The Tortuous Geometry of the Flat Torus !

List members , I am sure you will find this subject interesting due to it's myriad implications for the "reality" we experience . @Echo_on , I am keen to know your views on this , given your research in 2D for "squaring the circle" , whereas this article is about something similar in 3D...

However , anyhow - even those who don't have interest in mathematics or geometry , should try reading this :-

The Tortuous Geometry of the Flat Torus

In 2012, mathematics has given birth to a new baby. And she is beautiful! Her parents, Vincent Borrelli, Saïd Jabrane, Francis Lazarus, Boris Thibert and Damien Rohmer, who formed the Hévéa project, have named her the first C1 isometric embedding of the flat square torus. Sexy, right? Have a look at the first video ever taken of her:

OK, her name is a bit long so I’ll just call her the Hévéa Torus.

She is beautiful indeed… But what is she?

Amazingly, she was first imagined a century and a half ago, by her ancestors Carl Friedrich Gauss and Bernhard Riemann, in 1854. But it took a century for her great grandfather John Nash to actually prove that she could one day be conceived, although he didn’t specify how. Another 20 years later, her grandfather Benoît Mandelbrot laid the foundations of a new kind of geometry which hinted at an actual possible conception. But, weirdly enough, it took the advent of computers to finally procreate her, after 5 long years of top-level mathematics research! And yet, it all started with a very simple problem…

Bending the Flat Square Torus

These days, I’m spending way too much time playing the game on the right called Netwalk where a network needs to be built. The game is played within a square. One tricky aspect is that whenever you get out of the square on the right, you reappear on the left. Like in PACMAN, or in these awesome games.

What do these games have to do with the video above?

Amazingly, the network universe and the Hévéa Torus are geometrically identical!

You’re kidding, right? I mean, on the right we have a square, and the Hévéa Torus is… well, a weird twisted shape!

You’re right. From our perspective, these are very different. But, from the perspective of some being stuck within these 2D worlds, there is absolutely no difference! The Netwalk world and the Hévéa Torus are intrinsically mathematically (nearly) identical.

OK… I admit, there’s a slight difference in terms of accelerations in these geometries. We’ll get to that…

Humm… I have the biggest troubles imagining myself living in the Hévéa Torus…

You’re not the only one! It took two of the greatest giants of mathematics to figure out what it meant to live within a torus. The first one is Carl Friedrich Gauss, also known as the Prince of mathematics, who famously proved the Theorema Egregium, which you can learn more about by reading Scott’s article on non-Euclidean geometry. But, more importantly, it was Bernhard Riemann, Gauss’ disciple, who unlocked the wider geometry of so-called manifolds by providing a powerful new picture of what geometry could be.

What is this Riemann’s picture of geometry you’re talking about?

In short, a Riemannian manifold is a space, such that each local neighborhood of a point of that space looks flat. The torus of the video is an example of a 2-dimension manifold, also known as surface. Around each point, if you zoom sufficiently, then your surface will look like a 2-dimensional sheet of paper. And, on this sheet of paper, lengths and angles are the same as on actual sheets of paper!

More precisely, a scalar product must be defined on each local neighborhood. In the middle of the image above, we have a 2D section of a 6D Calabi-Yau manifold, which is widely studied by string theorists. Find out more about Riemannian geometry with my article on the spacetime of general relativity.

Now, involving lengths and angles makes things complicated… so let’s start with a simpler kind of manifold: The topological torus.

The topological torus?

Topology is the art of forgetting the geometrical concepts of angles and lengths. This means that topology allows stretching sheets of paper without effectively changing it.

Can you give an example?

Sure! Topologically, it is straightforward to transform the Netwalk square into a torus. We merely have to glue together opposite sides. This is what’s done below:

I’ve tried to make a torus with the Netwalk square but I miserably failed after hours of ridiculous attempts… Sorry for that!

This procedure of gluing is also described by this beautiful video by Geometric Animations:

Find out more about these gluing operations with my article on Poincaré conjecture. You can also learn more with my article on topology.

Nash’s Isometric Embedding

The trouble is that, as you can feel it while watching this video, it seems impossible to glue opposite sides without stretching the square we started with.

Is that much harder?

It is! This no-stretching requirement corresponds to an isometry. To get a sense of the constraint that isometry represents, check this awesome video by Colm Kelleher on TedEd, which explains how isometry can prevent the tip of your pizza from falling down:

As the video explains it, it’s impossible to bend a sheet of paper into a sphere, or into a potato chip. This is because of the curvatures of these shapes. And the classical smooth donut-like torus is highly curved too, so it’s impossible to bend the flat Netwalk world into that classical torus. An isometric bending of the Netwalk world requires to transform it into an non-curved torus!

Hummm… I’m now beginning to think it’s impossible to bend a square into a torus!

You’re not the only one! 60 years ago, most mathematicians even thought it was impossible! But no one could prove it. In fact, when deeply annoyed by his young, brilliant and pretentious colleague, MIT mathematician Ambrose Warren rashly retorted to him: “If you’re so good, why don’t you solve the [100-year-old isometric] embedding problem for manifolds“, which the bending of the Netwalk world with no stretching is an example of. Warren was hoping it would thereby keep his colleague busy for the rest of his life. But shortly later, that colleague was claiming he had solved it!

Mathematically, an embedding is an injective map of a manifold into some larger space, typically Rn, where n is at strictly greater than the dimension of the manifold. This map needs to be an immersion, which means that its derivatives (which are linear applications) must always be injective.

Who was that colleague?

He was the founder of game theory, future winner of a Nobel prize in economics and future hero of the movie A Beautiful Mind… John Forbes Nash.

And he immediately solved a 100-year-old problem?

Not really… As I said, the young Nash was a pretentious douchebag. After all, he had just completed a revolutionary 28-page PhD thesis, and he was visibly smarter than most of his colleagues. So, he figured out that only a prestigious problem was worthy of his time. One that would mark History. So, to test whether the isometric embedding problem was of that kind, Nash looked at the reactions of his colleagues as he told them he had cracked it. And surely enough, it seemed that solving the isometric embedding would mark History. So, Nash decided to give it a try.

Did Nash actually solve it?

Amazingly, he did! Nash might have been showy, he’s also definitely a world-class mathematician.

So what’s the answer? Can a square be bent into a torus?

The answer is yes. It can. Sort of… I’ve explained it all in this Science4All video:

I uploaded the above video the day before Nash sadly passed away in a car accident. Given how little praises I gave in the video above, I made another tribute video to honour this beautiful mind.

So, as it turned out, Nash proved that we could, provided the bent torus had an infinite number of points on which acceleration could not be defined!

What does that mean?

Here’s an illustration of what that means. Imagine a skater rolling down a slope:

In the first case, there is a huge discontinuity in the slope at the green point. In some sense, the slope cannot be defined at that point. In the third case, the slope is so smooth that the skater doesn’t feel anything special at this point. Finally, and most interestingly, in the second case, at the green point, the skater will suddenly feel a force acting on him which was not here before. Nash proved that this second case must happen at an infinite amount of times on the bent torus.

Mathematically, this second case corresponds to the green point having no second derivative. You can learn about the basics of derivatives in my article on differential calculus.

Nash’s claim means that Newton’s laws of mechanics would have no sense in the bent torus on an infinite amount of points, kind of like general relativity no longer makes sense at singularities better known as black holes!

Waw! That sounds sick! And hard to imagine!

I know! In fact, it was so sick that Nash couldn’t provide an example of a bent square torus! His proof was not constructive. He merely showed that there must be a way to bend a square into a torus, but didn’t say how…

Mandelbrot’s Fractals

Weirdly enough, it’s much easier to imagine a bent torus for which acceleration is undefinable everywhere.

Really? That sounds so weird…

To understand what that could mean, let’s look at some other sick geometrical objects: Non-differentiable continuous curves.

Do you mean curves which are continuous everywhere but differentiable nowhere? Is that even possible?

First examples of such curves were given in the 19th century. They were called monsters. I guess that, Historically, one of the first examples of a monster was that of the function ∑2−ncos(4nx). But the more popular example nowadays is the Koch snowflake, as beautifully explained in the awesome show Fractals – The Hidden Dimension by NOVA. Or, even better, the curve formed by the British coastline:

Oh! So this is what fractals are: Non-differentiable continuous curves!

Exactly! And it took another genius to unveil the magic of fractal geometry. This genius is Benoît Mandelbrot.

Learn more with Thomas’ article on fractals.

Curiously though, Mandelbrot’s ideas weren’t popular at first. His first papers on this new kind of geometry were so unusual that the mathematical community rejected them. Referees claimed that the papers, despite displaying pretty images, did not contain actual mathematics. Because the community was not ready for his new ideas, Mandelbrot decided to write a book on his own. This best-seller was so amazing that it quickly became iconic, not just for young mathematicians, but in popular culture as well!

Now, the thing with fractals is that it’s very easy to make a shape of any length. For instance, you can draw a “curve” that looks exactly like a circle, but actually has a length of 4. This is what Vihart did in this awesome video:

So, following Mandelbrot’s (and Vihart’s) ideas, it’s not too hard to imagine a torus whose lengths perfectly match that of the Netwalk world. In other words, through a fractal process, it’s (at least theoretically) possible to describe a simple procedure to fold the square of the Netwalk world into an origami torus-like “zigfinihedron” which glues opposite sides.

So why did it take 5 years to actually construct it?

Look carefully… The Hévéa Torus is not a zigfinihedron! What the Hévéa team was searching for was a smoother bending of the torus, which, although had no second derivatives, still had continuous tangent planes at all points! And the trouble of fractals is that they are precisely rough objects with no such continuous tangent planes…

So why did you bring that up?

Somehow we need a trade off, between the smoothness of classical surfaces and the roughness of fractals. This trade off is what the Hévéa team called smooth fractals.

The Hévéa Torus

The idea of smooth fractals isn’t far from what was hinted at by John Nash. His embedding theorem stated that if a manifold could be topologically embedded in a higher dimension space, as we have done it for the Netwalk world, then this embedding could be smoothly corrugated like Vihart’s zigfinigons to end up isometric, while having tangent planes at all points. What’s more, Nash proved that the amplitudes of the corrugations could be as small as we wanted.

Technically, Nash still required a C1 continuity, which means that tangent spaces must vary continuously. Moreover, the initial embedding needs to be a shrinking of the manifold. Furthermore, Nash only proved this result, provided that the dimension of the manifold was 2 smaller than the dimension of the space it was embedded in (and it would thus not work for our torus). The refinement to all embeddings is due to Nicolaas Kuiper, and is known as the Nash-Kuiper theorem. Finally, Nash also proved a Ck isometric embedding theorem, which says that we can still embed a manifold even with greater continuities, provided the higher space we embed our manifold in is of much greater dimension.

So, following Nash and Mandelbrot’s ideas, the Hévéa team has added several layers of corrugation to a classical torus embedding. This is what’s illustrated below, where the 3 first layers have been added sequentially:

All the images of this section are taken from the Hevea project webpage.

Let’s have a closer look!

Just like Vihart’s drawings, the Hévéa Torus is theoretically obtained by adding an infinite amount of such corrugations. But, computationally, the Hévéa team merely added 5 layers, as any greater layer would in fact be imperceptible given the resolution of the image.

But, is the Hévéa Torus really a Netwalk world with no stretching?

That’s hard to verify… but yes. Below is an image which displays the correspondence between lines in the Netwalk world and lines in the Hévéa Torus:

Amazingly, the black and green loops actually have the same lengths in both figures. Granted, it’s a bit less obvious on the Hévéa Torus, but that’s because the black loop has a more fractal-like structure, which makes it longer than it seems to be!

Wait… Is the Hévéa Torus really smooth?

Yes! The key to obtain the smoothness of our corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their “wavelengths”. By opposition, Vihart’s drawings had rather one-to-one ratios. For instance, at 3:50, she divided amplitudes of corrugations of her curves by 2 while she made twice many of them. Instead, by carefully adjusting the faster decrease of amplitudes of successive corrugations, the Hévéa team managed to guarantee the C1 continuity of the Hévéa Torus!

I’m not sure this convinces me…

One way to see that is to take the shape of the black loop of the figure above (which corresponds to a meridian), in the 3D space the Hévéa Torus is embedded in. Let’s compare that to a Koch snowflake.

Importantly, as opposed to the non-differentiable Koch snowflake, the meridian of the Hévéa Torus is still smooth enough to have (continuous) tangents at all points.

Let’s Conclude

Instead of recapitulating, I’ll let the great James Grime sum up all we’ve discussed here:

Now, the result of the Hévéa project is a spectacular achievement of over 160 years of mathematical pondering. Obviously, this article only mentions the few mathematicians who made the most stunning breakthroughs in this investigation. But, unfortunately, it also dramatically fails to account for the hundreds of other mathematicians who have shaped and reshaped our intuitions of curves and surfaces, in a deeper but less obvious fashion. For me, this amazing silent build up is the source of the magic of mathematics. Think about it. Thanks to many unsung mathematicians, we now have in front of our eyes an object that the great Gauss, Riemann, Nash and Mandelbrot couldn’t even imagine, even though they had been searching for it all along. I hope you feel privileged and dismayed by the following precious images of the Hévéa Torus

I should say that even though combining Nash and Mandelbrot’s ideas sound reasonably doable, it is actually a huge and difficult endeavor. Once again, the Hévéa team had to spend 5 years to come up with the Hévéa Torus, after several unsuccessful attempts. It was a tedious and tricky work, which involved very modern mathematics, including and especially Mikhaïl Gromov’s homotopy principle. This gives me one last occasion to stress how big an achievement the conception of the Hévéa Torus is. Congrats to the Hévéa team!

Let me leave you with my favorite image of the Hévéa Torus, taken from its interior, with light coming from the opposite side… Astonishing!

One last thought… Can you believe that cutting through the Hévéa Torus twice yields a square?

Regards

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Thanks @sidharthabahadur

Funny Sidhartha you should post this now!

As yesterday I was elaborating my topology of the Unity framework.

What I can say right now is that the torus is an affect - which is why it is observable to us. Its topology is the digital aspect, so what we observe in reality is the rolled up cylinder hole. And it's the numbers that create the observation of what the hole is.

M

Great to hear @Echo_on :)) This is such a fascinating area of research !

Regards

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@sidharthabahadur

UNITY number grids.pdf (44.8 KB)

Please download the attachment and study the point reflection of the number grid attachment and it will show you the answers. for this example the number 7 and 3 are factored into the unity framework. I have grouped specific numbers and colour coded them. They all share a manifold symmetry and an inverse duality. Numbers do really create games for a reason!

m

@Echo_on , on a different note , the image of that corrugated torus looks so much like a coiled up snake , doesn't it ? I am recalling the ancient Ouroboros imagery of a snake swallowing it's own tail...hmm !

Were our ancestors trying to explain this same concept using the symbolism of a coiled up snake - ultimately how a dynamic torus rolls back into itself...inside out , outside in ?

Regards

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@sidharthabahadur

As the number reveals I would say so the symbols. The more authentic data the more resolution and details we observe.

The ancients knew what we are now finding out here. How we interpret it all the more important.

But you are right, the snake could represent a coil or energy or spring etc. What goes round comes round...

M

@Echo_on , my intuition is strongly suggesting that this corrugated torus is actually the shape of our Universe which was symbolised in olden times by a coiled snake swallowing it's own tail...looking at it another way - even the Flat Earth concept can be converted into a 3 dimensional corrugated torus...the interior view of such a torus actually resembles a Tokamak reactor , a Dyson sphere , a car tyre or indeed even a Hollow planet !!

If we correlate with the mathematical solution for "squaring a circle" , I am reasonably sure , some interesting insight would emerge for converting that 2D circle to a 3 D Corrugated Torus...hmm ! What do you think ?

Regards

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@sidharthabahadur Is it possible?

(Tokamak and Torus is a Reactor! Or. reactor (n.)
"one that reacts," 1835, agent noun in Latin form from react. By 1915 in electricity as "coil or other piece of equipment which provides reactance in a circuit;" the nuclear sense is attested from 1945.

Sidhartha - today I learnt something and I apologise (sometimes I am "in the zone" and rash with comments. So please accept my sometimes harsh analysis! lol.

I know here on the Mindreach the Torus is a special thing and so I threw out my conceptions and looked at the sound unity framework to incorporate it into a visual model. Well, well, well!

It was something I missed previously but not today. So what is that is special about the Torus?

All platonic geometry creates below frequency <1 Not the Torus!
It can create at frequency above >1 at a higher the frequency to 1000hz. What this shows in my work is that it can create in our observable universe and is the only thing that can. It is probably why we all are fascinated by it. I tried to copy some videos directly here but the limit again of 1mb is extremely limiting.
So I will update a link tomorrow for you all to see.

M

@Echo_on , you have a very intriguing point of view...in one of your earlier posts you had also talked about the symmetry of space...which makes me think there must be some hidden equivalence between a square , a circle and a torus...within that correlation must be concealed the secrets of creation .

The Torus , as we've often discussed on this forum is a very fundamental shape in our Universe...besides being Holographic by design , it is in line with the Cosmic principle of "as above , so below" .

Therefore , I will eagerly look foward to your further thoughts on how sound and energy might be generated from a torus ! Maybe , an energy Vortex maybe a more appropriate term perhaps , than just a geometrical shape ?

Regards

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Hazrat Inayat Khan Quote: “He who knows the secret of sound, knows the mystery of the whole universe.”

I will look forward to post the Torus videos asap. I think you will be very excited..

M

What an "ensoundening" thought , @Echo_on :))

Please excuse me - I am having some fun by coining this new term , now that we know sound is more fundamental to the creation of this Universe , than even light !!

**Our two eyes are capable of seeing the light , but it is only our "third eye" , the pineal gland (if trained through years of intense meditation) , where our "sixth sense" resides & and not even our ear(s) , that is capable of "hearing" transcendental sound...

Man has forgotten how to use this spiritual "muscle" , or pineal gland - which is the root cause of so much turmoil in the world !

Regards

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@sidharthabahadur

Here you go guys! Oscillation from the Torus....Hope it gives some insight.

It is very difficult to show its true nature fully as the scaling, volume, energy, line thickness, intensities (ie variables) are constantly changing. So I have tried to average out the intensity etc so you all can see as much as possible - call it a preview!

I am in the process of trying out different software to record the data so am sure I will have much higher quality files for you guys to view in the coming months.

m

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Fantastic !! Wow ! What a show @Echo_on - mind boggling visualisations...during the 7:48 minutes of this video , my mind was able to "freeze frames" and spot all of the following patterns/designs :-

  1. Hollow Planets
  2. Hexagon on Saturn's North Polar Opening
  3. Swastika
  4. Yin Yang symbol
  5. The Black Sun
  6. A Supernova explosion
  7. The Toroidal Universe itself
  8. The Solar system
  9. Galaxies
  10. Atoms

Regards

1 Like

@sidharthabahadur Brilliant, I was thinking I was the only one! Lol! It is fascinating. Hopefully I can incorporate the sound which is missing currently to get a more concentrated stimulation of the pineal.

The video shows the Torus Oscillation from 0hz to 1000hz.

M

Yes , thanks @Echo_on - right through the video I kept getting the sensation one gets from watching a spiralling vortex , which entrains the consciousness...At one stage , I could even notice the outline of a nautilus shell with it's Fibonacci spiral (mathematical phi)...!

***There are notably some visuals that seemingly mimic the bioluminescence observed in deep sea creatures .

If I actually get down to attaching all the screenshots for each of those "frozen frames" , they might number more than 20 :))

Regards

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@Echo_on I'll see what we can do about the upload limits, but hosting cost vs allowing larger uploads and other resources have presented some cautious limits. We've contemplated donations in some fashion to help things further, but I'm not yet sure if that would be well received or how to properly structure it, etc.

As far as this video/model goes that you posted: Could you give us another explanation to clarify your method and means of generation in context of the previous discussion? It seems @sidharthabahadur may be intuiting this a little more ("getting it") in a seemingly more esoteric manner, but I'm a bit of a lower level thinker (engineer, pardon me) and seeing things like this sans more explanatory immediately connected context creates a void in appreciation and comprehension in my mind... unless I understand the whys and hows a little better.

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Thanks @Soretna for looking into the limits but as frustrating as it is I am happy to post a few links (no problem).

You mentioned @sidharthabahadur maybe getting it! and you need more explanatory explanations to comprehend. Not sure you remember or read any of my posts from Unity Framework. I mentioned that some would get it more than others and to stay with it! We all learn at different levels and stages, and some of us more technical than others are more visual etc.

I would very much also like to give you the instructions but this is also not the way to understand (feel) to 'Get It" It comes from within our Core! It is all from within so I fully comprehend where you are....

After the weekend I will post the mathematical framework to the Torus Video for all to see.( a direct relation to my Unity Work.) This is very exciting time to wish to understand.

Muchos gratitude

m

Well said , @Echo_on , Sacred Geometry does hold the key to many secrets of nature...there is a need to integrate "modern" geometry that we learnt in school & college , with the ancient or sacred geometry from the past . Such a unification will yield astonishing new insights .

Regards

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@Echo_on , for those members of our group who need a more visually oriented explanation , the necessary screenshots taken at various points of the video might be a good starting point...a list of all the important timestamps perhaps ?

Regards

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Good point, I will fit it in with my next upload @sidharthabahadur.

M