non-cubic toratopes?

Discussion of shapes with curves and holes in various dimensions.

Re: non-cubic toratopes?

Postby ICN5D » Tue Sep 01, 2015 8:34 pm

It's a tough one, I know. Slicing the 3-prong, we get a weird concentric hemi-circle surface, or 3 circles in triangle. The torus makes two concentric circles, and two displaced. But, a tiger as derived from a torus, has no concentric slices in 3D. So, by this principle, maybe a mantis has only two displaced slices, of the 3 tori.

And at the same time, a mantis would have the same structure of the the 3 hemi-circle frame (embedded with a torus in every point), which seems like it would have a weird hemi-torus concentric slicing.

I recently thought about trying to fabricate the mantis using the same method of how I approximated the 3-prong. The 3-prong can be defined as a tri-toroidal cassini system, then adjusted to self-intersection. At a certain value, the three tori converge to form a 3-prong.

I wonder if I took three tigers, multiplied them together, with a common minor diameter, into a tri-tigroidal cassini system. A self-intersecting version may approximate a mantis closer than anything I've made so far. Still not a true mantis, it would feature three perfectly formed tori in the fence arrangement, as a function of 4 variables, and may have the common hexagon of 6 circles trace array, under rotation.

This is the function that made the 3-prong:

((sqrt(y^2+z^2)-b)^2+(x-a)^2) * ((sqrt(((sqrt(3)x-y)/2)^2+z^2)-b)^2+((x+sqrt(3)y)/2+a)^2) * ((sqrt((-(sqrt(3)x+y)/2)^2+z^2)-b)^2+((x-sqrt(3)y)/2+a)^2) = c^6

Symmetrical Three-Prong Multitorus at a=2 , b=5 , c=2.5


Image
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Re: non-cubic toratopes?

Postby Keiji » Fri Oct 23, 2015 9:19 pm

Wow, that's an amazing looking figure! Thank you for posting it :)

I haven't really been following this thread, just happened to notice the image.

And that got me really interested in the possibilities...

Looking back at Marek's list:

Marek14 wrote:2D:
Circle (II)

3D:
Sphere (III)
Torus ((II)I)
Multitoruses ([II]n I) based on multiple semicircles attached to a common line.

4D:
Glome (IIII)
Torisphere ((III)I)
Multitorispheres ([III]n I) based on multiple half-spheres attached to a common circle.
Spheritorus ((II)II)
Multispheritoruses ([II]n II) based on multiple semicircles attached to a common line. This may include both 2D and 3D arrangements of semicircles, creating a mid-cut with circular or polyhedral arrangement of spheres.
Ditorus (((II)I)I)
Dimultitoruses (([II]n I)I) based on multitoruses.
Multiditoruses ([(II)I]n I) based on multiple half-toruses attached to common pair of concentric circles.
Multidimultitoruses ([[II]m I]n I) based on multiple halves of multitoruses attached to common group of circularly arranged circles.
Tiger ((II)(II))
Multitigers ([II]n (II)) based on multiple halves of duocylinder margins (circle x semicircle) attached to a common pair of parallel circles. Mantis would belong here.
Hypertigers ([II]m [II]n) based on multiple quarters of duocylinder margins (semicircle x semicircle). Their attachment seems more vague to understand.


I suppose we could say that sequence ([II]n I) = {(III), ((II)I), ([II]3 I), ([II]4 I), ([II]5 I), ...} follows the sequence of regular polygons including the initial degenerate cases Gn = {point, digon, trigon, square, pentagon, ...}

As I see it, the most direct extension of the 3D torus into higher dimensions is the sequence ((II)I), ((III)I), ((IIII)I), ... - so the most direct extension of this new sequence of multitoruses into higher dimensions is as ([II]n I), ([III]n I), ([IIII]n I), ... - this case seems simple enough to understand.

If we now look at the opposite case - the spheritorus instead of the torisphere, and the related sequence ((II)I), ((II)II), ((II)III), ... - I believe these would be arrangements of semicircles around a common something - not necessarily a line (digon) as Marek states. For the ((II)II) case I imagine any of the Gn sequence could replace the line. Therefore ([II]n II) isn't specific enough to represent the full array of possibilities. I believe ([II]n II) would represent the n-prisms, while other possibilities would be non-prismatic 3D figures, such as a dodecahedron, an Archimedean solid, or maybe even some (all?) of the Johnson solids.

If the above is true then the multitoruses and multitorispheres (and their analogues in higher dimensions) are perhaps better represented by replacing "n I" with "Gn" - and then the multispheritoruses can be represented by replacing "n II" with "X" - any qualifying 3D polytope. So ([II]5 I) is now ([II] G5), and potential multispheritoruses might be ([II] +G5) (pentagonal prism arrangement), ([II] Ko6) (truncated octahedral arrangement), etc.

Now, I'm a fan of powertopes, so you know where I'm going with this...

For the ((II)III) case, the family elements are identified by a 4D polytope. I imagine a duoprism would qualify. Would a duotegum also qualify? How about a square octagoltriate?

In general, what defines whether a multitorus exists for a given polytope? Or do they all exist?

Are there any restrictions on combining different qualifying polytopes at different nesting levels? For example, does ([[II] X] Y) exist for all valid ([II] X) and ([II] Y)?

As if that didn't raise enough questions already, I notice that my replacement of "n I..." with "X" doesn't work for tigers, because the "I" is not there. So, is ([II] X [II] Y) a construction that even makes sense? Or is some even more elaborate notation required to fully describe the higher-dimensional hypertigers?
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Re: non-cubic toratopes?

Postby ICN5D » Sun Oct 25, 2015 12:10 am

It's a neat object, isn't it? I've been checking out how it intercepts a 2-plane, to get a better understanding of what to expect from mantis, or the multiditorus. This 3-prong multitorus in 3D makes a different intersection in each coordinate 2-plane:

    xy : the triangle array of 3 circles, with a reverse orientation after 180 degree flip
    xz : a balloned up letter C, with a reverse orientation after 180 flip : the concentric circle pair cut in half, with smaller hemi-circles closing off the open ends
    yz : two disjoint ellipses , when none of the arms are in 2D

These sections have smooth and symmetrical transformations under rotation, something I've been meaning to animate. A few things I noticed about them:

The 3 circles have two distinct, reflected arrangements, which do not transform by a 90 degree rotation, but a 180 degree flip.

When a single C-shaped arm is in the real plane, there are two more that stick off at plus/minus 120 degrees into the complex plane. This is when just one of the hemi-circle lobes sits in the 2-plane.

After studying the 3-prong like this, I think I've figured out what the other sections of mantis will look like. And, I feel that the notation needs to be more elaborate to define them, if it's supposed to include hyperplane intersections as well as the whole shape.

The mantis should have something analogous to the ballooned up C , which would look like what you'd get when taking the vertical column of tori from tiger cut, chopping them in half into a column of hemi-toruses, then connecting the two together with smaller hemi-toruses at the open ends, into one continuous loop. This will be one of the hemi-circle edges sticking into the 3-plane, with a torus embedded into it in a tiger-like fashion.

I do not believe the 3 torus-in-a-triangle arrangements of a mantis are transformable into each other by a 90 degree rotation, without passing by one of the single arm sections, as described above. I base this solely on how the common frame of the 3 hemi-circle edges intercept 2D. If and when a mantis gets defined and rendered into being, it will be utterly bizarre, compared to what we see with regular toratopes!

Anyways, that's what I think I've figured out, so far.
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Re: non-cubic toratopes?

Postby Teragon » Mon Jan 04, 2016 10:55 am

ICN5D wrote:Yes. Here's an animation of a=2 , b=2 , d=0 , t=0 , and animating c from 0 to 6.28. It's among the wilder things this equation makes. This is also the first gif I made using a screencapture program. The timing is a little tricky, but it's still much easier than manually one by one.

Image


Wow! Those rotatopes are crazy. What exactly is shown in this animation? A single rotation with a changing plane of rotation?
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Re: non-cubic toratopes?

Postby ICN5D » Mon Jan 04, 2016 6:29 pm

Well, firstly, this is just a wild concoction I made one day, as part of an experiment. I was trying to see if I could make a particular theoretical 4D torus called the Mantis. As you may have seen around this forum, there's a 4D torus called the tiger. This one will intercept a 3-plane as a vertical column of 2 toruses, which divide/merge when sliding along 4D. The Mantis will intercept as 3 toruses, inward-facing, in a triangular arrangement. This happens because when you set one of the 4 variables to zero, the cross-section equation becomes reducible, into perfect cube roots of a torus, in that exact arrangement.

My idea was first to approximate the Mantis, by defining a 4D tiger in a hyperspherical coordinate system (sphere x line), with the ability to adjust periodicity, and the general plane of rotation. The plane of rotation can be changed to morph from a 3-torus to a tiger (from a side-by-side pair of 2 toruses to the vertical column). If we multiply the periodicity, we'll get even numbers of inward-facing tori, like the Mantis (but not odd, which was the goal). There are two directions to change the periodicity, which will multiply the amount of inward-facing tori spaced over the surface of a sphere. That's what is shown in these pictures on this thread.

So, this animation is when both directions of periodicity are mult by 2, and rotating the general rotation plane by 360 degrees, continuously. It basically morphs between a 3-torus equivalent to a tiger equivalent. The output of the graph is pretty wild, and there's much more to it than what I made.
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Re: non-cubic toratopes?

Postby Teragon » Tue Jan 05, 2016 9:06 pm

I've read a bit about what a mantis is expected to be. It's a far more complicated to imagine than a tiger. If you know about one cross-section, you don't know about all the others automatically, let alone how the 3D shape that a 4D beeing would perceive looks like and how it transforms under rotations. Anyway cross-sections give a very limited view on 4D objects. My intuition tells me the odd toratopes are no more complicated than the even toratopes, it seems to be a problem arising from the derivation starting from the tiger. Theoretically it would be enough to stretch the whole object's angular dependence in the azimuth direction by a factor of 1.5, which is obviously not what the parameter "a" does. It's hard to figure out where the azimuth-dependence is, with all those rotations in the equation.

ICN5D wrote:So, this animation is when both directions of periodicity are mult by 2, and rotating the general rotation plane by 360 degrees, continuously. It basically morphs between a 3-torus equivalent to a tiger equivalent. The output of the graph is pretty wild, and there's much more to it than what I made.


Ok, so you're not changing the point of view, but morphing the object between tiger- and the 3-torus-configuration?
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Re: non-cubic toratopes?

Postby ICN5D » Fri Jan 08, 2016 5:24 am

Actually, I think I've figured out the mantis slices. The common geometry between a mantis and a 3-prong multitorus is a product of three half-disk edges, the tri-edge (the edges of a trigonal hosohedron), joined at two poles over the surface of a sphere. So, there should be some parallels in the slice morphings.

A 3-prong can be defined as a circle embedded into the tri-edge frame. A multi-ditorus is when you embed a torus into the tri-edge, in a similar orientation as the circle. A mantis is the other way to position a torus, when it gets embedded. The two different positions of the torus are the differences between the 3-torus and the tiger. The multi-ditorus and the mantis are analogous in this way.

After studying the rotating 2D slices of a 3-prong, you will see a finite number of distinct transformations that are unique to the symmetry of a tri-edge frame. That's the key idea behind interpreting the slices. Static images of slices on coordinate planes aren't that good of a representation. Too much handwaving is involved to make the picture clearer.

But, the dynamic rotation morph of a slice will, in fact, reveal far more information than you might expect. So long as we can indirectly interpret them correctly, to fill in the missing details. That's what I've been doing for a while. I'll probably animate the 3-prong as graphed in 2D someday, since it's interesting to see. I'm not yet sure how to manually 'build' the other proposed mantis slices, without drawing them. Maybe I can use CSG in POVray to do the trick. Describing all of these relations are best done visually.

Teragon wrote:Ok, so you're not changing the point of view, but morphing the object between tiger- and the 3-torus-configuration?


Yes, by rotating the stationary plane of rotation. This plane of rotation is the one defined in the construction process, when the starting torus was rotated into 4D around in a circle.


Theoretically it would be enough to stretch the whole object's angular dependence in the azimuth direction by a factor of 1.5, which is obviously not what the parameter "a" does. It's hard to figure out where the azimuth-dependence is, with all those rotations in the equation.


The problem with trying to use 1.5 this way, is from changing the coordinate system to approximate a shape. The periodicity in polar, cylindrical, and spherical only work with whole integers to make unbroken graphs of toruses.

There is only one rotation parameter in the equation, that rotates the 3D slice. All of the rest of the trig functions are for the stationary plane adjustment for tiger/3-torus interchange, and the spherical coordinate change.


EDIT : The periodicity in polar, cylindrical, and spherical only work with whole integers to make unbroken graphs of toruses
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