Polychoral Hypertorus

[Hodge8]

I have been searching for any work done on a polychoron equivalent of the hypertorus, and I have found nothing.

I have created such an object:





More details on my web-site:
https://sites.google.com/site/polychoralhypertorus/

I would be grateful for any comments, and I wonder what (if any) other work has in fact been done in this area.

Fixed your images. ~Keiji

[Klitzing]

First of all, I wanna tell you that your Images semm not to come through (at least with me).
(Might be that it is based upon the mixture of http and https.)

The first picture at your website then can be interpreted as if you were after a convex segmentoteron, using a (qyrated) square atop a tesseract. This one could even given a lace prism notation: xo4ox ox ox&#x. - The height of that segemtoteron then could be calculated as sqrt[(sqrt(2)-1)/2] = 0.45509.

Btw. the second picture kind of looks wrong: Those square corners are not connected to the outermost cube.

But your element description at the bottom of your page seem that you were after a polychoron rather than a polyteron?

--- rk

[Hodge8]

Thanks for your comment, Klizing.

Sorry about the images; I do not know the reason for that.

the second picture kind of looks wrong

Here are pictures with the cells shown (also added to the web site):

But your element description at the bottom of your page seem that you were after a polychoron rather than a polyteron?

Yes, definitely aiming at a polychoron (numers of elements ammended; 12 tets and 12 pyramids).

[Klitzing]

Now I found that even the first image was wrong: the 4 edges from the small top square down to the small bottom square are missing.

From the polyteron, I described, which surely is uniform, it then follows, that your mere polychoral Interpretation cannot be uniform. The former lacing edges have to get shortened in the projection.

Here is in detail what my 5D interpretation would be:

Code: Select all

xo4ox oo4ox&#x
({4} || gyro tes)

o.4o. o. o.    | 4  * | 2  8  0  0 | 1  8  4  8 0  0 0 | 4  8  4 2 0 0 | 4 2 1 0
.o4.o .o .o    | * 16 | 0  2  2  2 | 0  1  2  4 1  4 1 | 1  2  4 2 2 2 | 2 1 2 1
---------------+------+------------+-------------------+---------------+--------
x. .. .. ..    | 2  0 | 4  *  *  * | 1  4  0  0 0  0 0 | 4  4  0 0 0 0 | 4 1 0 0
oo4oo oo4oo&#x | 1  1 | * 32  *  * | 0  1  1  2 0  0 0 | 1  2  2 1 0 0 | 2 1 1 0
.. .x .. ..    | 0  2 | *  * 16  * | 0  0  1  0 1  2 0 | 1  0  2 0 2 1 | 2 0 1 1
.. .. .. .x    | 0  2 | *  *  * 16 | 0  0  0  2 0  2 1 | 0  1  2 2 1 2 | 1 1 2 1
---------------+------+------------+-------------------+---------------+--------
x.4o. .. ..    | 4  0 | 4  0  0  0 | 1  *  *  * *  * * | 4  0  0 0 0 0 | 4 0 0 0
xo .. .. ..&#x | 2  1 | 1  2  0  0 | * 16  *  * *  * * | 1  2  0 0 0 0 | 2 1 0 0
.. ox .. ..&#x | 1  2 | 0  2  1  0 | *  * 16  * *  * * | 1  0  2 0 0 0 | 2 0 1 0
.. .. .. ox&#x | 1  2 | 0  2  0  1 | *  *  * 32 *  * * | 0  1  1 1 0 0 | 1 1 1 0
.o4.x .. ..    | 0  4 | 0  0  4  0 | *  *  *  * 4  * * | 1  0  0 0 2 0 | 2 0 0 1
.. .x .. .x    | 0  4 | 0  0  2  2 | *  *  *  * * 16 * | 0  0  1 0 1 1 | 1 0 1 1
.. .. .o4.x    | 0  4 | 0  0  0  4 | *  *  *  * *  * 4 | 0  0  0 2 0 2 | 0 1 2 1
---------------+------+------------+-------------------+---------------+--------
xo4ox .. ..    | 4  4 | 4  8  4  0 | 1  4  4  0 1  0 0 | 4  *  * * * * | 2 0 0 0   squap
xo .. .. ox&#x | 2  2 | 1  4  0  1 | 0  2  0  2 0  0 0 | * 16  * * * * | 1 1 0 0   tet
.. ox .. ox&#x | 1  4 | 0  4  2  2 | 0  0  2  2 0  1 0 | *  * 16 * * * | 1 0 1 0   squippy
.. .. oo4ox&#x | 1  4 | 0  4  0  4 | 0  0  0  4 0  0 1 | *  *  * 8 * * | 0 1 1 0   squippy
.o4.x .. .x    | 0  8 | 0  0  8  4 | 0  0  0  0 2  4 0 | *  *  * * 4 * | 1 0 0 1   cube
.. .x .o4.x    | 0  8 | 0  0  4  8 | 0  0  0  0 0  4 2 | *  *  * * * 4 | 0 0 1 1   cube
---------------+------+------------+-------------------+---------------+--------
xo4ox .. ox&#x | 4  8 | 4 16  8  4 | 1  8  8  8 2  4 0 | 2  4  4 0 1 0 | 4 * * *   {4}||gyro cube
xo .. oo4ox&#x | 2  4 | 1  8  0  4 | 0  4  0  8 0  0 1 | 0  4  0 2 0 0 | * 4 * *   squippypy
.. ox oo4ox&#x | 1  8 | 0  8  4  8 | 0  0  4  8 0  4 2 | 0  0  4 2 0 1 | * * 4 *   cubpy
.o4.x .o4.x    | 0 16 | 0  0 16 16 | 0  0  0  0 4 16 4 | 0  0  0 0 4 4 | * * * 1   tes


In fact, that one could be derived as follows: take the tesseractic antiprism, i.e. the tesseract || hexadecachoron. Next apply a (5D) bidiminishing at two opposite vertices of the hexadecachoron. Thus you remain with tesseract || octahedron. Next apply a further (5D) bidiminishing at 2 opposite vertices of the octahedron, and you would be left with what I was providing. - An alternate, but harder to be visualized creational process for this solid would be to build the duoprism of a square and a square-antiprism.

--- rk

[wendy]

A good deal of work has been done on polychoral torii. It's quite interesting.

There are toratopes, to which a whole subforum viewforum.php?f=24 is given to. There are things called tigers, which are interesting too. Keiji has done a good deal of work here.

There's a comb product, which is the 'repetition of surfaces'. One use of this is to produce 'polytope torus skins'.

In four dimensions, the skin of a polytope is 3d. So if you take a 2d skin (say a dodecahedron), and a 1d skin (say a decagon), you can multiply these in cartesian product to get a 3d skin which covers two different torii.

Suppose you take a hollow dodecahedron (ie its skin), and take a prism-product of a ten-link chain (say a decagon-skin). You then have a kind of pipe in 4d, like a dodecahedral prism stack, ten high, with the dodecahedra blown out: in effect, you have 120 pentagonal prisms.

You can join top to bottom in the 'hose' fashion, by bending the tower around in a C fashion, and then link top to bottom. What's inside the pipe will end up inside the torus.

You can join top to bottom in the 'sock' fashion. What this does, is you make the top bigger, and roll it down the pipe, rather as you might roll a sock down your leg. When the top gets to the bottom, you then join it. What's inside the sock ends up outside the torus.

You can make a 3d skin out of three 1D skins, so what you get is a torus whose section is a torus. You take a 3d rectanguloid of rubbery cubes, and you roll it into a cylinder, and then the top to bottom into a torus. So you have a torus-shaped pipe. Which ever way you do it, sock or hose, you end up with the same figure.

[Hodge8]

Thanks for your comments, Wendy, and for pointing me in the right direction. Seems I posted in the wrong sub-forum.
In the mean time, I have found (I think) an easy way to make polychoral hypertorii in Stella4D. Simply open a polyhedral torus and create a 4D prism on it.