Are planets and stars actually toroidal , not spherical in shape ?

List members , we have heard that Earth is an oblate spheroid , i.e. it is bulging at the equator and flattened at the Poles . The Sun and all other planets/planetoids of our Solar system and even exoplanets have been found to be oblate spheroids .

Why is it that nobody has EVER found a perfectly spherical planet or star ?? Could it be that stars and planets are actually compressed "double torus" shaped merely appearing as somewhat spherical ? Some of the unexplained aspects of stars and planets can actually be better understood by imagining them as toroidal in shape !

http://www.aleph.se/andart/archives/2014/02/torusearth.html

Can toroid planets exist?

Torus–Earth

One question at Io9 that came up when they published my Double Earth analysis was "What about a toroidal Earth?" This is by no means a new question, and there has been some lengthy discussions online and earlier modelling. But being a do-it-yourself person I decided to try to analyze it on my own.

Can toroid planets exist?

It is not obvious that a toroid planet is stable.

For all practical purposes planets are liquid blobs with no surface tension: the strength of rock is nothing compared to the weight of a planet. Their surfaces will be equipotential surfaces of gravity plus centrifugal potential. If they were not, there would be some spots that could reduce their energy by flowing to a lower potential. Another obvious fact is that there exists an upper rotation rate beyond which the planet falls apart: the centrifugal force at the equator becomes larger than gravity and material starts to flow into space.

The equilibrium shapes of self-gravitating rotating ellipsoidal planets have been extensively analyzed. Newton started it (leading to some early heroic expeditions to ascertain the true shape of Earth), Maclaurin refined it, Jacobi discovered that for high rotation rates ellipsoids with unequal axes were more stable than the oblate ellipsoids of Maclaurin. Chandrasekar has a nice history of the field. Since then computers have become available, and analytical and numerical calculations of more complex or the relativistic case have been done.

Similarly, equilibrium states of self-gravitating toroid shapes have been examined by Poincare, Kowalewsky and Dyson (Dyson 1893, Dyson 1893b). Indeed, one can at least in theory spin up an ellipsoidal planet into a ring, although there is plenty of potential for complex wobbles that destabilizes the whole system and it looks like there is a “jump” to the ring state. The ring may itself be unstable, in particular to a “bead” instability where more and more mass accumulates at some meridians than others, leading to breakup into two or more orbiting blobs. Dyson analysed this case and found it relevant when the major radius / minor radius > 3 – thin hoops are unstable. There is also a lower rotation rate where the ring become unstable to tidal forces and implodes into a “hamburger” or ellipsoid. So the total mass and angular momentum needs to be in the right region from the start.

It looks like a toroid planet is not forbidden by the laws of physics. It is just darn unlikely to ever form naturally, and likely will go unstable over geological timescales because of outside disturbances. So if we decide to assume it just is there, perhaps due to an advanced civilization with more aesthetics than sanity, what are its properties?

Directions

I will call the two circles along the plane of rotation the equators (the inner and outer). When it does not matter which one I talk about I will just call it “the equator”. As for the poles, they are the circles furthest away from the equatorial plane.

Hubward is towards the rotation axis, rimward is away from it. Planewards is towards the equatorial plane. North is towards the closest part of the North Pole circle, south towards the closest part of the South Pole circle.

Toroid gravity

How does gravity work on a toroid planet?

The case of a very large main radius torus is essentially a cylindrical planet. In this case the gravitational force falls off as 1/r, where r is the distance from the axis. The total force on any section will be proportional to the total mass (proportional to R, the major radius) and the gravitational force (proportional to 1/R), so the overall force will be constant as we increase R. Adding some rotation will balance it. The surface gravity is 2G rho/r, where rho is the mass per unit length. So as long as the surface gravity is big enough (by having a small r) this will overcome the centrifugal acceleration and stuff will indeed stay down. But things are much harder to guess for small radius torii.

I decided to use a Monte Carlo method to estimate the equilibrium shape. Given the total planetary mass and angular momentum, I start out by distributing a number of massive but infinitely thin rings (with the potential borrowed from this physics exercise - it is a good thing electric and gravitational potentials look the same in classical physics). I calculate their joint potential and added a centrifugal potential. This allows me to approximate equipotential surfaces and "fill" the potential near the center of the torus with more and more rings until their mass correspond to the planetary mass. I recalculate the angular speed based on the new mass distribution. Then I repeat the process until the planet either flies apart, implodes into a ball or enough iterations go by. This is not the most elegant way of doing it (the literature uses series expansions in toridal harmonics), but it works for me.

The main result is that toroid planets look feasible for sufficiently large enough angular momentum and mass. The cross-section is neither circular nor elliptic but rather egg-shaped, with a slightly sharper inside curvature than on the outside.

[ Why doesn't the planet get squashed into a plane disk? The rotational pull tries to flatten the planet, but it must act against the local gravity field which tries to turn it into a ball (or cylinder).]

While these planets are stable in my simulation, the range of feasible values is not huge: most combinations of mass and angular momentum are unstable. And I have not examined the tricky issue of bead instability.

I will look at a chubby toroid of one Earth mass and a small central hole (“Donut”), and a wider hoop-like toroid with 6 Earth masses but more earth-like gravity (“Hoop”).

Donut

Figure 1: Local gravitational acceleration (m/s2) around Donut, as experienced by a co-rotating object.

Donut has a hubward/interior equator 1,305 km from the center, and a rimward/exterior equator 10,633 km away. The equatorial diameter is 9,328 km.

The planet extends 1,953 km from the equatorial plane, with a north-south diameter of 3,906 km. The ratio of the diameters is 2.4.

The north-south circumference is 21,587 km (0.54 times Earth), while the east-west circumference is 66,809 km (1.7 of Earth).

The total area 8.2108 km2, 1.6 times Earth. The total volume is 1.11012 km3, within 1% of Earth (after all, Donut was selected as a roughly one Earth mass world). The Volume/area = 1300, 61% of Earth: there is more surface per unit of volume.

One day is 2.84 hours long.

Hoop


Figure 2: Local gravitational acceleration around Hoop, as experienced by a co-rotating object.

Hoop has a hubward/interior equator 8,633 km from the center, and a rimward/exterior equator 19,937 km away. The equatorial diameter is 11,304 km.

The planet extends 4,070 km from the equatorial plane, with a north-south diameter of 8,141 km. The cross-section has roughly the 4:3 ratio of an old monitor.
The center of mass circle is 14,294 km from the center.

The north-south circumference is 30,794 km (0.77 of Earth) while the east-west circumference is 125,270 km (3.1 times Earth). The total area is 2.5109 km2, 4.9 times Earth, and the total volume 6.51012 km3, 6 times Earth. The volume/area = 1500, 70% of Earth.

The day is 3.53 hours.

Environment

So, what is life on these torus-Earths?

Gravity

The surface gravity depends on location. It is weakest along the interior and exterior equator, while strongest slightly hubward from the "poles". This can be a fairly major difference.

Donut


Figure 3: Surface gravity (m/s2) of Donut.

Donut has just below 0.3 G gravitation along the equators and 0.65 G along the poles. The escape velocity is not too different from Earth, 11.4 km/s.

The geosynchronous orbit of Donut is very close to the outer equator, less than 2,000 km up. A satellite orbiting there will stay over one spot, but unlike on Earth it will not be able to cover a hemisphere with transmissions, just a smaller region.

On the other hand, the circumferential velocity at the equator is 6.5 km/s, making launches easier. Launching east a rocket needs just 4.9 km/s velocity to escape.

There is a central unstable Lagrange point at the middle of the hole. A satellite will be attracted to the equatorial plane, but any deviation outwards will be amplified.

Hoop


Figure 4: Surface gravity (m/s2) of Hoop.

Hoop has 1.1 G gravity along the poles but just 0.75 G along the rimward equator. The hubward equator has slightly higher gravity, 0.81 G.

Escape velocity is 19 km/s (remember, the planet weighs in at 6 earth masses). Rimward equator velocity is 9.9 km/s – a rocket will need to provide 10 km/s to escape if it launches eastward.

Note again that having a low gravity equator and high gravity poles does not mean stuff will roll or drift towards the poles: as mentioned before, the surface is an equipotential surface, so gravity (plus the centrifugal correction) is always perpendicular to it.

But an air mass flowing towards the pole will be squeezed together. In fact, the different gravities will create horizontal pressure differences that are going to interact with temperature differences to set up jet streams in nontrivial ways.

Light

First, the nights and days of these worlds are very short. There is not much time for the environment to cool down or heat up during the diurnal cycle. What really matters is how much light they get over longer periods like seasons. Assuming these worlds orbit at an Earth-like distance from a Sun-like star, these are long enough to matter.

[If the torus-worlds orbited closer, tidal forces would really start to bite and before long the planets would become unstable. Since luminosity grows roughly as the fourth power of star mass and the life zone radius scales as the square root of luminosity, in the life zone the experienced tidal forces scale as M/(√(M4))3=1/M5. That is, bright stars have far less tidal effect on habitable planets: maybe Donut and Hoop better orbit some blue-white F star rather than a G star like the sun to be really safe. ]

Torus-shaped worlds have an outer rim that is not too different from a normal ellipsoidal planet. Days occur with a sunrise at the eastern horizon and a sunset at the western horizon. The sun moves along a great circle that slowly shifts north and south over the year, giving seasons. However, on the interior side things are different. Here other parts of the planet can shadow the sun: to a first approximation we should expect far less solar energy.

We can look at three different cases: zero axial tilt, 23 degrees (like Earth) and 45 degrees.

Zero tilt

For zero tilt the hubward side will never get any sunlight: the sun is always hidden below the horizon or by the arc of the world. At the poles the sun is moving just along the horizon, and slightly inwards there will be a perennial dawn/dusk. The temperature difference will be big, with the interior at subarctic temperatures: this is not entirely different from a tidally locked world, and we should expect water (and maybe carbon dioxide) to condense permanently here. The end result would be an arid (but perhaps not super-hot) outer equator, possibly habitable twilight polar regions, and an iced-over interior.

23 degree tilt


Figure 5: Seasons on Donut during spring, summer, autumn and winter.

For a terrestrial 23 degree tilt spring and autumn will be like the zero tilt case: light along the equator, dark inside the hole. But during summer and winter the sun has a chance to shine past the rim and onto the opposite side of the hole. Also, there will be large regions with midnight sun or perpetual night in summer and winter, respectively. On Earth the Polar Regions are small, but here they are at the very least long contiguous circles.

The spring dawns and autumn twilights on the hubward side would have some amazing deep colors, since the sun would be rising past the atmosphere of the other side (already pre-dawned or pre-twilighted, you could say). This would be added to the local atmospheric optics, producing some very deep reds and color gradients. Just before or after sunrise/sunset parts of the corona would also be visible.

These sights would be more impressive if they weren't so brief. On Earth, the sun moves close to 15° per hour: at its fastest, the sun moves one diameter in 2.1 minutes. On Donut solar motion is 127° and on Hoop 102°: a sunrise takes 15 or 19 seconds, respectively. Coming in at a slanted angle and the delaying effects of atmospheric refraction would prolong things a bit, but to an Earthling it would still look very brief.

Standing on the hubward surface looking up, the other side will be about 20 degrees across on Hoop and 30 degrees on Donut – an enormous arc across the sky.

[Why is Donut not much wider? Donut is very flat, so the world is seen very foreshortened in the sky. Incidentally, this means that when sunlight refracts through the atmosphere on the other side to hit the hubward side during a dawn or twilight it will be far deeper red than on Hoop.]

On the inside, having lit parts of the other side would light things up like moonlight. But the total area could potentially be much larger, making for some very bright (if still nightly) nights. For Hoop, this is potentially 16,000 times stronger than Earth moonlight (8000 lux) when the entire opposite side is lit (assuming an Earthlike albedo), making a night as bright as an overcast day. On Donut this reaches low daylight levels (12000 lux). However, this is the “full opposing side” situation: near the equinoxes only a thin sliver is visible.


Figure 6: Averaged insolation over a day on Donut during spring, summer, autumn and winter for the 23 degree case.

In the case of Donut, the rather flat surface means that the northern or southern hemisphere will also catch a lot of sunlight: the total heating on the planet is larger during these seasons than in spring and autumn, unlike on Earth where it is constant since the receiving area stays constant. There are also slightly nontrivial effects due to the angle between the surface and the sunlight, making the temperate zones get slightly less energy than the Polar Regions and tropics.

The rimward tropics have a fairly constant inflow of solar energy. As we go towards the poles seasonality becomes stronger: at the tropics there is more energy coming in during summer than ever happens at the equator. But the winters are of course equally darker. At the poles and beyond on the peak-gravity hubward side there is sun for half a year followed by polar night. Here the climate truly swings: the rimward tropics at least have brief 1.5 hour nights, but here they last 6 months. Finally, close to the hubward equator in the hole day and night return even in winter (plus extra light reflected from the other side), making it a bit more temperate


Figure 7: Averaged insolation during different seasons on Donut, as a function of latitude in the 23 degree case. 0 denotes the rimwards equator, 90 the north pole, 180 the hubward equator in the hole, 270 the south pole.

The rather big difference in energy deposited at the sunlit summer side of the hole and the dark winter side of the hole will tend to drive some strong weather – but as we will see, due to the other peculiarities of these worlds evening out the energy differences is harder than on Earth.

Overall, the total energy deposited is 2.5 times higher in the rimward equatorial area than in the temperate and polar areas, and the inside of the hole has about a fourth less energy than the surroundings.


Figure 8: Energy received across a year for different latitudes on Donut.

Hoop has less self-shadowing. More importantly, it is not as flattened as Donut.


Figure 9: Average insolation during a day on Hoop, 23 degree case.


Figure 10: Averaged insolation during different seasons on Hoop, as a function of latitude in the 23 degree case. 0 denotes the rimwards equator, 90 the north pole, 180 the hubward equator in the hole, 270 the south pole.

The seasons at first look like what one would expect. A spring and autumn where the hubward regions are in shadow, summers and winters where one polar circle gets a lot of sunlight and the other far less while the hubward regions get light. Note that this produces a seasonal cycle in the hubward area that is at double frequency of the rimward regions (this is true for Donut too): the warm weather happens in “July” and “January”.


Figure 11: Energy received across a year for different latitudes on Hoop.

Somewhat non-intuitively compared to Donut, here the hubward equator does get more sunlight across the year than the Polar Regions . We can hence expect the climate to be a bit like on Earth, with colder Polar Regions and warmer equatorial regions. The rimward equator still gets 60% more energy, though.

45 degree tilt

Perhaps the most surprising thing is that for high enough axial tilt we get four cold zones and four warm!

The easiest way of understanding this is to consider a spherical planet with 90 degree axial tilt like Uranus. For half of the year the North Pole is turned towards the sun and most of the hemisphere has constant daylight. As equinox approaches the axis points sideways, so the planet gets evenly irradiated. The end result is that the poles get more energy than the equator. On a torus world the same dynamics holds true, but now the Polar Regions are circular too.


Figure 12: Energy received across a year for different latitudes on Hoop in the 45 degree case.

For Hoop the difference is not enormous, about 10% in total insolation. The rimward equator is mildly hotter than the Polar Regions and the hubward equator.

Donut slightly larger differences but in practice most of the surface is dominated by the mildly warm polar regions. The rimward equator is only slightly warmer than the cooler rimward temperate areas.

Geosphere

The surface area is larger than on Earth, and the volume/area ratio is smaller (For Donut the ratio is 1,300 km, for Hoop 1,500 km, for Earth 2,124 km). One might hence suspect that more thermal energy is leaking out, reducing volcanism and plate tectonics. However, even a small amount of tidal heating due to influences from the sun might release plenty of energy stored in angular momentum. In the case of Hoop there are also 6 times more radioisotopes inside the planet than on Earth but only 5 times more surface area.

Continental drift would be affected by the different inner and outer radii. A circle r km inwards from a circle of radius R will be just 2pir km shorter, and the relative change will be r/R. So for Hoop a continental plate drifting from the outer equator across a pole to the inner equator will have to shrink to 43% of its original width to fit. On Donut the effect is much bigger: it becomes 12% of its original width! Hence continental plates moving hubwards on the inside will tend to experience folding, while plates moving rimwards on the inside will experience rifting. Expect some rugged landscape and archipelagoes near the hubward equator.

Gravity affects the height of mountains. On Hoop the difference is not enormous compared to Earth, but on Donut mountains at the poles can be 1.5 times higher (maximum around 12 km) and near the equators 3 times higher (24 km). Combined with the ruggedness near the hole this might make for some dramatic landscapes.

The fast rotation will likely produce a strong magnetic field; unlike on Earth the polar regions will not have auroras since the field lines will not intersect the surface
 I think – figuring out dynamo currents in a toroid iron core sounds fun but is beyond me.

Atmosphere

We have seen that the light levels change a lot, and that would make us suspect plenty of wind transporting heat from hot sunlit areas to cool shadowed areas. However, the high rate of rotation means that the Coriolis Effect will influence air and water flows to a large degree.

The Coriolis Effect makes air moving towards or away from the rotational axis bend away, since it has more or less velocity than the ground. A parcel of air “at rest” near the equator has a lot of actual momentum since the equator is moving fast around the rotation axis: if that air were to flow pole-wards it would now have a noticeable velocity eastwards or westwards. This is why the global airflow is not just simple convection cells from the equator towards the poles: as heat is transferred using air polewards the air flow gets twisted around, producing trade winds.

On torus worlds the rotation rate is 8 times faster than on Earth and the velocity differences are larger. Air hence tends to be twisted around far more, producing a more banded zonal climate than on Earth. Exactly how banded is hard to tell without detailed atmospheric calculations, but it is likely more like on Jupiter than on Earth. This in turn means that heat transfer is less effective: the temperature differences between the hot and cold regions will be bigger.

It is likely that there will be inter-tropical convergence zone (ITCZ, alias the doldrums or equatorial lows) around the rimward equator, where winds approaching from north and south will blow westwards (trade winds) while warm air rises, moves away from the equator, cools and descends at a higher or lower latitude (where we should expect major deserts). The big seasonality changes especially on Donut will make the ITCZ shift north and south, triggering monsoons in some regions. However, the rapid rotation will make the Hadley cell thinner than Earth’s 30 degree size (exactly how much thinner is slightly tricky to estimate, since it also depends on the latitude-varying gravity).

Big temperature differences over short distances are going to power plenty of weather, even if it is hard to predict exactly how it is going to look. Especially near the hole on Donut seasonal weather will be wild: warm air from the sunlit side will flow through it in a big vortex, balanced by cool winds from the dark side circulating in the opposite direction.

The scale height, how quickly pressure drops off with altitude, is proportional to gravity. Hence clouds will be 3 to 1.5 times taller on Donut, while Hoop clouds will be more Earth-like.

Like on Earth cyclones can form at the mid-latitudes. Stronger Coriolis forces would make tighter hurricanes, about four times smaller. However, they would tend to last longer on Donut (since the high scale height gives them far more air to play with). Wind speeds depend on the temperature difference between the top of the atmosphere and the ocean, which could vary a great deal across the year.

Hydrosphere

The amount of water on either world is not vastly different from Earth, although Hoop’s 6 times greater mass with merely 5 times greater area would provide it with 20% more water volume from the initial accretion (so for the same coverage the oceans would be 20% deeper). The higher mass might also accumulate more cometary infall, but it is hard to judge how much this would be.

The big seasonal temperature swings will be more pronounced far from the moderating influence of oceans: continents near the poles will be more extreme than equatorial ones. Whether they can maintain ice caps throughout polar summer depends on their layout and the background temperature; since ice reflects away sunlight effectively and the Coriolis Effect can keep air from warming them it is likely. The same for sea ice, although here there is potential for warming sea currents from hubwards or rimwards. Since the flow of water in oceans is constrained by the shape of the basins, the Coriolis Effect will merely drive gyres rather than prevent north-south flow; large oceans like the Pacific will be more east-westerly than narrow north-south oceans like the Atlantic.

The low gravity near the equator will make some tall waves on Donut: they can be expected to be three times taller than on Earth. Waves at Donut’s poles are still 150% of the ones on Earth. Hoop is closer to normal (133% taller at the equator, 90% height at the poles). The wild hubward summer-winter weather on Donut will likely drive some amazing storm waves.

Biosphere

From these considerations, it seems likely that one could have a fairly Earth-like biosphere on Donut and Hoop. Storms, severe weather and long winters are things species on Earth have adapted to fine. There might be interesting differences in ecosystems based on latitude, since there are more variation between different bands than on Earth (gravity, seasonality, temperatures etc.). Also, at least on Hoop each band has a much larger surface area: there is more room for species diversity within each eco-zone.

Moons

Would these worlds be able to keep moons?

A moon orbiting exactly in the equatorial plane in a circular orbit it would just feel a potential looking like it came from a spherical planet of some intermediate density. However, if it orbited in slightly eccentric orbit things would change. The potential field close to the planet falls of more slowly than 1/r (the answer for normal spherical planets): the Kepler ellipse is no longer the right solution. And as soon as the orbit becomes slightly tilted things turn even more complicated – now the moon will feel the flatness.

In many ways this is the problem facing satellite designers already: Earth is oblate enough that orbits are affected. This problem was dealt with in the earliest days of spaceflight (see Wikipedia, (Tremaine & Yavetz 2013) or (Nielsen, Goodwin,& Mersman 1958)).

Basically, the main effect is that an elliptic orbit precesses – it slowly changes direction in space, for Earth largely depending on the inclination. Eccentricity can also drift, which is a bigger deal. In any case, for a toroid world these effects are far larger: the multipole moments (measures of just how non-spherical the field is) are of course enormous. In fact, they are so big that the standard methods no longer work and we need to do computer simulations.

However, I feel confident that moons in sufficiently remote and circular orbits will be pretty stable. Most likely they will precess so that their orbit is more of a rosette than an ellipse, but they will not go crazy. Of course, moons in close orbits are another matter


Running a simulation (where I did not use the full torus potential, but rather a ring of 30 masses) demonstrate some of the possibilities. Indeed, an equatorial elliptic orbit looks nice and stable but precesses into a rosette.

A nearly polar orbit has even more precession, not just making it rosette around in a plane but also slowly precessing the plane. The moon could appear in the sky in any constellation.

What about orbits through the hole? As mentioned earlier, the exact center is an unstable Lagrange point. Place a moon there, and any kick will make it fall out. But there are orbits through the center that look stable (or rather, give them a kick and they turn into another similar-looking orbit rather than fall down). The simplest is just a moon bobbing up and down through the hole:

In fact, one can have a moon bobbing up and down over a particular longitude in a bent rectangular region.

And given some longitudinal velocity, it will move around the hole, filling out a wobbly hyperboloid of one sheet (a “vase orbit”?).

What about orbits that actually go through the hole in just one direction? It turns out that there are plenty of “figure 8”-like orbits that go through, precessing to form a larger torus-shaped tangle.

Note that the orbit is a bit “elliptic”, with a lopsided figure 8. From “apogee” above the rimward equator it will go through the hole and turn over the opposite side, where it will have a “perigee” near the antipode of its starting point. Then it will go through the hole, coming out near where it started – but precession will make it wind along the torus. Hence the two-sheet appearance of the entire orbit.

These simulations should be taken as first sketches, since the real case requires quite a bit of computational care. My numerical precision is not good enough to tell what the long term stability truly is. Hoop and Donut have even messier gravity fields since they are flattened, and there will of course be perturbations due to the sun and other planets.

Tidal forces

Tidal forces are an issue. Imagine a moon orbiting equatorially outside a torus world. It causes a bulge of water and rock beneath it. The rapid rotation will tend to push the bulge ahead of the moon (assuming the moon orbits in the same direction the planet turns and is above geostationary orbit). The gravity of the bulge will hence drag the moon forward, imparting a slightly faster motion – which in space means the moon moves outward to a slightly higher orbit. This is how Luna has absorbed a fair deal of Earths angular momentum, slowing Earth’s rotation and drifting further away. In the case of wild rotation like on a torus-world this effect is bigger: moons will tend to be pushed away and possibly lost.

What happens to close moons, orbiting below the geostationary orbit? They actually move faster than the bulge, and now it slows them. That means a lower orbit. Soon they spiral inwards and become giant meteors. The same happens for retrograde moons when they are too close. Of course, if the moon is big enough it might break up due to tidal forces into a ring.

The wilder orbits through the hole are likely to be destabilized by tidal forces. The bobbing orbits will tend to acquire angular momentum from the bulge, and turn faster and faster – until they crash into the planet or are lost. Some figure-8 orbits might be in the right resonance to gain and lose energy equally, but I suspect they generally have the same problem. So sadly, I suspect torus-worlds will lack the truly exotic moons. However, artificial satellites with a bit of station-keeping are still possible. Those bobbing orbits might be good for communications satellites for the hubward surface.

**Nassim Haramein has proposed a double torus shape for all stars and planets . He believes , celestial objects look like spheres but are actually toroidal . In mathematics , sphere is a special case of toroid . That means all spheres are also toroids though all toruses are not spheres . It means that sphere is just a subset of torus which is a more fundamental shape in the universe .

Regards

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@sidharthabahadur I had an interesting discussion about this with a friend recently and he believes the universe is toroidal.

I strongly think that earth is the form of platonic solid. The platonic solid inscribed sphere... created possibly from a benzine ring. Why you see hexagon north poles. This could also provide a clearer picture of gravity and the Electric Theory etc. I disagree that the earth is an oblate spheroid because of earths magnetic polarity. And no proof or reason why...

In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere.

In the research of earth, orbits, and geometry Kepler is fascinating... I am not convinced physics can answer your question as you will need exact maths to formulate which you do not.
You will understand much more with platonic solds and geometry with added tesla, energy, frequency, vibration added to the equation...

Stars of outerspace meanwhile are conscious data...

M

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We really must have a way to observe the "ether" because I think that that has a huge bearing on the understanding of this shape. I believe that the "jets" at the center of "blackholes" (which I do not believe exist / are not actually blackholes as currently described), galaxies, and the "universe" reveal something of the geometry.

It is very possible that the ether itself is MORE dense than we are and we, being currently composed of a more gross or coarse (less dense) matter, are not currently thinking or observing in the proper manner to manifest this (perhaps similar - but not really an apropos comparison - to how it is hard for fish to understand water as the medium in which they're swimming).

Robert Distinti makes an interesting conjecture on this topic here:

It may also be worth reviewing the first (and maybe second) chapter of this book "Ye Are Gods" by Annalee Skarin in that it tends to give this same impression / conclusion.

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@Soretna If we can't observe ether (yet we know it exists) and do not have correct data (from mainstream science) it is not through science answers will be revealed.

To me its common sense that if we live in a 3d world (analogue) then all matter is built from platonic geometry within an inscribe sphere else it flattens out into 2d.

https://www.mathsisfun.com/geometry/platonic-solids-why-five.html

Only 5 geometric shapes can exist in 3d plane or as some would say the real world.

@Echo_on , @Soretna , I echo your thoughts on this deep subject .

Nassim Haramein , a Swiss scientist , has done tremendous work in understanding sacred geometry with it's arcane concepts like platonic solids , dodecahedron , icosahedron and so many other complex and fractal shapes in nature .

The "Coriolis effect" observed on all stars and planets can only be explained , if we take the "double torus" approach to their structure . This is also in line with esoteric literature .

Regards

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You know what...the only way to explain Hollow planets and stars is the "dynamic" dual torus model with it's inherent "hollowness" in the center and continuous inside out energy "flow" pattern . From the smallest sub-atomic realm to the cosmic scale , this same shape can explain all...

The perfect sphere model can only yield solid billiard ball type planetary structures , which is so obviously not the case :))

Regards

List members , here are a series of 5 incredible videos , the first 4 created by Nassim Haramein and the 5th by David Allen LaPoint - a plasma physicist who has come up with a path breaking theory which is giving sleepless nights to mainstream cosmologists :)) Enjoy !

I can assure you , if you go fully through all these 5 videos , it will forever change your existing views about our planet and your existing view about the reality of our universe itself :

Double Torus Dynamic - Nassim Haramein

Double Torus Dynamic - Nassim Haramein

Watch THRIVE for FREE at: http://www.ThriveMovement.com/the_movie Stay informed, subscribe to our mailing list: ...

The above video explains the Coriolis effect . This argues that Earth & all other planets are actually a double torus "disguised" as a sphere , meaning they just look spherical but at closer examination are not !

Nassim Haramein Double Torus

Nassim Haramein Double Torus

With thanks & gratitude to The Resonance Project & Nassim Haramein

The Universal Pattern: TORUS -

The Universal Pattern: TORUS -

The Universal Pattern: TORUS - Animation by Cristian Bredee Explore an interactive online learning community wit...

Tetragrammaton, Toroide, Spirale Aurea, DNA (Nassim Haramein)

Tetragrammaton, Toroide, Spirale Aurea, DNA (Nassim Haramein)

O Modelo Estrutural do tudo, de todos elementos em todas proporçÔes, o Duplo Tórus. Uma bela trilha sonora com b...

David Allen LaPoint's explanation of electromagnetism in terms of Primer fields which also take the dual Torus shape

The Primer Fields - Part 1

The Primer Fields - Part 1

In this video series the currently accepted theories of physics and astrophysics are shaken to the core by a radical...

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This idea of Toroid as the fundamental shape in our cosmos was represented in ancient mythology by the pervasive symbol of the Ouroboros (known by different names) - the snake that eats it's own tail . It also symbolises the universal esoteric principle of "as above , so below" :-

https://www.crystalinks.com/ouroboros.html


Symbolic representation of coming full circle (cycle)

The Ouroboros is an ancient symbol depicting a serpent or dragon eating its own tail. The name originates from within Greek language; (oura) meaning "tail" and (boros) meaning "eating", thus "he who eats the tail".

The Ouroboros represents the perpetual cyclic renewal of life and infinity, the concept of eternity and the eternal return, and represents the cycle of life, death and rebirth, leading to immortality, as in the Phoenix.

The current mathematical symbol for infinity - may be derived from a variant on the classic Ouroboros with the snake looped once before eating its own tail, and such depictions of the double loop as a snake eating its own tail are common today in fantasy art and fantasy literature, though other conjectures also exist.

It can also represent the idea of primordial unity related to something existing in or persisting before any beginning with such force or qualities it cannot be extinguished. The ouroboros has been important in religious and mythological symbolism, but has also been frequently used in alchemical illustrations, where it symbolizes the circular nature of the alchemist's opus. It is also often associated with Gnosticism and Hermeticism.

Showing itself primarily in ancient Gnostic texts, the Ouroboros is any image of a snake, worm, serpent, or dragon biting its own tail. Generally taking on a circular form, the symbol is representative of many broad concepts. Time, life continuity, completion, the repetition of history, the self-sufficiency of nature, and the rebirth of the Earth can all be seen within the circular boundaries of the Ouroboros.

Societies from throughout history have shaped the Ouroboros to fit their own beliefs and purposes. The image has been seen in ancient Egypt, Japan, India, utilized in Greek alchemic texts, European woodcuts, Native American Indian tribes, and by the Aztecs. It has, at times, been directly associated to such varying symbols as the Roman god Janus, the Chinese Ying Yang, and the Biblical serpent in the Garden of Eden.

Geometry - Creation

Simulation Hypothesis - Holographic Universe Theory

Tube Torus


Flower of Life


12 Around 1 - Alchemy Wheel


The Ouroboros and the Tree of Life

Origins of the Ouroboros

Egypt


Papyrus of Dama Heroub Egypt, 21st Dynasty

The serpent or dragon eating its own tail has survived from antiquity and can be traced back to Ancient Egypt, circa 1600 B.C.E. It is contained in the Egyptian Book of the Netherworld. The Ouroboros was popular after the Amarna period.

In the Book of the Dead, which was still current in the Graeco-Roman period, the self-begetting sun god Atum is said to have ascended from chaos-waters with the appearance of a snake, the animal renewing itself every morning, and the deceased wishes to turn into the shape of the snake Sato ("son of the earth"), the embodiment of Atum.

The famous Ouroboros drawing from the early alchemical text The Chrysopoeia of Cleopatra dating to 2nd century Alexandria encloses the words hen to pan, "one is the all". Its black and white halves represent the Gnostic duality of existence. As such, the Ouroboros could be interpreted as the Western equivalent of the Taoist Yin-Yang symbol. The Chrysopoeia Ouroboros of Cleopatra is one of the oldest images of the Ouroboros to be linked with the legendary opus of the Alchemists, the Philosopher's Stone.

Greece

Plato described a self-eating, circular being as the first living thing in the universe - an immortal, mythologically constructed beast. The living being had no need of eyes when there was nothing remaining outside him to be seen; nor of ears when there was nothing to be heard; and there was no surrounding atmosphere to be breathed; nor would there have been any use of organs by the help of which he might receive his food or get rid of what he had already digested, since there was nothing which went from him or came into him: for there was nothing beside him.

Of design he was created thus, his own waste providing his own food, and all that he did or suffered taking place in and by himself. For the Creator conceived that a being which was self-sufficient would be far more excellent than one which lacked anything; and, as he had no need to take anything or defend himself against any one, the Creator did not think it necessary to bestow upon him hands: nor had he any need of feet, nor of the whole apparatus of walking; but the movement suited to his spherical form was assigned to him, being of all the seven that which is most appropriate to mind and intelligence; and he was made to move in the same manner and on the same spot, within his own limits revolving in a circle.

All the other six motions were taken away from him, and he was made not to partake of their deviations. And as this circular movement required no feet, the universe was created without legs and without feet. In Gnosticism, this serpent symbolized eternity and the soul of the world.

Middle East

Because the Albigenses came from Armenia, where Zoroastrianism and Mithra worship were common, it may be that the symbol entered their iconography via the Zoroastrian Faravahar symbol, which in some versions clearly features an ouroboros at the waist instead of a vague disc-shape.

In Mithran mystery cults the figure of Mithra being reborn (one of the things he is famous for) is sometimes seen wrapped with an ouroboros, indicating his eternal and cyclic nature, and even references which do not mention the ouroboros refer to this circular shape as symbolizing the immortality of the soul or the cyclic nature of Karma, suggesting that the circle retains its meaning even when the details of the image are obscured.


The Double Triangle of Solomon

India

Ouroboros symbolism has been used to describe Kundalini energy. According to the 2nd century Yoga Kundalini Upanishad, "The divine power, Kundalini, shines like the stem of a young lotus; like a snake, coiled round upon herself she holds her tail in her mouth and lies resting half asleep as the base of the body" (1.82). Another interpretation is that Kundalini equates to the entwined serpents of the Caduceus, the entwined serpents representing commerce in the west or, esoterically, human DNA.

The Kirtimukha myth of Hindu tradition has been compared by some authors to Ouroboros.

Ouroboros... the dragon circling the tortoise which supports the four elephants that carry the world.

China


Chinese Ouroboros from Chou dynasty, 1200 BC.

The universe was early divided into Earth below and Heaven above. These, two as one, gave the idea of opposites but forming a unity. Each opposite was assumed to be powerful and so was their final unity. For creation of the universe they projected reproduction to conceive creation. Now reproduction results in the union of two opposites as male and female.

Correspondingly, the Chinese believed Light and Darkness, as the ideal opposites, when united, yielded creative energy. The two opposites were further conceived as matter and energy which became dual-natured but as one. The two opposites were yin-yang and their unity was called Chhi. Yin-Yang was treated separately in Chinese cosmology which consisted of five cosmic elements.

Since Chinese alchemy did reach Alexandria probably the symbol Yin-Yang, as dual-natured, responsible for creation, was transformed into a symbol called Ouroboros. It is a snake and as such as symbol of soul. Its head and anterior portion is red, being the color of blood as soul; its tail and posterior half is dark, representing body.

Ouroboros here is depicted white and black, as soul and body, the two as "one which is all." It is cosmic soul, the source of all creation. Ouroboros is normally depicted with its anterior half as black but it should be the reverse as shown here. With the name Chemeia taken to Kim-Iya, the last word would take Ouroboros to Yin-Yang.

Japan


Pre 1400 Japan

Mesoamerica

The serpent god Quetzalcoatl is sometimes portrayed biting his tail on Aztec and Toltec ruins. A looping Quetzalcoatl is carved into the base of the Pyramid of the Feathered Serpent, at Xochicalco, Mexico, 700-900 AD.

Seven-segmented Aztec Ouroboros

South America

It is a common belief among indigenous people of the tropical lowlands of South America that waters at the edge of the world-disc are encircled by a snake, often an anaconda, biting its own tail.

Native American

Norse

In Norse mythology, it appears as the serpent Jormungandr, one of the three children of Loki and Angrboda, who grew so large that it could encircle the world and grasp its tail in its teeth. In the legends of Ragnar Lodbrok, such as Ragnarssona patter, the Geatish king Herraud gives a small lindworm as a gift to his daughter Pora Town-Hart after which it grows into a large serpent which encircles the girl's bower and bites itself in the tail. The serpent is slain by Ragnar Lodbrok who marries Pora. Ragnar later has a son with another woman named Kraka and this son is born with the image of a white snake in one eye. This snake encircled the iris and bit itself in the tail, and the son was named Sigurd Snake-in-the-Eye.

Christians

Christians adopted the Ouroboros as a symbol of the limited confines of this world (that there is an "outside" being implied by the demarcation of an inside), and the self-consuming transitory nature of a mere this-worldly existence following in the footsteps of the Preacher in Ecclesiastes.

It could very well be used to symbolize the closed-system model of the universe of some physicists even today.'

Rome


Earthly Ouroboros from Alciato's Emblems


Oceanic Ouroboros from Alciato's Emblems


Janus 1608

Haiti

In 1812, the Republic of Haiti under President Alexandre Petion issued its first locally minted coinage which featured an image of a serpent biting its own tail.

West Africa

Snakes are sacred in many West African religions. The demi-god Aidophedo uses the image of a serpent biting its own tail. The Ouroboros is also seen in Fon or Dahomean iconography as well as in Yoruba imagery as Oshunmare.

Freemasonry

The ouroboros is displayed on numerous Masonic seals,
frontispieces and other imagery, especially during the 17th century.

Theosophical Society

The Ouroboros is featured in the seal of the Theosophical Society
along with other traditional symbols.

Tarot and Watermarks

The Ouroboros symbol appears in both 14th- and 15th-century Albigensian-printing watermarks and is also worked into the pip cards of many early (14th-15th century) playing cards and tarot cards. Watermarks similar to those used by the Albigensians appear in early printed playing cards, suggesting that the Albigenses might have had contact with the early authors of tarot decks.

A commonly used early symbol - an ace of cups circled by an ouroboros - frequently appears among Albigensian watermarks. It is conceivable that this is the source of some of the urban legends associating this symbol with secret societies, because the Albigenses were closely associated with the humanist movement and the inquisition it sparked.

Alchemy

Alchemically, the ouroboros is also used as a purifying glyph. Ouroboros was and is the name for the Great World Serpent, encircling the Earth.

The word Ouroboros is really a term that describes a similar symbol which has been cross-pollinated from many different cultures. Its symbolic connotation from this owes to the returning cyclical nature of the seasons; the oscillations of the night sky; self-fecundation; disintegration and re-integration; truth and cognition complete; the Androgyny; the primeval waters; the potential before the spark of creation; the undifferentiated; the Totality; primordial unity; self-sufficiency, and the idea of the beginning and the end as being a continuous unending principle.

Ouroboros represents the conflict of life as well in that life comes out of life and death. 'My end is my beginning.' In a sense life feeds off itself, thus there are good and bad connotations which can be drawn. It is a single image with the entire actions of a life cycle - it begets, weds, impregnates, and slays itself, but in a cyclical sense, rather than linear.

Thus, it fashions our lives to a totality more towards what it may really be - a series of movements which repeat. "As Above, So Below" - we are born from nature, and we mirror it, because it is what man wholly is a part of. It is this symbolic rendition of the eternal principles that are presented in the Emerald Tablets of Thoth.

The Ouroboros connects the Above and Below


Connection between Man and God

Carl Jung

Swiss psychologist Carl Jung interpreted the Ouroboros as having an archetypal significance to the human psyche. It makes its way into our conscious mind time and time again in varying forms as the basic mandala of alchemy. Jung defined the relationship of the ouroboros to alchemy:

The alchemists, who in their own way knew more about the nature of the individuation process than we moderns do, expressed this paradox through the symbol of the ouroboros, the snake that eats its own tail. In the age-old image of the ouroboros lies the thought of devouring oneself and turning oneself into a circulatory process, for it was clear to the more astute alchemists that the prima materia of the art was man himself.

The ouroboros is a dramatic symbol for the integration and assimilation of the opposite, i.e. of the shadow. This 'feed-back' process is at the same time a symbol of immortality, since it is said of the ouroboros that he slays himself and brings himself to life, fertilizes himself and gives birth to himself. He symbolizes the One, who proceeds from the clash of opposites, and he therefore constitutes the secret of the prima materia which [...] unquestionably stems from man's unconscious'. ( Collected Works, Vol. 14 para.513)

Other References

The Jungian psychologist Erich Neumann writes of it as a representation of the pre-ego "dawn state", depicting the undifferentiated infancy experience of both mankind and the individual child.

The 19th century German chemist named Kekule dreamed of a snake with its tail in its mouth one day after dosing off. He had been researching the molecular structure of benzene, and was at a stop point in his work until after waking up he interpreted the dream to mean that the structure was a closed carbon ring. This was the breakthrough he needed.

Organic chemist August Kekule claimed that a ring in the shape of Ouroboros that he saw in a dream inspired him in his discovery of the structure of the benzene ring.

Today the Ouroboros is often found as a tattoo.

The X-Files' Dana Scully chose the Ouroboros to be tattooed on her back because she felt it represented the progression of her life. It seems that the Ouroboros is a powerful archetypal symbol, a part of our Spiritus Mundi, the collective unconscious which thrives within each soul.

Crop Circles

Regards

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I couldn't help but laugh at the 2nd to last image: someone was playing with an Amiga too much when they rendered that crop circle:

Any interesting post nonetheless...

@Soretna , you maybe surprised that this symbol of AMIGA OS , like so many other Corporate logos , could have actually been picked up from an esoteric source (!!) By observing carefully , it looks like there are a total of 64 squares on the complete sphere , only half of which being visible from one side...!

Even a chessboard has 64 squares because it too is derived from esoteric symbolism around the number 64 in sacred geometry . Please sample this :-

GRID OF LIFE

64-tetrahedron

The star tetrahedron is the basis of the 64 grid tetrahedron also known as the Grid of Life. This is the masculine equivalent to the Flower of Life pattern. As the name suggests, this shape is made up of 64 tetrahedrons. Which as you know is one of the platonic solids.

When the 64-tetrahedron is overlaid on the flower of life this represents a powerful concept; the spheres represent space and the lines lines represent where space and time come together.

One of the most fascinating aspects of the 64 tetrahedron is the number 64 itself and how it appears throughout the natural world.

Here are a few examples:

  • There are 64 codons in human DNA
  • 64 is the number of cells we have before the cells start to bifurcate (differentiate) after conception.
  • 64 is the fundamental number in computer memory and bits of code
  • 64 is encoded in the Tetragrammaton within the Hebrew Bible, which is the four letter theonym YHWH that means God in Hebrew.
  • 64 is the number of generations from Adam until Jesus
  • 64 manifestations of Lord Shiva in Hinduism
  • 64 squares in chess and checker boards
  • 64 hexagrams in the I Ching
  • 64 tantras which is a form of Hinduism
  • 64 letters in the Sanskrit Alphabet
  • 64 is the maximum number of strokes in any Chinese character

This metaphysical symbol can produce many trees of life which radiate inward, this is why the 64 tetrahedron is symbolic of linking up ones individual paths in harmony with our soul family. It is said that connecting with your soul / star family will amplify each individual's energy to accomplish and manifest one's goals. A true symbol of the mastermind principle of harmony with among a group of individuals.

Regards

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Folks , here are abstracts of two research papers - the first is from CERN , that talks about the so called "oblateness" of extrasolar planets in the larger universe , beyond our own Solar system . The second pertains to the so called "oblateness" of stars . The key point I am trying to highlight here - NOT even one star or planet has been found to be of spherical shape - why ?? That is what science has been missing out on !

Now by applying the principle that "the same laws of science must apply to all stars and planets in the universe" , we find that in fact , ALL planets and stars are toroidal (oblate) in shape .

**For some of the stars that are similar to our Sun , equatorial radius can be upto 17% larger than the polar radius - now would anyone in their right mind , call that a spherical shape ??

A perfectly spherical star or planet (if it could even exist !) , would be nothing but a sterile billiard ball floating through space :))

Measuring the Oblateness and Rotation of Transiting Extrasolar GiantPlanetsJason W. Barnes and Jonathan J. FortneyDepartment of Planetary Sciences, University of Arizona, Tucson, AZ, [email protected], [email protected]

ABSTRACT

We investigate the prospects for characterizing extrasolar giant planets by measuring planetary oblateness from transit photometry and inferring planetary rotational periods. The rotation rates of planets in the solarsystem vary widely, reflecting the planets' diverse formational and evolutionary histories. A measured oblateness,assumed composition, and equation of state yields a rotation rate from the Darwin-Radau relation. The lightcurveof a transiting oblate planet should di er signi cantly from that of a spherical one with the same cross-sectionalarea under identical stellar and orbital conditions. However, if the stellar and orbital parameters are not knownapriori, tting for them allows changes in the stellar radius, planetary radius, impact parameter, and stellar limbdarkening parameters to mimic the transit signature of an oblate planet, diminishing the oblateness signature.Thus even if HD209458b had an oblateness of 0.1 instead of our predicted 0.003, it would introduce a detectabledeparture from a model spherical lightcurve at the level of only one part in 105. Planets with nonzero obliquitybreak this degeneracy because their ingress lightcurve is asymmetric relative to that from egress, and their best-case detectability is of order 10−4. However, the measured rotation rate for these objects is non-unique due todegeneracy between obliquity and oblateness and the unknown component of obliquity along the line of sight.Detectability of oblateness is maximized for planets transiting near an impact parameter of 0.7 regardless ofobliquity. Future measurements of oblateness will be challenging because the signal is near the photometric limitsof current hardware and inherent stellar noise levels .

https://iopscience.iop.org/article/10.3847/0004-637X/830/1/45

CONVECTION IN OBLATE SOLAR-TYPE STARS

Abstract

We present the first global 3D simulations of thermal convection in the oblate envelopes of rapidly rotating solar-type stars. This has been achieved by exploiting the capabilities of the new compressible high-order unstructured spectral difference (CHORUS) code. We consider rotation rates up to 85% of the critical (breakup) rotation rate, which yields an equatorial radius that is up to 17% larger than the polar radius. This substantial oblateness enhances the disparity between polar and equatorial modes of convection. We find that the convection redistributes the heat flux emitted from the outer surface, leading to an enhancement of the heat flux in the polar and equatorial regions. This finding implies that lower-mass stars with convective envelopes may not have darker equators as predicted by classical gravity darkening arguments. The vigorous high-latitude convection also establishes elongated axisymmetric circulation cells and zonal jets in the polar regions. Though the overall amplitude of the surface differential rotation, ΔΩ, is insensitive to the oblateness, the oblateness does limit the fractional kinetic energy contained in the differential rotation to no more than 61%. Furthermore, we argue that this level of differential rotation is not enough to have a significant impact on the oblateness of the star.

Regards