Tegum polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: Tegum polytopes

Postby wendy » Sun Jun 18, 2017 9:14 am

To imagine the groups here.

Take an octahedron from x3o4o3o. This is divided by three mirrors in the planes perpendicular to the diagonals. So you divide the octahedron into 8 triangle-pyramids. These are the cells of the symmetry o3o3o o3*b. There is a 60-60-60 triangle, and three 90-45-45 ones. We take the 60-60-60 triangle, and can apply different symmetries. The order is 8 * 24 = 192.

Dividing the triangle, so A = B gives the symmetry [3,3,4] or the tesseract symmetry. This can be done in three different ways, giving the compound of three tesseract in x3o4o3o. The order is 2 * 192 = 384.

The second symmetry of the triangle is to rotate A to B to C to A. This converts the octahedra into pyritohedral symmetry 3*2. The relevant symmetry is 3/*/2, that is, a rotated triangle with three legs, like the manx thing. This is the symmetry of the s3s4o3o. The o3o bit are the preserved mirrors, the s3s bit are the rotational-only bit. This is [3,4,3+].

The third symmetry is full permutations of A,B,C, which leads to a subgroup of 6, or a group of 1152. This is the full ico symmetry [3,4,3] by Coxeter.
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Re: Tegum polytopes

Postby Klitzing » Sun Jun 18, 2017 1:54 pm

Okay then, the hex symmetry is [31,1,1].
The 3-hex compound has hull = ico and thus has full symmetry [3,4,3]. That is, it ought cycle AND reflect.
The ex has symmetry [3,3,5].
But the common subsymmetry of [3,3,5] and [3,4,3] is NOT [3,4,3] itself, even so a f-scaled ico is vertex inscribable into ex, it rather is [3,4,3]+ only.
Therefore it IS clear that sadi = ico-dim. ex has a symmetry which just cyclically changes the legs of [31,1,1].

This is like in 3D too: [3,5] and [3,4] have for common subsymmetry [3,4]+, the pyritohedral one, even so a cube is vertex-inscribable into doe.

 

But now onto the other part of my mail:
The according hi representation within [31,1,1] symmetry still seems to be open.

--- rk
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Re: Tegum polytopes

Postby student91 » Sun Jun 18, 2017 7:05 pm

Klitzing wrote:[...]But now onto the other part of my mail:
The according hi representation within [31,1,1] symmetry still seems to be open.

--- rk

That was the main point of this post (click on the [31,1,1]-link, then look at number 14, in binary 14 is 1110, which translates to o3o3o5x)
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Re: Tegum polytopes

Postby Klitzing » Sun Jun 18, 2017 7:52 pm

Ah yes, forgot about that one.
Thus you tell:
hi = Coo|foB|xoF|f-3-ooo|ooo|fff|x-3-oCo|oBf|oFx|f  *b3-ooC|Bfo|Fxo|f-&#zx
where C = 2f+2x, B = 2f+x, F = f+x, and f : x=1.618 : 1.

Great, thanx!
--- rk
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Re: Tegum polytopes

Postby Klitzing » Tue Jun 20, 2017 4:03 pm

Fiddled it out, finally! :nod: :] 8)

This is, how quidex could be designed directly bottom-up as tegum sum, rather than the top-down description so far known only (eg. as dual of sadi, as quatri-ico-diminished ex, as ico-stellated hi):
Code: Select all
ooV|xoF|o-3-ooo|ooo|F-3-Voo|Fxo|o *b3-oVo|oFx|o-&#z(v,f,F)   → existing heights=0
                                                               V = 2f = 3.236067977
                                                               F = ff = 2.618033989
                                                               v = 1/f = 0.618033989
                                                               f = 2 cos(36°) = 1.618033989

o.. ... .-3-o.. ... .-3-o.. ... . *b3-o.. ... .            | 8 * *  *  *  *  * |  4  0  0  0  0  0  0 |  6  0  0  0  0  0 |  4  0  0
.o. ... .-3-.o. ... .-3-.o. ... . *b3-.o. ... .            | * 8 *  *  *  *  * |  0  4  0  0  0  0  0 |  0  6  0  0  0  0 |  0  4  0
..o ... .-3-..o ... .-3-..o ... . *b3-..o ... .            | * * 8  *  *  *  * |  0  0  4  0  0  0  0 |  0  0  6  0  0  0 |  0  0  4
... o.. .-3-... o.. .-3-... o.. . *b3-... o.. .            | * * * 32  *  *  * |  1  0  0  3  0  0  0 |  3  0  0  3  0  0 |  3  0  1
... .o. .-3-... .o. .-3-... .o. . *b3-... .o. .            | * * *  * 32  *  * |  0  1  0  0  3  0  0 |  0  3  0  0  3  0 |  1  3  0
... ..o .-3-... ..o .-3-... ..o . *b3-... ..o .            | * * *  *  * 32  * |  0  0  1  0  0  3  0 |  0  0  3  0  0  3 |  0  1  3
... ... o-3-... ... o-3-... ... o *b3-... ... o            | * * *  *  *  * 24 |  0  0  0  4  4  4  8 |  2  2  2  8  8  8 |  4  4  4
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. o.. .-3-o.. o.. .-3-o.. o.. . *b3-o.. o.. .-&#v        | 1 0 0  1  0  0  0 | 32  *  *  *  *  *  * |  3  0  0  0  0  0 |  3  0  0  v-edges
.o. .o. .-3-.o. .o. .-3-.o. .o. . *b3-.o. .o. .-&#v        | 0 1 0  0  1  0  0 |  * 32  *  *  *  *  * |  0  3  0  0  0  0 |  0  3  0  v-edges
..o ..o .-3-..o ..o .-3-..o ..o . *b3-..o ..o .-&#v        | 0 0 1  0  0  1  0 |  *  * 32  *  *  *  * |  0  0  3  0  0  0 |  0  0  3  v-edges
... o.. o-3-... o.. o-3-... o.. o *b3-... o.. o-&#f        | 0 0 0  1  0  0  1 |  *  *  * 96  *  *  * |  1  0  0  2  0  0 |  2  0  1  f-edges
... .o. o-3-... .o. o-3-... .o. o *b3-... .o. o-&#f        | 0 0 0  0  1  0  1 |  *  *  *  * 96  *  * |  0  1  0  0  2  0 |  1  2  0  f-edges
... ..o o-3-... ..o o-3-... ..o o *b3-... ..o o-&#f        | 0 0 0  0  0  1  1 |  *  *  *  *  * 96  * |  0  0  1  0  0  2 |  0  1  2  f-edges
... ... .   ... ... F   ... ... .     ... ... .            | 0 0 0  0  0  0  2 |  *  *  *  *  *  * 96 |  0  0  0  1  1  1 |  1  1  1  F-edges
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. x.. o   ... ... .   ... ... .     ... ... .-&#(v,f)t   | 1 0 0  2  0  0  1 |  2  0  0  2  0  0  0 | 48  *  *  *  *  * |  2  0  0  kite
... ... .   ... ... .   .o. .x. o     ... ... .-&#(v,f)t   | 0 1 0  0  2  0  1 |  0  2  0  0  2  0  0 |  * 48  *  *  *  * |  0  2  0  kite
... ... .   ... ... .   ... ... .     ..o ..x o-&#(v,f)t   | 0 0 1  0  0  2  1 |  0  0  2  0  0  2  0 |  *  * 48  *  *  * |  0  0  2  kite
... ... .   ... o.. F   ... ... .     ... ... .-&#f        | 0 0 0  1  0  0  2 |  0  0  0  2  0  0  1 |  *  *  * 96  *  * |  1  0  1  golden triangle
... ... .   ... .o. F   ... ... .     ... ... .-&#f        | 0 0 0  0  1  0  2 |  0  0  0  0  2  0  1 |  *  *  *  * 96  * |  1  1  0  golden triangle
... ... .   ... ..o F   ... ... .     ... ... .-&#f        | 0 0 0  0  0  1  2 |  0  0  0  0  0  2  1 |  *  *  *  *  * 96 |  0  1  1  golden triangle
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. xo. o-3-o.. oo. F   ... ... .     ... ... .-&#(v,f,f)t | 1 0 0  3  1  0  3 |  3  0  0  6  3  0  3 |  3  0  0  3  3  0 | 32  *  *  tower a-d-g-e
... ... .   .o. .oo F-3-.o. .xo o     ... ... .-&#(v,f,f)t | 0 1 0  0  3  1  3 |  0  3  0  0  6  3  3 |  0  3  0  0  3  3 |  * 32  *  tower b-e-g-f
... ... .   ..o o.o F   ... ... . *b3-..o o.x o-&#(v,f,f)t | 0 0 1  1  0  3  3 |  0  0  3  3  0  6  3 |  0  0  3  3  0  3 |  *  * 32  tower c-f-g-d

Obviously this incidence matrix shows up the same cyclical symmetry of the legs of this symmetry representation as in the according representation sadi does. That is, both are [3,4,3]+ symmetrical figures, which is the common subgroup of [3,3,5] and [3,4,3].

The first 6 vertex types display tetrahedral vertex figures, the 7th then is a dodecehardal one. The first 3 face types are a kite, which happens to be nothing but a mono-stellated regular pentagon. The other 3 then are a large obtuse golden triangle, which happens to be a bistellated regular pentagon. The cells then are all tri-trigonally diminished icosahedra, or, conversely described, a mono-trigonally stellated dodecahedron. A picture of that cell looks like this:
Image

Quidex is the one but last member of that multi-ico-diminishing of ex:
  0-idex = ex,
  1-idex = sadi,
  2-idex = bidex (noble, cells being teddies),
  3-idex = tridex,
  4-idex = quidex,
  5-idex = hi.
Moreover we have: k-idex is dual to (5-k)-idex ! (More details on that can be found here (scroll a bit down).

--- rk
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