3D spiral bundle of circular toruses?

Discussion of shapes with curves and holes in various dimensions.

3D spiral bundle of circular toruses?

Postby quickfur » Wed Aug 24, 2022 5:39 pm

Just picking the brains of y'all toratope geniuses that lurk here: in 3D, is it possible to decompose a torus (say of major radius 3 and minor radius 1) into a bundle of equivalent circles? Or equivalently, is it possible to take a set of n circles of the same radius and interlock them in such a way that they are inscribed inside a common torus?

If so, what would be a mathematical formulation of such a construct? What would be the equations of said circles? I'm particularly interested in symmetric constructions, where, say, intersecting the bundle with a plane perpendicular to the plane of the inscribing torus would produce the vertices of two regular polygons where the plane intersects the circles.

Even more specifically, I'm wondering under what conditions the following construction would yield circles: we know the intersection of a torus with a plane perpendicular to the plane of its major radius is two circles. Suppose we inscribe a regular n-polygon in one of these circles, then rotate it around the origin inside the torus. This would trace out an n-gonal torus. Its vertices would trace out circles, but of diverse radii. But what if we also rotate the polygon in its plane as we sweep it around the torus, such that the polygon is rotated by some multiple of 2*pi/n by the time it returns to its original position, say it's rotated by k*2*pi/n? Under what values of n,k would the vertices of the polygon trace out circles of identical radius, if any?
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Re: 3D spiral bundle of circular toruses?

Postby Plasmath » Wed Aug 24, 2022 9:19 pm

quickfur wrote:Just picking the brains of y'all toratope geniuses that lurk here: in 3D, is it possible to decompose a torus (say of major radius 3 and minor radius 1) into a bundle of equivalent circles? Or equivalently, is it possible to take a set of n circles of the same radius and interlock them in such a way that they are inscribed inside a common torus?

Could something like this be what you're thinking of? This would be the example for n=7.
Torus rings.png
Torus rings.png (73.32 KiB) Viewed 4801 times

Torii also have Villarceau circles, which could be used to construct something similar. These circles and circles perpendicular to the ones shown above should be the only circles that exist on the torus.

quickfur wrote:If so, what would be a mathematical formulation of such a construct? What would be the equations of said circles? I'm particularly interested in symmetric constructions, where, say, intersecting the bundle with a plane perpendicular to the plane of the inscribing torus would produce the vertices of two regular polygons where the plane intersects the circles.

I believe the symmetric construction you mentioned is what is shown above. Instead of a regular polygon, you can do it with any cyclic polygon by varying the angles between the circles.
Figuring out the equation of the circles is more difficult. The solution I used utilizes vectors, but there may be other simpler solutions.
Let the origin be at the center of the torus, V be the vector [Vx Vy 0]to the center of the given circle, r be the minor radius, and R be the major radius. Then the parametric equation for the circle is:
Code: Select all
r*V/(R+r) sin(t) + [0 0 r] cos(t) + V

for -π < t < π.

quickfur wrote:I'm wondering under what conditions the following construction would yield circles: we know the intersection of a torus with a plane perpendicular to the plane of its major radius is two circles. Suppose we inscribe a regular n-polygon in one of these circles, then rotate it around the origin inside the torus. This would trace out an n-gonal torus. Its vertices would trace out circles, but of diverse radii.
But what if we also rotate the polygon in its plane as we sweep it around the torus, such that the polygon is rotated by some multiple of 2*pi/n by the time it returns to its original position, say it's rotated by k*2*pi/n? Under what values of n,k would the vertices of the polygon trace out circles of identical radius, if any?

Given the image above, the first thing you're talking about would be the bracketoratope ({II}n*I) , definitions found here.
I'm not sure what exactly the second shape would be, however the vertices would always trace out coil shapes instead of circles unless there is no rotation. This still sounds like an extremely cool-looking and interesting shape, and it kind of reminds me of a Mobius strip, where instead of connecting a strip of paper to itself with an added turn, we connect a prism to itself with an added turn. I wish I could make a visualization of this but I don't know how.
Last edited by Plasmath on Wed Aug 24, 2022 10:38 pm, edited 1 time in total.
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Wed Aug 24, 2022 10:32 pm

Thanks, Villarceau circles are what I was looking for. Basically, pick one of n pairs of Villarceau circles equally spaced apart and non-intersecting, and you get a pattern of circles that wrap around the body of the torus. If both of each pair of circles are picked, you get an intersecting pattern of cells that span the surface of the torus.

P.S. Basically, the construction I have in mind is to take one Villarceau circle, and make (n-1) rotated copies of it spaced at angles of 2*pi/n radians apart (which would also be Villarceau circles). This produces a symmetric pattern of congruent circles equally spaced around the torus.
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Re: 3D spiral bundle of circular toruses?

Postby Plasmath » Wed Aug 24, 2022 10:50 pm

This makes me wonder if higher-dimensional toratopes have analogs. Could the torisphere have Villarceau spheres? Would the ditorus have Villarceau torii?
This might also be an image of something similar to what you’re talking about, although this image has twice the circles.
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Wed Aug 24, 2022 11:18 pm

I suspect the duocylinder would have something similar.

Well in fact, the projection of the Hopf fibration of the 3-sphere into 3D maps the Hopf fibers into Villarceau circles.
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Thu Aug 25, 2022 12:59 am

Whoa, I just realized something interesting about Villarceau circles: say you have n Villarceau circles equally spaced around the major axis of the torus. Then two adjacent circles trace out the boundaries of a double-twist Möbius strip. :o :o_o: :lol: I haven't verified whether the width of this Möbius strip is constant, though. If it is, it would have very interesting implications indeed!
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Re: 3D spiral bundle of circular toruses?

Postby Plasmath » Thu Aug 25, 2022 2:32 am

Maybe it's possible to create a single-twist strip! I suspect that might create a klein bottle or real projective plane.

Also, I feel like the fact that torii map to circles in the hopf fibration with each point in the circle corresponding a Villarceau circle in the glome can't be a coincidence. There's got to be deeper mathematics here. I also wonder if there are Hopf fibration analogs from the 4D torii to the torus... then there would definitely be higher-dimensional Villarceau shapes (however I'm not really sure how the Hopf fibration works, which I would need to look more into to find out).
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Thu Aug 25, 2022 3:43 am

Plasmath wrote:Maybe it's possible to create a single-twist strip! I suspect that might create a klein bottle or real projective plane.

I doubt a single-twist strip is possible, because that would imply that a strip of the torus's surface is non-orientable, both sides of which are on the outside, which is a contradiction.

Also, I feel like the fact that torii map to circles in the hopf fibration with each point in the circle corresponding a Villarceau circle in the glome can't be a coincidence. There's got to be deeper mathematics here.

Of course it's not a coincidence. The existence of the Hopf fibration is the reason Villarceau circles exist. :P :lol:

I also wonder if there are Hopf fibration analogs from the 4D torii to the torus... then there would definitely be higher-dimensional Villarceau shapes (however I'm not really sure how the Hopf fibration works, which I would need to look more into to find out).

Take a look at projections of spidrox. If you follow the description, its surface consists of a number of rings (great circles): 12 rings consisting of alternating prisms and antiprisms, and 20 rings consisting of a spiralling chain of square pyramids. Each of these rings wrap around the surface of the polytope in a spiralling, interlocked fashion. Now imagine if you reduce each of these rings into great circles. These great circles are a subset of the Hopf fibration of the 3-sphere. Now imagine if you fill out the rest of the surface of the 3-sphere with other great circles in between these ones, in the same spiralling and interlocking manner, then you have a decomposition of the surface of the 3-sphere into congruent circles. These circles are the Hopf fibration of the 3-sphere.

Of course, spidrox isn't the only polytope sporting this structure. Actually, all the convex regular and uniform polytopes as well as a number of CRF polytopes have this structure, only, it's obscured by the higher degree of symmetry that they sport. :P Another polytope where this structure becomes obvious is bidex. There it sports a smaller subset of the Hopf fibration, but there's enough of it for you to discern the general pattern.

One could argue that the Hopf fibration is the 4D analog of the Villarceau circles in the 3D torus. :lol: They are both decompositions of a surface into interlocked, intertwined circles at an oblique angle that isn't readily apparent from a cursory inspection of the surface. In fact, the two are homeomorphic, because the Hopf fibres on a toroidal cross-section of the 3-sphere exactly maps to the Villarceau circles of the projection of the toroid into a 3D torus.

As for whether higher-dimensional analogues exist, I'm not aware of them, but I suspect there may well be many more instances of this interesting phenomenon up there.
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Thu Aug 25, 2022 3:51 am

Here's a model of Villarceau circles of a torus that I made in POVRay:

Image
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Thu Aug 25, 2022 2:46 pm

Apparently I was wrong about embedded regular polygons tracing out circles. Well, at least, they wouldn't trace out Villarceau circles if the polygon was regular. Instead, the distance between Villarceau circles change as you go around the torus. Here's an animation that shows this effect: a rotating torus with 5 Villarceau circles (themselves "fattened" into toruses for better visibility of their distance relationships), cut away by a vertical plane that bisects the torus so that you can see the inside.

Image

As you can see, the polygon traced by the Villarceau circles is not regular, and it changes shape as it rotates around the body of the torus.
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Re: 3D spiral bundle of circular toruses?

Postby Plasmath » Thu Aug 25, 2022 9:04 pm

quickfur wrote:Image

As you can see, the polygon traced by the Villarceau circles is not regular, and it changes shape as it rotates around the body of the torus.


These are really amazing animations! There's just something about them that's really mesmerizing to look at. I have a feeling that these polygonal distortions would no longer exist if we looked at Villarceau circles on a Clifford torus instead of a normal 3D one, although I don't know how to confirm this.
Also on the topic of Clifford torii, I think that a Hopf fibration-like mapping from one of the 4D torii to a Clifford torus is probably likely, considering the Clifford torii's fundamental domain is a square.

quickfur wrote:I doubt a single-twist strip is possible, because that would imply that a strip of the torus's surface is non-orientable, both sides of which are on the outside, which is a contradiction.

I meant connecting Mobius strips together instead of double-twisted Mobius strips to try to create a surface (although looking back I wasn't exactly very clear that I was talking about that). I think it would create a Klein bottle but I'm not sure.
Last edited by Plasmath on Fri Aug 26, 2022 12:17 am, edited 1 time in total.
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Re: 3D spiral bundle of circular toruses?

Postby quickfur » Fri Aug 26, 2022 12:09 am

Plasmath wrote:[...]
These are really amazing animations! There's just something about them that's really mesmerizing to look at. I have a feeling that these polygonal distortions would no longer exist if we looked at Villarceau circles on a Clifford torus instead of a normal 3D one, although I don't know how to confirm this.

You're absolutely right. The Clifford torus is a member of a set of decompositions of the 3-sphere into concentric toroids. Villarceau circles in 3D map to great circles on the 3-sphere that lie on the Clifford torus. Great circles that are equally spaced around the Clifford torus remain equidistant at all points; so yes, they absolutely form regular polygon cross-sections. One may thus view the distortions in 3D as an artifact of projecting the 4th dimension into one of the first 3: in 4D, there is no distortion because the curvature of the torus lies entirely in the plane orthogonal to its cross-section (in 4D, the perpendicular of a 2D cross-section is a plane, not just a line). In 3D, however, this curvature is mapped to the plane of the major axis of the torus. Because this now involves one of the axes where the polygons lie, distortion is inevitable.

Look again at the projections of spidrox. The 12 rings of alternating prisms, for example, divide into 1 + 5 + 5 + 1, corresponding to the faces of a regular dodecahedron. Within each group of 5, the rings are always equidistant to its neighbours, and form a spiralling regular pentagonal configuration around each other. Replace the rings with great circles through the middle of the alternating prisms, and you have 5 great circles in pentagonal formation on a Clifford torus. Project this to 3D, and you get a 3D torus with 5 Villarceau circles. In 4D, the pentagon connecting the 5 great circles is regular and has no distortion; when projected into 3D, though, it exhibits the same distortion we see in the above animation.

Also on the topic of Clifford torii, I think that a Hopf fibration-like mapping from one of the torii to a Clifford torus is probably likely, considering the Clifford torii's fundamental domain is a square.

Not only it's likely, it's actually so. :D

quickfur wrote:I doubt a single-twist strip is possible, because that would imply that a strip of the torus's surface is non-orientable, both sides of which are on the outside, which is a contradiction.

I meant connecting Mobius strips together instead of double-twisted Mobius strips to try to create a surface (although looking back I wasn't exactly very clear that I was talking about that). I think it would create a Klein bottle but I'm not sure.

Ahh I gotcha. You're right, if you glue Möbius strips together side-by-side until it closes up, you'd get a Klein bottle. You can't actually do this physically in 3D, though, because in 3D the resulting surface necessarily intersects itself. In 4D, however, this construction can be carried out without any self-intersection, and you get the Klein bottle, which is a knotted 2-sphere.

Note, however, that the "bottle" part of the name is actually a misnomer: in 4D, the Klein bottle actually cannot hold any liquid, because a 2D manifold in 4D is insufficient to hold liquid in place. You need a 3D manifold in order to contain a liquid in 4D, so the Klein bottle actually isn't a bottle; it's just a 4D equivalent of a twisted hyper-string. I.e., a 4D Mobius strip.
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