are not circles around it, nor even ellipses, but helices (a helix is a spiral that turns about the same diameter, instead of constantly moving away from a central point — the difference, say, between a corkscrew and a helter-skelter slide). Astronomers have long known about these movements of the solar system — I have an astronomy encyclopedia from 1959 which describes them with a diagram — but they have never been emphasised to the general public. It is hard to enivisage when we are so conditioned to the usual flat-plane presentation of the solar system in science books and websites.

The concept of the ‘vortex solar system’ was spelled out by the Indian ethnobotanist Dr Pallathadka Keshava Batt in his 2008 book, Helical Helix: Solar System a Dynamic Process, which proposed helical movements for all objects in the universe, but his work in this area has had little impact in the West. Yet it is indicative immediately of how the universe and its process comprise vortices — probably double vortices, which can be likened to two pyramids rotating apex to apex, as this is what scientists are beginning to think governs electromagnetic fields.

Torsion fields (also called spin fields, or axion fields) are envisaged as twists, screw or helix-like manifestations of subtle energy which can curve to the left or right. Studies by the Russian astronomer and astrophysicist Nikolai Kozyrev, who died in 1983, and other studies by Russian scientists subsequently, have suggested that matter harnesses torsion waves to sustain its existence.

Such research, however, suggesting a theory for the behaviour of space and the nature of consciousness, is regarded as ‘fringe’ science in the West, especially because, from time to time, it has been used to propose faster-than-light travel and to explain extra-sensory perception, levitation and other paranormal phenomena, as well as ‘spiritual healing’ and dowsing.

Russian researchers (S V Avramenko, G F Ignatjev, G A Sergejev, S N Tarakhtiy, for example), created and tested torsion generators. A team at the Physics Institute of the Ukraine Academy of Sciences, and at Chernovitsky University, researched such generators, as well as the ‘cavity structural effect’. Since every atom throws off torsion waves as it vibrates, so the thinking goes, certain shapes can help harness and direct the energy flow more than others. The Platonic Solids would be in this category, but other shapes can harness the energy, too, as discovered by Dr Victor Grebennikov.

Torsion generators

The Chernovitsky group was particularly interested in the effects that cones of different sizes had on various processes. They found that the best ‘passive’ torsion generators were formed by cones shaped into the phi, or Golden, ratio of 1:1.618, revealing that torsion waves were phi spirals, as a cone duplicating this pattern harnessed the waves best.

Pyramidal structures can also harness torsion energy: Dr. John De Salvo, of the Giza Pyramid Research Association, published the Russian and Ukranian pyramid research of A Golod, V Krasnoholovets and associates from the Institute of Physics in Kiev, Ukraine, one of the leading scientific and military research centres in the former USSR. Dr N B Egorova, at the Mechnikov R&D Institute of the Russian Academy of Medical Science, tested the pyramid’s ability to harness torsion waves using living organisms placed inside, with positive results.

According to the Russian research, torsion wave energy pervades space at varying degrees of concentration. As star and planetary systems move through, and rotate with, the galaxy they encounter different concentrations in specific time intervals, in cycles varying from thousands to millions of years. It is speculated that a high density of torsion waves could have transformative effects on DNA on Earth, causing more highly evolved forms of life to replicate more rapidly than less evolved forms. Evolution certainly seems to have taken place in sudden leaps and bounds rather than in a gradual and uniform way.

Now, as is well known, Earth has a 26,000-year cycle known as the precession of the equinoxes, and it has been estimated that torsion wave energy peaks about every 75,000 years, or each third time round. It is said that we are now at such a peak; when torsion energy peaks, it is alleged, DNA can be restructured, resulting in an evolutionary advance.

Only about three per cent of human DNA is required for genome purposes, the other 97 per cent being referred to by scientists rather dismissively as ‘junk DNA’, as they do not understand the purpose of it. However, it is the ‘junk’, evidently, that will be reorganised by the influx of torsion energy waves.

Some Russian scientists have gone so far as to say this spiralling torsion energy could be the substance of the human soul, and the precursor of the DNA molecule, in the sense that a torsion wave emanating from the galactic centre, and passing through our solar system at the moment of a person’s birth, influences the DNA uniquely, this ‘energy signature’ subtly altering the DNA inherited from our parents.

Universal consciousness

A causal chain thus connects each of us to a universal consciousness through our DNA. Theorists speculate that spin interactions can be transmitted through and/or by space in a manner akin to electromagnetic waves but holding neither energy nor mass, only information, which links it to the idea of a universal consciousness.

If torsion fields and consciousness are inter-related, then it is of great significance for the various field theories of consciousness which have been put forward in recent years; such theories, in my view, offer the best explanation of the nature of consciousness in the current state of our knowledge — and not just because of the obvious spiral implications, my belief being that the spiral curve is the shape of time and the trajectory of consciousness, and a key to the riddle of existence and the inner essence of reality.

The idea is that consciousness, and not matter, is the ground of all existence, and that our minds participate in it as a universal informational continuum, rather than enfold a small part of it discretely. This is the standpoint of the ‘new mysterians’, who regard an understanding of consciousness as being beyond the scope of physical theory. Consciousness, after all, is the point of intersection between the cosmic dimension and the human dimension, between timelessness and time, where we receive the numinous and the ‘life-force’. It is a truism that invisible energy fields conduct life, in the sense of managing or directing it.

In discussing the goal of human wholeness, of self-actualisation, Carl Gustav Jung, in Psychology and Alchemy (Collected Works, vol. 12), said the way was not straight (or linear) but went in spirals. The spiral nature of this development, or individuation, to use his word, was induced by the complementarity of past and future, expressed in the concept of ‘regressive renaissance’, the idea that we can move forward only by locating a symbol from our past in order to transform, to move to a higher turn of the spiral.

I believe the universe is becoming aware of itself through the consciousness — our consciousness — that it is evolving, indicating that it has purpose and intelligence, although this possibility has been hidden from us for millennia.

Today, however, in many spheres of life, a number of related trends are appearing, constituting what I might term a ‘new reality’: a changing understanding of what ‘reality’ is, of what consciousness is, and of what our role in the universe is, a universe much more abundant in its complexity and magicality than hitherto realised. Complementing the Jungian idea of personal individuation, it involves an understanding of cosmic coherence at a new level on the evolutionary curve towards integration and wholeness.

Folks , it is just mesmerising to see the stunning analogies between an electromagnetic field , Toroidal/Helical geometry and the characteristics of a fluid like water - who would have thought that's even mathematically possible ??...at a deeper level of reality , there must exist some ETHERIC correlation between these apparantly "disparate" phenomena...hmm !

@Soretna , for last many years I have been exploring a geometric linkage to all the phenomena of physics that we study...in order to derive these phenomena from "First Principles" , I believe geometry could be an excellent starting point .

***Especially , for electromagnetism - any electrical engineer would agree that there is some DEEP and intrinsic relationship between electromagnetism and the donut/toroid shape . In that regard , I found this enclosed article very useful . Please take a look . You may also want to read the pdf document which is more to do with Plasma Physics and Nuclear fusion in Tokamak reactors :-

Calculations in toroidal geometry with full MHD (Magnetohydrodynamics) equations -

Theory and applications of toroidal moments in electrodynamics: their emergence, characteristics, and technological relevance

Nahid Talebi, Surong Guo and Peter A. van Aken

Cite this

Abstract

Dipole selection rules underpin much of our understanding in characterization of matter and its interaction with external radiation. However, there are several examples where these selection rules simply break down, for which a more sophisticated knowledge of matter becomes necessary. An example, which is increasingly becoming more fascinating, is macroscopic toroidization (density of toroidal dipoles), which is a direct consequence of retardation. In fact, dissimilar to the classical family of electric and magnetic multipoles, which are outcomes of the Taylor expansion of the electromagnetic potentials and sources, toroidal dipoles are obtained by the decomposition of the moment tensors. This review aims to discuss the fundamental and practical aspects of the toroidal multipolar moments in electrodynamics, from its emergence in the expansion set and the electromagnetic field associated with it, the unique characteristics of their interaction with external radiations and other moments, to the recent attempts to realize pronounced toroidal resonances in smart configurations of meta-molecules. Toroidal moments not only exhibit unique features in theory but also have promising technologically relevant applications, such as data storage, electromagnetic-induced transparency, unique magnetic responses and dichroism.

Although the history of Maxwell equations and light-matter interaction started as early as 1861 [1], electromagnetism has been the field of most challenging and rival concepts such as understanding the actual velocity of the information transfer [2], the actual wave function of photons [3], [4], [5], realization of cloaking, negative refraction, and transferring of the data beyond the diffraction limits using plasmons and metamaterials [6], [7], [8], [9], [10], [11], and the emergence of the Abraham-Minkowski controversy and the actual linear momentum of light [12], [13]. At the heart of our understanding of light-matter interaction, there exist multipole-expansion sets that tell us how to construct the extended electromagnetic sources according to the localized sources with well-known electromagnetic field and radiation patterns. This concept, for example, forms the basics to understand the quasi-static dipole and quadrupole localized plasmon resonances in individual [14] or chains of metallic nanoparticles arranged in the forms of converter-coupler gratings [15], waveguides [16], [17], [18], or resonators [19], [20] (the so-called longitudinal and transverse resonances). The coefficients of such an expansion set known as moments were conventionally ordered as appearing in the expansion set according to the electric and magnetic multipole terms.

In 1958, however, Zeldovich [21] reported on the possibility of elementary particles under breakdown of spatial parity to interact with the electromagnetic wave in a peculiar form considering the interaction energy Hint=−→S⋅→∇×→H,Hint=−S→⋅∇→×H→, where →SS→ is the spin and →HH→is the magnetic field. Considering this problem in classical electrodynamics, in 1967, Dubovik noticed the possibility of introducing a new class of moments being excluded from the family of electric and magnetic moments, with different time-space symmetries but appearing in similar orders in the expansion set to the magnetic moments [22], the so-called toroidal moments. From those early days, discussions about toroidal moment have created an impetus in both solid-state physics and electrodynamics.

What is so interesting about toroidal moments? This question is partly entangled with the human curiosity to find new states of matter and ordering and partly related to the technological applications. Toroidal ordering in solid states can open up possibilities for a new kind of magnetoelectric phenomenon [23], with applications in data storage and sensing. Whereas toroidal moments have been reviewed in Ref. [23], with a focus on static spin-based toroidal moments and magnetoelectric effect (ME) in condensed-matter physics, we will put the emphasis of our review on different expansion sets leading to the toroidal moment and on electrodynamic toroidal moments in metamaterials and free space. Ref. [24] has provided a concise review in this aspect. Our review is intended to provide a more complete answer to this question and to invoke new directions in light-matter interactions involving toroidal moments. For this purpose, we start with the theoretical investigation of the multipole expansion sets in Section 2 leading to toroidal moments and time-space symmetry characteristics of moments, respectively. We furthermore discuss in Section 3 the toroidal moments in solid-state and photonic systems, the origin of asymmetric magnetoelectric tensors in materials, the electromagnetic interaction between external light and (meta-)materials that sustain toroidal moments, and finally the electromagnetic field and radiation pattern associated to isolated toroidal moments. In Section 4, we further consider the possibility of the excitation of materials with toroidal moments with different polarization states of light and also with relativistic electrons. In Section 5, we explore the coupling of toroidal moments to other classes of moments like electric dipole to form radiation-free sources of electromagnetic fields and also discuss the characteristics of toroidal metamaterials. We continue the review to cover the technological applications of toroidal moments and finally provide a conclusion and an outlook.

2 Families of multipoles in electrodynamics

Mathematical expansion sets are essential for our understanding of the physical properties of systems and matter. Not only such expansion sets help us to analytically find solutions to simple problems, but they also help us to simplify or find approximate solutions to complicated systems. Among the so-called expansion sets that physicists routinely use are Rayleigh expansion sets [10], [25], Eigenmode expansion sets [11], [26], [27], and multipole expansion sets [28]. The multipole expansion, which is based on the expansion of either potentials [29], [30], [31] or fields [32], [33], has several applications in classical electrodynamics, such as finding solutions to inverse problems [34], [35], [36], reconstruction of images [37], and in general, decomposition of induced charges into a set of localized charge distributions with well-known near-field and far-field characteristics. Moreover, semianalytical methods based on multipole expansion sets of magnetic and electric vector potentials, such as the method of auxiliary sources [38], [39], [40], the multiple multipole method [41], [42], [43], or the generalized multipole technique [11], [18], [44], [45], [46], [47], [48], [49], are exponentially convergent efficient methods in numerical electromagnetics [50]. In the following, we consider the family of multipole expansions for static and dynamic systems, considering the Taylor expansions of sources, potentials and fields.

2.1 Multipole expansions for potentials and fields

A direct Taylor expansion of charge and current densities proposed by Dubovik in 1990 already demonstrated the existence of a toroidal moment described as →T=(1/10c)∫(→r→r⋅→J(→r)−2r2→J(→r))d3r,T→=(1/10c)∫(r→ r→⋅J→(r→) − 2r2J→(r→)) d3r, which originates from the currents flowing along the meridians of an infinitesimal sphere as →JT=→∇×(→∇×(→Tδ(→r))).→J(→r)J→T=∇→ × (∇→ × (T→δ(r→))). J→(r→)then continues as the inner flow by extending from one pole to the other pole of the sphere [22], [51] (see Figure 1A). Moreover, toroidal moment was shown to be a multipole order originated from the transverse part of the current density →J⊥(→r,t),J→⊥(r→, t), as the longitudinal current (→J∥(→r,t))(J→∥(r→, t)) is not independent from the charge density, because of the charge conservation criterion →∇⋅→J∥(→r,t)=−∂ρ(→r,t)/∂t∇→⋅J→∥(r→, t)=−∂ρ(r→, t)/∂t and hence is related to the multipole expansion of the scalar potential. Here, the longitudinal current is associated with the irrotational part of the current density vector field and points out from the charge density ρ(→r,t)ρ(r→, t) toward a direction normal to any surface inclosing the charge distribution, whereas →J⊥(→r,t)J→⊥(r→, t) is parallel to such a surface. Obviously, a configuration of the current density distribution in the form of →JTJ→T is topologically similar to the poloidal currents flowing at the surface of a torus, hence leading to toroidal magnetic fields and the so-called polar toroidal moment (see Figure 1B). Interestingly, the poloidal current distribution leads to a diminishing of the magnetic quadrupole moment and hence an increase in the detectability of toroidal moment [52]. As a simplification, a ring of either static [23], [53] or dynamic [54], [55], [56], [57] magnetic moments is considered as a configuration for the excitation of a polar toroidal moment or the so-called magnetic toroidal moment. A dual configuration to the polar toroidal moment is an axial toroidal moment (or electric toroidal moment), which is composed of a ring of electric dipolar configurations [51], [58]. However, in contrast to a polar (magnetic) toroidal moment, the axial (electric) toroidal moment does not violate time and parity symmetry. In this paper, we mainly focus on the polar (magnetic) toroidal moment.

Current density and magnetic moment distributions associated with the toroidal moment.

Poloidal currents excited at the surface of (A) a sphere and (B) a toroid. (C) Equivalently, a ring of magnetic moments is also attributed to the excitation of a toroidal moment.

More often, the solutions to potentials rather than field components are considered. This is based on the fact that solutions of the Helmholtz equations, known as wave potentials, are well established in arbitrary coordinate systems. Moreover, it is possible to form a Helmholtz equation for potentials in complicated materials like chiral and topological materials [59], which is not in general feasible for field components. In addition, these are the vector →A(→r,t)A→(r→, t) and the scalar φ(→r,t)φ(r→, t) potentials, which more often appear in the quantum mechanical Hamiltonian of the interaction of charged particles with light. However, scalar and vector potentials are also not independent, where a gauge theory should be applied to pursue a physically relevant wave solution for electromagnetic fields [60]. In practice, considering each gauge, theory dictates a relation between the scalar and vector potentials, meaning that they are not independent. Hence, the multipoles obtained by expanding them are also not independent. Considering the Lorentz gauge as →∇⋅→A(→r,t)=−(1/c)∂φ(→r,t)/∂t,∇→⋅A→(r→, t)=−(1/c)∂φ(r→, t)/∂t, the longitudinal part of the vector potential is also determined by the scalar potential. In other words, multipole expansion sets related to the scalar potential or the longitudinal part of the vector potential, along with the transverse part of the vector potential, form a complete basis of multipoles. Following Vrejoiu [29], the reduction of the multipole tensors in the form of symmetric traceless ones demonstrates the existence of a toroidal moment, which has the same order as the magnetic quadrupole moment, but sustains different symmetry rules. This is, however, in contrast to the static case, for which no toroidal moment is obtained even after a tedious reduction of the tensors according to the symmetry groups. Therefore, the family of toroidal moments is composed of hybrid moments, which only come to existence by expanding the transverse part of the vector potential. In this regard, the toroidal dipole and quadrupole tensors in Cartesian coordinate and in Gaussian units are obtained as

where i, k∈(x, y, z) and δik is the Kronecker-delta function. It should be noted that, often, an equivalent definition of the toroidal moment is considered as →T=(1/6c)∫→r×(→r×→J(→r,t))d3rT→=(1/6c)∫r→×(r→×J→(r→, t)) d3r (the toroidal moment is the moment of the dipole-magnetic moments), which has an equivalent time-averaged quantity to the one which was introduced in Eq. (1).

Additionally, a direct expansion of fields may be also exploited, for which the outgoing-wave Green’s function is expanded versus spherical harmonic multipoles [61]. Incorporating the Helmholtz decomposition of the current density function into the longitudinal and transverse toroidal and poloidal terms [31], the resulting electric field has three components as →E(→r,ω)=−→∇eL(→r,ω)−→r×→∇eT(→r,ω)−→∇×(→r×→∇eP(→r,ω))E→(r→, ω)=−∇→eL(r→, ω)−r→×∇→eT(r→, ω)−∇→×(r→×∇→eP(r→, ω)) where eL, eT and eP are its Debye scalar potentials associated with longitudinal, toroidal and poloidal currents, respectively. In this regard, toroidal currents produce toroidal electric and poloidal magnetic fields, and poloidal currents produce poloidal electric and toroidal magnetic fields.

In a different approach, Spaldin et al. [62] have introduced the magnetization density →μ(→r)μ→(r→) in a solid state system and its interaction energy with the external magnetic field as Hint=−∫→μ(→r)⋅→H(→r)d3r,Hint=−∫μ→(r→)⋅H→(r→)d3r, where →μ(→r)μ→(r→) can have contributions from both spin and orbital momentum. They further have expanded the magnetic field in powers of the field gradient to conclude moment distributions in the form of (i) the net magnetic moment of the system (→m=∫→μ(→r)d3r),(m→=∫μ→(r→)d3r), (ii) the pseudo-scalar quantity called magnetoelectric monopole (a=(1/3)∫→r⋅→μ(→r)d3r),(a=(1/3)∫r→⋅μ→(r→) d3r), (iii) the toroidal moment vector dual to the antisymmetric part of the tensor (→t=(1/2)∫→r×→μ(→r)d3r),(t→=(1/2)∫r→×μ→(r→) d3r), and (iv) the quadruple magnetic moment of the system (qij=(1/2)∫[riμj+rjμi−(2/3)δij→r⋅→μ(→r)]d3r),(qij=(1/2)∫[riμj+rjμi−(2/3)δij r→⋅μ→(r→)] d3r), to name only the first few terms in the expansion set. The interaction energy can be further expanded as Hint=−→m⋅→H(→r=0)−a(→∇⋅→H)→r=0−→t⋅(→∇×H)→r=0−∑ijqij(∂iHj+∂jHi)→r=0,Hint= −m→ ⋅ H→(r→=0) − a(∇→⋅H→)r→=0− t→⋅(∇→×H)r→=0−∑ijqij(∂iHj+∂jHi)r→=0, where i, j are Cartesian directions and δij is the Kronecker-delta function.

As a summary, all kinds of multipole expansion techniques can be introduced, but decomposition techniques should be exploited to extract the toroidal order, with the symmetry specifications discussed in the following chapter.

2.2 Classification of multipoles according to symmetry rules

In a Maxwell-Lorentz system of equations in which the only sources are charges and currents related to each other as →J=∑ei→viδ(→r−→ri),J→=∑ei v→i δ(r→−r→i) , electric, magnetic and toroidal moments form a complete system of moments and there is no place for further generalizations. However, for the sake of completeness and symmetry, consideration of magnetic charges is recommended as well [63], as →∇⋅→B(→r,t)=ρg(→r,t)∇→⋅B→(r→, t)=ρg(r→, t) and →∇×→E(→r,t)=−(∂→B(→r,t)/∂t)−→Jg(→r,t),∇→×E→(r→, t)=−(∂B→(r→, t)/∂t)−J→g(r→, t), where ρg(→r,t)ρg(r→, t) and →Jg(→r,t)J→g(r→, t) are magnetic charge and magnetic current density distributions, respectively and →E(→r,t)E→(r→, t) and →B(→r,t)B→(r→, t) are the electric and magnetic flux components, respectively, at time t and position shown by the displacement vector →r.r→. Moreover, in contrast to the electric current distribution, which is a vector, the magnetic current distribution is an axial vector (pseudo-vector). Generalizing the multipole expansion schemes to include magnetic sources, a new class of poloidal magnetic currents is unraveled, which gives rise to the axial (electric) toroidal moments, which is an exact dual to the polar (magnetic) toroidal moment discussed previously. In analogy to the work carried out in Ref. [51], the axial (electric) toroidal moments are denoted here as G. Under space-time parity operations (space inversion and time reversal), the whole class of multipoles behaves as depicted in Table 1. The symmetry rules should be applicable to the vector fields acting on the multipoles, as the free energy of the system should be invariant upon space-time inversions. In this regard, the free energy of the interaction of an arbitrary system with the electromagnetic field in free-space is given by →Hint=−→P⋅→E−→T⋅(∂→D/∂t)−−→M⋅→H−→G⋅(∂→B/∂t)H→int=−P→⋅E→−T→⋅(∂D→/∂t)−M→⋅H→−G→⋅(∂B→/∂t) [64]. Moreover, Spaldin et al. [23] have argued that the cross-product →P×−→MP→×M→ sustains similar space-time symmetries as a toroidal moment, but in fact, it is not a toroidal moment, as it interacts with the electromagnetic field according to the free-energy term (→P×−→M)⋅(→E×→H).(P→×M→)⋅(E→×H→). We only mention here that the local distribution of the vector →r×−→M(→r)=→r×(→r×→J(→r))r→×M→(r→)=r→×(r→×J→(r→)) can be described as the distribution of the toroidal moment, when only the time-averaged quantity of this vector is anticipated and compared to (→r→r⋅→J(→r)−2r2→J(→r)),(r→ r→⋅J→(r→)−2r2J→(r→)), as described in Section 3.1. Moreover, as it will be discussed later in this review, a linear ME might occur in a system with a net toroidal moment, which means that the free-energy function contains the contribution →T⋅(→E×→H)T→⋅(E→×H→) [22].

Table 1:

Transformation properties of electric (P), magnetic (M), polar toroidal (T), and axial toroidal (G) moments under space inversion and time reversal operations.

Multipole Parity
Space Time P – + M + – T – – G + +

Although parity considerations in space-time is believed to provide a complete class of multipoles, a more general picture beyond the quasi-static limits has to be considered when the retardation effects are taken into account [65], in which the Taylor expansion versus the wave number (k) is included as well. As this expansion is always ruled out versus k 2=(ω/c)2 in free space, there is no room to discuss momentum-frequency symmetries in classical electrodynamics. However, considering induced currents in materials, anisotropic media, and also material losses, it is well known that time-reversal symmetries might be violated in special cases [66]. In this regard, a complete picture including the k-symmetry has to be developed.

To conclude this section, we mention that despite several experimental realizations of toroidal moments in both solid-state and metamaterials systems, there are still debates about an individual class of toroidal moments, to be excluded from the magnetic quadrupole terms. An interesting study in this regard is presented in Ref. [67], where the author has considered multipole expansions at both potential and field levels, but also in both Cartesian and spherical coordination.

3 Characteristics, interaction, and radiation of toroidal moment in materials

3.1 Toroidization and ME in materials

A static toroidal moment exists in various materials both microscopically and macroscopically, covering transition metal ions [68], biological and chemical macromolecules [69], [70], [71], [72], [73], [74], bulk crystals [75], [76], [77], [78], [79], [80], and glasses [81]. In macroscopic condensed matter, formation of toroidal moments in materials plays a vital role in the asymmetric ME. ME describes a phenomenon of a spontaneous magnetization (polarization) induced by an external electric (magnetic) field. The effect can be linear or nonlinear (higher order), having symmetric or antisymmetric tensors, and usually depends on temperature. An antisymmetric tensor is described by [ξ]T=−[ξ], where T is transpose operator. Moreover, an antisymmetric tensor is a special case of an asymmetric tensor where the latter is described by ξij ≠ ξji. One consequence of the antisymmetric linear ME in materials is toroidization, which is denoted by forming an order of vortices of the spontaneously induced magnetization [80] or spin [23].

In solid-state systems where the magnetic fields are induced by localized spins (→Sα)(S→α) at sites →rα,r→α, the toroidal moment can be derived as →T=(gμB/2)∑α→rα×→Sα,T→=(gμB / 2)∑αr→α×S→α, with the gyromagnetic ratio g and the Bohr magneton μB. However, toroidal moments have origin-specific values related to the choice of the lattice [82]. By shifting the origin from →rr→ to →r′=→r+→R0,r→′=r→+R→0, the toroidal moment changes as −→T′=→T+(gμB/2)(→R0×∑α→Sα).T′→=T→+(gμB / 2)(R→0×∑αS→α). In contrast to the moment, changes in toroidization (toroidal moment per volume of unit cell) are uniquely defined values. Nevertheless, as for the ferroelectric materials, spontaneous toroidization can be defined in materials with net toroidal moments, which describes the rate of change in toroidal moments when a vector field is applied. In general, symmetry considerations hint at the contribution of toroidal moments to the magnetoelectric polarization when considering the free energies of systems with spin ordering and the electrodynamics Hamiltonian of systems showing the linear ME. However, toroidal moments are not the only source of the ME. It is almost impossible to provide a direct relation between toroidization and magnetoelectric coupling [23]. Nonequilibrium electrodynamic consideration can describe the differences in the off-diagonal magnetoelectric tensors below a critical temperature, which is related to the coupling between the spontaneous toroidization, magnetization, and polarization [83], [84], [85]. In general, the antisymmetric contribution to the magnetoelectric tensor is an indication for the presence of toroidization in materials.

Ferrotoroidicity describes a spontaneous, long-range alignment of toroidal moments in materials. It has been of major interest to investigate whether the ferrotoroidicity consisting of magnetic toroidal moments can be defined as an individual class of ferroic state, in addition to ferroelastic, ferromagnetic, and ferroelectric orders [23], [53], [80], [86], [87]. The reason is that the ferrotoroidicity complements the others in terms of space-time-inversion symmetry, whose transformation property is characterized by changing sign under both time reversal and spatial inversion. In this sense, ferrotoroidic state in materials is expected to be dominant like other primary order parameters. Zimmermann et al. [53] demonstrated for the first time the hysteretic switching of ferrotoroidic domains in LiCoPO4, where they simultaneously applied crossed static magnetic and electric fields. They concluded that the ferrotoroidic order is a primary order parameter as other ferroic states.

It has been further discussed that the toroidal moment might be merely a vortex-like arrangement of magnetic moments from antiferromagnetic state. Tolédano et al. [80] proposed a case study of LiFeSi2O6 in this aspect. All the magnetoelectric tensor components during the symmetry-breaking transition from monoclinic to triclinic magnetic phase at 18 K in LiFeSi2O6 were measured. In terms of crystallographic symmetry, either a biquadratic coupling of the toroidal-moment vector components or the antiferromagnetic vector components is permitted and is able to explain the observed inversion symmetry breaking. However, in terms of microscopic physics, only the free energy of the toroidal coupling term can be large enough to drive the transition in a single step. This is due to the fact that the free-energy contribution from the antiferromagnetic vector components is geometrically attenuated. Therefore, to distinguish ferrotoroidicity from ferromagnetic order in a material, it is necessary to examine both the crystallographic symmetry and the collective response of the associated moments.

As mentioned previously, in systems with nonvanishing magnetic moments the toroidal moment depends on the choice of origin [82]. It was suggested to choose an origin for which the system is centrosymmetric and nontoroidal in the initial configuration. The change of toroidal moment is then derived from a symmetry-breaking structure distortion at the final state and thus can be interpreted as the spontaneous toroidal moment of the system [82]. Ferrotoroidicity has been demonstrated by materials whose magnetic order breaks spatial inversion symmetry. It is waiting for exploration in a variety of material systems and holds a great potential for future applications [86].

Experimental characterization of ferrotoroidic states requires a probe and a detection mechanism that violates space-time symmetry. Obviously, the measurement of magnetoelectric coefficients is an appropriate but indirect method for investigating toroidal moments in materials, as was achieved for Co3B7O13Br [83] and Cr2O3 only when it starts to be driven into the spin-flop phase [88]. Moreover, the magnetoelectric tensor of Cr2O3 sustains also a relativistic-invariant pseudoscalar term [89], which is associated with the axionic ME [90] and the so-called Tellegen term [91].

Considering the magnetic insulator LiCoPO4 and the XY-like spin glass NixMn1−xTiO3 (x≈0.42) as examples, LiCoPO4 has a net toroidal moment T (0, Ty, Tz) ≠ 0 below 21.8 K [53]. The predominant component Tz of the toroidal moment is induced by two pairs in Co2+ (each pair highlighted in purple or green in Figure 2A), while the other component Ty caused by spin rotation away from the y-axis is approximately two orders of magnitude smaller and thus can be neglected [53]. For the ilmenite structure of the XY-like spin glass NixMn1−xTiO3 (x≈0.42), the Ni2+ and Mn2+ ions are randomly distributed in the magnetic (Ni, Mn) plane and form a honey comb lattice. At the presence of the cross-product of the electric and magnetic field components, a net toroidal moment is polarized in the direction perpendicular to the honeycomb lattices during the spin-freezing process (in Figure 2B) [81].

Figure 2:

Examples of toroidal moment in materials.

(A) Z-component of the toroidal moment in LiCoPO4 originated from two spin pairs in Co2+ shown for its magnetic unit cell (rectangle) [53]. Because of the different radii ˜r<r,r˜<r, the clockwise and counterclockwise contributions from these two pairs do not cancel. (B) Crystal structure of NixMn1−xTiO3 projected along the hexagonal c axis [001] [81]. (C) Structural model of the interface type (-Fe-FeO-TiO2-BaTiO3) between Fe and BaTiO3 [92]. (D) An array of vortex-antivortex pairs present in each PbTiO3 layer indicated by polar displacement vectors (yellow arrows) [93]. Figures reproduced with permission from (A) Ref. [53], NPG; (B) Ref. [81], NPG; (C) Ref. [92], NPG; (D) Ref. [93], NPG.

Apart from toroidization in bulk material at low temperatures, toroidization might be achieved at room temperature by interface engineering due to structural modification, charge transfer and strain effects [92], [93]. Figure 2C shows one possible interface type of (-Fe-FeO-TiO2-BaTiO3) between Fe and BaTiO3, which results in a spontaneous magnetization and polarization at room temperature. Polar vortices for electric toroidal moment are observed in oxide superlattices (SrTiO3 and PbTiO3) [93], as shown in Figure 2D. However, in the experiments mentioned here, where the spatial-temporal symmetry was broken with a strong correlation across the interface, it is the quantity →P×−→MP→ × M→ which contributes to the detected time-averaged signal.

3.2 Interaction of optical waves with toroidized materials

Optical waves are used to detect the long-range order of toroidization in materials, but not for producing toroidization. Moreover, if any detection method should be used for recognizing ferrotoroidic orders, it should sustain odd parity and odd time-reversal symmetries. An example of an optical effect with such symmetry considerations is gyrotropic birefringence [94], [95], which is also called nonreciprocal directional dichroism (NDD) [96] (see Figure 3A). NDD appears in materials with ferrotoroidic ordering and is described as a phenomenon that the electromagnetic absorption spectrum depends on the direction of the wave vector of the incident light relative to the toroidal moment →TT→ (or →P×−→M;P→×M→; see discussions in Section 2.2) [97]. However, Di Matteo et al. [98] have shown that NDD has contributions from a magnetic quadrupole, a toroidal polar moment and magnetic octupole moments, in contrast to the natural circular dichroism, which is a measure of only the axial toroidal quadrupole moment of a system.

Figure 3:

Examples of the interaction between optical waves and toroidal materials.

(A) NDD signal measured from a GaFeO3 crystal for a polarization of the incident X-rays being aligned as Eω||b and Eω||c, where a and b are depicted in the inset [96]. (B) Ferrotoroidic domains of a LiCoPO4 (100) sample at 10 K imaged with SHG light at 2.197 eV [78]. Black and red lines indicate the different ferrotoroidic domain walls caused by Tz and Ty, respectively. Inset: ferrotoroidic domain movement caused by a temperature cycle. Figures reprinted with permission from (A) Ref. [96], APS; (B) Ref. [78], NPG.

Another mechanism of observation by optical effects is based on the space-time symmetry of the second-order nonlinear susceptibility of materials with toroidal ordering. The second-harmonic generation (SHG) effect has been exploited to image the ferrotoroidic domain in LiCoPO4 [53], [78]. An incident light impinges the crystal and induces emission of electromagnetic waves from it, including the SHG wave. The nonlinear optical response of the crystal is determined by the crystal symmetry and microscopic structure. In addition, the long-range ordering and other additional order parameters affect the symmetry in different ways. Therefore, the SHG in general is differently polarized by different ordering parameters. As an example, the ferrotoroidic domains caused by different toroidal ordering Tz and Ty in a LiCoPO4 (100) sample was imaged with SHG light at 2.197 eV [53], [78] (see Figure 3B).

3.3 Dynamic toroidal moments in artificial metamolecules and dielectric nanostructures

In addition to static toroidal moments in various condensed materials, engineered toroidal moments using the concepts in plasmonic and metamaterials [99] cause growing interest, as such moments can be excited by and interact with electromagnetic fields more efficiently than the toroidal moments in condensed matter [100]. This helps us to understand the basis of dynamic toroidal moments interacting with light as well as modifying the optical properties of materials.

However, the toroidal dipole response in electrodynamics is often masked by more dominant electric and magnetic multipoles at similar energies. Therefore, artificial toroidal metamaterials are initially designed to amplify toroidal moments and suppress the competing electric and magnetic multipoles. This interesting field was set off by mimicking toroidal coils [52], [100] at microwave frequency to explore the toroidal dipole response in a great variety of structures as well as entering the optical regime by scaling structures down to the nanoscale [57]. Figure 4 gives a brief but not complete overview of different investigated toroidal metamolecules in catalogues of split-ring resonators and their variants, magnetic resonators, apertures, plasmonic cavities and structures, and dielectric nanostructures. It should be mentioned that the classification of metamaterials is not restricted to a single catalogue, for example, metallic double disks can be either considered as magnetic resonators [106], [117] or plasmonic cavities [118]. There are also other novel designs, for example, vertically assembled dumbbell-shaped apertures and split ring resonators resulting in a horizontal toroidal response in the optical region [119], [120].

Figure 4:

A brief overview of the investigated metamolecules with toroidal dipolar responses.

Split-ring resonators (SRR): (A) Schematic drawing of 3D SRR constituted by four rectangular metallic wire loops embedded in a dielectric slab [101]. (B) A combined SRR by sharing a central connecting bridge [102]. (C) An asymmetric SRR-based planar toroidal metamaterial [103]. (D) Planar conductive metamaterials formed by two symmetrical split rings [104]. Magnetic resonators: (E) an optical toroidal structure composed of a gold hexamer and metallic mirror separated by a dielectric layer [105]. (F) Three magnetic resonators consisting of two metallic rods and a dielectric spacer [106]. (G) An infrared toroid metamaterial composed of asymmetric double bars [107]. (H) A THz flat-ring-dimer (metallic double disks) toroidal metamaterial [108]. Apertures: (I) Toroidal metamaterial arrays consisting of dumbbell-shaped apertures manifest the destructive interference between electric and toroidal dipole moments leading to scattering transparency [54]. (J) Electric toroidal dipolar response has been achieved by metamaterial based on sun-like aperture element at microwave frequency [109]. Dielectric nanostructures: (K) Dielectric nanoparticle [110]. (L) Dielectric nanodisk with illustration of toroidal electric field distribution [111]. (M) Dielectric cylinders [112]. (N) Dielectric nanotubes [113]. Plasmonic cavities: (O) Core-shell nanoparticles support toroidal dipole excitation by a plan wave [114]. (P) Plasmonic oligomer nanocavities with seven nanoholes in metallic films sustain toroidal responses at visible wavelengths [19]. (Q) Toroidal modes are sustainable in the infrared and visible regime by sidewall-coated plasmonic nanodisk antenna [115]. (R) Circular V-groove array supports plasmon toroidal mode at optical frequencies [116]. Figures reprinted with permission from (A) Ref. [101], APS; (B) Ref. [102], Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim; (C) Ref. [103], APS; (D) Ref. [104], APS; (E) Ref. [105], NPG; (F) Ref. [106], by courtesy of Jing Chen; (G) Ref. [107], AIP; (H) Ref. [108], Elsevier B.V.; (I) Ref. [54], NPG; (J) Ref. [109], AIP; (K) Ref. [110], OSA; (L) Ref. [111], NPG; (M) Ref. [112], APS; (N) Ref. [113], OSA; (O) Ref. [114], Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim; (P) Ref. [19], AIP; (Q) Ref. [115], ACS; (R) Ref. [116], OSA.

The far-field radiation patterns of a toroidal moment are virtually identical with those of dipolar electric and magnetic moments, even though these moments are fundamentally different [24], [55] (see Section 3.4). The destructive interference between their radiation patterns has been exploited to construct a nonradiating scatterer. This approach is called the anapole excitation [104], scattering transparency [114], or analogous electromagnetic-induced transparency [102]. Such an intriguing design has promising potential applications in designing low-loss, high-quality factor cavities for sensing, lasers, qubits, and nonscattering objects for cloaking behavior. The anapole configuration is described in detail in Section 5.1.

In parallel, the dielectric metamaterial [121] was proposed to overcome dissipation loss of metals as encountered in metal-based toroidal metamaterial [112]. It has been pointed out that dissipation losses in metals originating from the ohmic resistance can hinder the excitation of toroidal multipoles, especially in the optical regime, resulting in weak coupling to external fields.

A fantastic characteristic of toroidal metamaterials as aforementioned is the feasibility in tuning toroidal responses via size, shape, material [114], and spatial arrangement and symmetry [56] of the constitutive elements [122]. So far, the major existing toroidal metamolecules are designed to achieve pronounced magnetic toroidal dipole response due to their peculiar property of asymmetry in both spatial inversion and time reversal. In this case, introducing space-inversion asymmetry, from either the geometry or the excitation source, is necessary for toroidal moment excitation, while breaking time-reversal symmetry has already been fulfilled by the magnetic dipole intrinsically. These toroidal artificial structures open an avenue to study the interaction with electromagnetic radiation in both the far-field and the near-field.

3.4 Electromagnetic fields and radiation patterns associated with toroidal moments

The electromagnetic field of a toroidal dipole can be suitably derived by using the retarded Green’s function [65], [123], [124]. Considering an infinitesimal toroidal moment with arbitrary orientation as →TT→ oscillating at the angular frequency ω, the time-harmonic field components at the position →r=Rˆnr→=Rn^ is given by

The vector-field profile around the toroidal moment has also a toroidal configuration, as shown in Figure 5A. Interestingly, the electric field component of an oscillating toroidal dipole has three components depending on R −3, R −2 and R −1, respectively, dissimilar to the electric and magnetic dipoles, which depend on R −3 only. This fact has consequences on the interaction between toroidal dipoles and other classes of dipoles, as the interferences between the different parts causes the cancellation of free energy at specific distance-frequencies, at which the free energy becomes equal to zero. The interaction between two toroidal moments →T1T→1 and →T2T→2 in free space is given by Hint∝ωRe{−i→T2⋅→E∗1(ω,R2)}.Hint ∝ ωRe{−iT→2⋅E→1*(ω, R2)} . Using Eq. (3), the interaction energy is calculated as

Toroidal field profile and the interaction energy between two toroidal dipoles.

(A) Electric-field configuration associated with a toroidal dipole moment positioned at the origin of the coordinate system. Another toroidal dipole moment positioned at the proximity of the original toroidal dipole moment can couple with it. (B) The interaction energy of two coupled toroidal moments versus the polar angles θ 12 and θ and (C) versus the wavelength and the distance between them. Positive and negative interaction energies produce the higher- and lower-energy states in a hybridization picture, respectively. The arrows show the orientation of toroidal moments. The color bar is identical for (B) and (C).

in which it is assumed that both multipoles oscillate at the same frequency and R 12 is the distance between two multipoles. The interaction energy between two multipoles oriented as →T1=T1ˆzT→1=T1 z^ and →T2=T2cos(θ)ˆz+T2sin(θ)ˆxT→2=T2cos(θ)z^+T2sin(θ)x^ positioned at a distance R 12<c/ω versus the polar angles θ 12 and θ is shown in Figure 5. In a hybridization picture, a negative interaction energy between the moments produces the lower-energy state of a system of coupled toroidal moments, whereas a positive interaction energy causes the higher-energy state. The interaction energy between two toroidal dipoles sustains an extremum at configurations depicted in Figure 5B; i.e. when the two toroidal dipoles are oriented either parallel or antiparallel in a transverse or longitudinal arrangement. This is very similar to the coupling between two electric dipoles [125]. Dissimilar to the case of coupled electric dipoles, however, the interaction energy between two toroidal dipoles can be quite different at distances R 12<c/ω and R 12>c/ω, even for a fixed arrangement and orientation of toroidal dipoles. It is already obvious from Eq. (4) that at R 12=c/ω, H int=0 (see Figure 5C). This fact causes switching of the eigenenergies for symmetric and antisymmetric coupling at distances ranging from R 12<c/ω to R 12>c/ω. It should also be mentioned that for localized toroidal moments this, switching might not be an obvious fact, as the critical distance R 12=c/ω is comparable to the wavelength, which is usually much larger than the scale of a toroidal dipole moment source. For a nonlocalized metamaterial-based toroidal moment coupling configuration which is comparable to the scale of the wavelength, this switching becomes better observable.

The radiation pattern associated to a toroidal dipole moment is given by P=(c/4π)R2ˆn⋅Re{→E(→r,t)×→B∗(→r,t)},P=(c/4π)R2 n^⋅Re{E→(r→, t)×B→*(r→, t)}, which, after some simple algebraic manipulations, is simplified to (ω6/4πc5)(∣∣∣→T∣∣∣2−(ˆn⋅→T)2).(ω6/4πc5)(|T→|2− (n^⋅T→)2). This is quite similar to the radiation pattern of an electric dipole moment [24]. Particularly, they share the same multipolarity and angular radiation pattern and show different frequency dependencies. This holds for the higher-order multipoles too and is very important when they coexist and compete. Interestingly, different contributions to the near-field profile of the toroidal moment cancel themselves in the far-field pattern, and only the last term in Eq. (3) contributes to the radiation pattern.

4 Excitation of toroidal moments

4.1 External light

Whereas the detection of natural spin-based toroidal moments of solid states needs complicated methods like X-ray gyrotropy and circular dichroism [98], [126], X-ray Compton scattering [127], nonreciprocal reflection [128], or second harmonic generation [53], the excitation and detection of dynamic metamaterial-based toroidal moments are apparently much simpler. This is because each element of the metamaterial configuration is designed in an intelligent way to (i) couple effectively with the incident light and (ii) sustain an effective ring-like orientation of magnetic moments as shown in Figure 1C, which can be arranged in arbitrary numbers of magnetic moments in the ring affecting directly the strength of toroidal moments.

Depending on the polarity of light and the spatial symmetry of the metamaterial structures, different strategies are used to excite toroidal moments in metamaterials. In the literature, external light, with either linear [129], radial [105], [130], or circular polarization [101], [109], has been used to illuminate the metamaterials normally, laterally, or obliquely (see Figure 6).

Figure 6:

Different excitation strategies for toroidal moments in metamaterials via polarization and incidence directions.

Dashed boxes denote no published result yet. Figure reproduced from (A) Ref. [111], Creative Commons CC-BY license; (B) Ref. [130], OSA; (C) Ref. [112], APS; (D) Ref. [101], APS; (E) Ref. [116], OSA.

In general, the polarization of the electric field of the incident light induces effective current loops in the structure, which generates magnetic dipoles. By carefully arranging the structure elements and the polarization direction of the incident light, a vortex of magnetic dipoles, i.e. a toroidal moment, is thus achieved. For instance, split-ring resonators usually are illuminated from a lateral direction.

4.2 Relativistic electrons

Relativistic electrons possess a unique spatial structure of the electromagnetic field attached to their vicinity. Distinct to far-field light sources, relativistic electrons act not only as a near-field excitation source, but also share certain overlapping excitation features with various optical excitation field geometries in an ultra-broad band frequency range, such as a radial and circular far-field [131], and a dipolar near-field (Figure 7) [19], [132]. Such combined electronic and optical excitations allow a comprehensive study of toroidal metamaterials.

Recently, electron energy loss spectroscopy (EELS) and energy-filtered transmission electron microscopy (EFTEM) with relativistic electrons have been used to study toroidal moments at high spatial resolution [132]. EELS and EFTEM use focused and parallel electron beams, respectively, which can interact with the optical modes of the sample. Due to this inelastic interaction, the electron loses energy, which can be detected using an electron spectrometer. The energy loss suffered by the electron is an indirect probe of the optical resonances of the sample [133], [134], [135], [136], [137].

Figure 7:

Electromagnetic-field components surrounding a relativistic electron. The excitation of relativistic electrons shares certain symmetric features with that of a radial and circular far-field, and of a dipolar near-field. Reprinted with permission from Ref. [132]. Copyright (2012) American Chemical Society.

5 Coupling of toroidal moments to other classes of moments

5.1 Formation and discovery of anapoles

Nonradiating field distributions and sources have attracted attention from time to time, mainly due to their connection with the discovery and characterization of elementary charged particles. A very prominent example of such a category is an electron, with the well-known Coulomb field associated with it, which even at a uniform velocity maintains its evanescent field profile. Classically, several current distributions were heuristically introduced with vanishing far-field radiation. An interesting work in this field, initiated by Devaney and Wolf [138], has considered the multipole expansion of the fields generated by time-harmonic current distributions inside an enclosed region specified by r<R, to conclude that the radiation multipole moments are dependent only on the transverse components of the spatial-harmonic current density distribution given by ˜→JT(K,ω)=(→K×˜→J(→K,ω))×→K/K2.J→˜T(K, ω)=(K→×J→˜(K→, ω))×K→/K2. This statement is valid only for those components for which →K=k0=ω/cK→=k0=ω/c [138]. In other words, a nonradiating source should have a vanishing ˜→JT(K=k0,ω)J→˜T(K=k0, ω) component [138], [139], [140]. However, the classification of multipole moments as provided by Devaney and Wolf excludes the toroidal moment class. For including toroidal moments, a complete analysis as discussed above is necessary.

An example of a nonradiating source is known as an anapole. An anapole is composed of a toroidal dipole moment and an electric dipole moment. As the radiation pattern associated to these two classes of moments are quite similar to each other, even in the near-field (see discussions in Section 3.4), it is possible to precisely tune the strength of these two moments by means of geometry [111], [141] (Figure 8A and B), in order to cancel the radiation pattern, even though the field components of a toroidal dipole moment being proportional to R −2 and R −3 will still contribute to the near-field profile. In fact, the anapole moment was first introduced theoretically by Zeldovich [142]. For quite a long time, however, discussions regarding the existence of such a moment were just theoretical. The first experimental evidence for the existence of an anapole in the cesium atomic nuclei has been published in 1997 by Wood et al. [143]. Among the applications of the anapole moment are its relation to dark matter in the universe [144], [145], enhancing light absorption by metamolecules with an anapole moment [146], and enhancing efficiency of higher harmonics generation [141], have been already discussed in the literature.

Figure 8:

The combination of a magnetic toroidal dipole moment and an electric dipole moment in a dielectric (A) nanorod [141] and (B) nanodisc can create an anapole moment [111]. Figure reproduced from (A) Ref. [141], OSA; (B) Ref. [111], Creative Commons CC-BY license.

5.2 Toroidal metamaterials

Despite the fact that different configurations of toroidal metamolecules have been so far intensively investigated for their intriguing possibilities to sustain toroidal moments, still the coupling between adjacent elements have not been thoroughly studied yet. In fact, due to the exotic near-field contribution, such a coupling and hybridization of toroidal moment can obey quite different selection rules in comparison with electric and magnetic dipoles. Moreover, even neglecting the coupling between adjacent elements, an arrangement of toroidal moments in a photonic crystal lattice leads to interesting new behaviors such as resonant transparency and Fano excitations within the effective-medium approximation [52], [54], [102].

6 Applications of toroidal moments

The formation of toroidal moments in a spatially confined hybrid electric-magnetic configuration leads to unusual properties, which opens a horizon of potential applications in data storing [147] and designing of low-loss metamaterials or metadevices [129], [148], [149], [150], such as ultrasensitive sensors and diagnostic tools [101], [151], polarization twister [109], near-field lasing [152] and lasing spacer [153].

For long-range toroidal moments in condensed-matter systems, their inherent ME is of technological interest since it provides an alternative and convenient way for manipulation of data in for example storage discs. Indeed, data storing at the nanoscale can be realized by switchable, ordered electric toroidal moments in low-dimensional ferroelectric structures [147], [154]. Moreover, the switching of toroidal moments requires no electrode contact at the nanoscale but is conveniently controlled by time-dependent magnetic fields. Additionally, it exhibits no noticeable cross-talk [154]. As for the case of LiCoPO4, the generation and manipulation of ferrotoroidic domains were implemented by simultaneously applied magnetic and electric fields, not solely by either of the fields alone [53]. It gives the material a promising potential to reject external disturbance in data storage, as an illustration. Moreover, the intrinsic linear ME of ferrotoroidic materials is anticipated to be a source of giant MEs via effective enhancement [87].

Benefiting from more confinement in real space, dynamic toroidal dipolar excitation in metamaterials usually achieves higher quality factors by carefully designing the geometry [129], [148], [149] and the chemical component [152]. It paves the road for applications as low-loss, high-Q cavities in metadevices [103], [104], [148], [149], [150] and laser spacer [153]. Through destructive or constructive interference between toroidal multipoles and electric or magnetic multipoles, astonishing electromagnetic phenomena can be achieved, such as manipulation from non- to super-radiating [130], [155], resonant transparency [54], [114], [156], extremely high Q-factors [104] and nanolasering in the near-field [152].

Instead of local toroidal excitations, a propagating toroidal excitation in free space, as a peculiar form of delivering an electromagnetic field, is intriguing. This concept was introduced in 1996 as focused doughnuts, a single-cycle, broad-bandwidth pulse with a spatially localized toroidal field configuration and longitudinal field components [157]. It was numerically shown that such pulses allow to excite a prominent toroidal dipole mode in a dielectric nanosphere [158]. The challenge of realizing such a focused doughnut lies in simultaneously controlling frequency and spatial dispersion over a wide bandwidth [24]

Another peculiar feature of toroidal moments is circular dichroism [101], [151]. It is seemingly challenging the common interpretation based on electric and magnetic quadrupoles and also might have a potential in diagnosing and sensing fields [101]. Moreover, an effective conversion between left-handed to right-handed circularly polarized waves at same frequency has been demonstrated via arrays of sun shape apertures [109]. It can be potentially applied as polarization twister at the microwave region.

Folks , just a thought & this is an original one from me - "If Mathematics or numbers are the language of the Universe , then Geometry is the design and layout of the Universe ."

I have been exploring this very question...
And it's a good question.... coming from a previous discussion...a person stated that geometry is not fundamental as it's a branch of maths....
For me, Numbers are the heart of geometry and geometry the art of the universe.