Names for the duals of the uniform polychora

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

Names for the duals of the uniform polychora

Postby username5243 » Sun Mar 26, 2017 9:56 am

I was just wondering: has anyone made a set of names for the duals of the convex uniform polychora? As in, actual names, not just calling them "dual of rectified tesseract" etc?
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Re: Names for the duals of the uniform polychora

Postby wendy » Sun Mar 26, 2017 10:31 am

By the wythoff symbol, these are the figures, and their duals. You insert the schlafli symbol of the base figure here.

x3x3o4o truncated 16choron o3o3m4m surtegmated 16choron
x4x3o4o truncated tesseract o4o3m3m surtegmated 16 choron

The dual comes by reversing the x/m. So x3x3o4o truncated 16ch is dual of m3m3o4o surtegmated tesseract.

oxoo rectated- oomo surtegmate
xxoo truncated oomm apiculated
oxxo bitruncated ommo bi-apiculated
xxxx omnitruncate mmmm vaniated.
xoox runcinated moom strombiated
xoxo cantelated omom twice surtegmated #
xxxo cantetruncad ommm apicusurtegmated
xxox cantetruncated momm #

# No name has been devised for this.
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Re: Names for the duals of the uniform polychora

Postby Klitzing » Mon Mar 27, 2017 6:46 am

wendy wrote:...
x3x3o4o truncated 16choron o3o3m4m surtegmated 16choron
x4x3o4o truncated tesseract o4o3m3m surtegmated 16 choron

The dual comes by reversing the x/m. So x3x3o4o truncated 16ch is dual of m3m3o4o surtegmated tesseract.

oxoo rectated- oomo surtegmate
xxoo truncated oomm apiculated
...

Kind of looks like several typos here:

> x4x3o4o truncated tesseract o4o3m3m surtegmated 16 choron
ought probably
x4x3o3o truncated tesseract o4o3m3m surtegmated tesseract
(or is it the other way round? i.e. first line tesseract, second 16choron?)

Cannot decide whether surtegmate or apiculate is the one to be used for mmoo resp. omoo
(this at least got inconsistently mixed up within your examples...)

--- rk
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Re: Names for the duals of the uniform polychora

Postby Mercurial, the Spectre » Tue Jun 20, 2017 1:50 pm

Hmm... if we can call them by their cell types and the number of cells... yeah...
For example,
the dual of the bitruncated 24-cell = tetradisphenoidal 288-cell (since it has 288 congruent tetragonal disphenoid cells) and also the intersection of two dual 24-cells, and
the dual of the grand antiprism = hexatetrago-tetrapentagohedral 100-cell (the hexatetrago-tetrapentagohedron, which is closely related to the dodecahedron, is the dual of the verf of the grand antiprism, basically a topological sphenocorona with the tetragon-shared edge length being phi). It would contain some of the 120-cell's vertices along with additional vertices, but I don't know for sure what it looks like.

You could simply drop out the naming convention (such as n-kis for dual truncation or join for dual rectification) when dealing with duals of uniform polychora, except when dealing with duals of duoprisms (duopyramids) and polyhedral prisms (catalan tegums or bipyramids). Simply list the cell type (often one type) and then number of cells within the polychoron.

Sorry if I posted this months later, I basically came up with a way of naming the 4D catalans. Maybe I'm the first one, for sure :D
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Re: Names for the duals of the uniform polychora

Postby wendy » Wed Jun 21, 2017 11:10 am

Welcome to the forum.

The names i made for them is based on a simple construction. You imagine a figure, like x3o4o3o, and then its dual o3o4o3x.

The figure is covered by a skin. Now imagine the dual increasing in size from zero. Eventually the vertices of o3o4o3x will start to raise peaks on the faces of x3o4o3o, or 'apiculate' it. This continues until the peaks are high enough that the sloping faces align by pairs. This is the surtegmate, or surface tegum of the triangle-margins of x3o4o3o, and the edges of the o3o4o3x.

As the o3o4o3x gets biger, its edges are now higher than the triangles of x3o4o3o, and the triangle-tegum just created, now divides along the line into three tetrahedra. This is the second apiculation, and it continues until the tetrahedra fall by threes around the triangles of o3o4o3x.

And so on.

It is the dual process of truncation by intersection. Given that the greek word 'prisma' means off-cut, as ye might make a prisms by cutting a hexagonal bar, the general intersection of figures could as much be called the 'prism-sum', as the tegum-sum represents the covering of intersecting frames by a skin.

The m3o4m3o and o3m4m3o both have 288 faces, and you run into the same number too many times in catalans. For example, m3m3m5o, m3m3o5m, m3o5m3m, and o3m3m5m, the respective duals of the figures with x in the same position, all have 7200 faces. Likewise, o3o3m5m and m3o3o5m both have 2400 faces.
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Re: Names for the duals of the uniform polychora

Postby username5243 » Wed Jun 21, 2017 11:47 am

Mercurial, the Spectre wrote:Hmm... if we can call them by their cell types and the number of cells... yeah...
For example,
the dual of the bitruncated 24-cell = tetradisphenoidal 288-cell (since it has 288 congruent tetragonal disphenoid cells) and also the intersection of two dual 24-cells, and
the dual of the grand antiprism = hexatetrago-tetrapentagohedral 100-cell (the hexatetrago-tetrapentagohedron, which is closely related to the dodecahedron, is the dual of the verf of the grand antiprism, basically a topological sphenocorona with the tetragon-shared edge length being phi). It would contain some of the 120-cell's vertices along with additional vertices, but I don't know for sure what it looks like.

You could simply drop out the naming convention (such as n-kis for dual truncation or join for dual rectification) when dealing with duals of uniform polychora, except when dealing with duals of duoprisms (duopyramids) and polyhedral prisms (catalan tegums or bipyramids). Simply list the cell type (often one type) and then number of cells within the polychoron.

Sorry if I posted this months later, I basically came up with a way of naming the 4D catalans. Maybe I'm the first one, for sure :D


Sad to say this still creates ambiguity.

Simple example: Take prit (= x4o3x3x) and proh (= x4x3o3x). Their duals both have 192 kite pyramid cells, and I don't know how to tell them apart in their names.

I think Klitzing posted more about the dual of gap in the "Tegum polytopes" thread (in addition to the dual of sadi).
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Re: Names for the duals of the uniform polychora

Postby Mercurial, the Spectre » Wed Jun 21, 2017 12:10 pm

wendy wrote:Welcome to the forum.

The names i made for them is based on a simple construction. You imagine a figure, like x3o4o3o, and then its dual o3o4o3x.

The figure is covered by a skin. Now imagine the dual increasing in size from zero. Eventually the vertices of o3o4o3x will start to raise peaks on the faces of x3o4o3o, or 'apiculate' it. This continues until the peaks are high enough that the sloping faces align by pairs. This is the surtegmate, or surface tegum of the triangle-margins of x3o4o3o, and the edges of the o3o4o3x.

As the o3o4o3x gets biger, its edges are now higher than the triangles of x3o4o3o, and the triangle-tegum just created, now divides along the line into three tetrahedra. This is the second apiculation, and it continues until the tetrahedra fall by threes around the triangles of o3o4o3x.

And so on.

It is the dual process of truncation by intersection. Given that the greek word 'prisma' means off-cut, as ye might make a prisms by cutting a hexagonal bar, the general intersection of figures could as much be called the 'prism-sum', as the tegum-sum represents the covering of intersecting frames by a skin.

The m3o4m3o and o3m4m3o both have 288 faces, and you run into the same number too many times in catalans. For example, m3m3m5o, m3m3o5m, m3o5m3m, and o3m3m5m, the respective duals of the figures with x in the same position, all have 7200 faces. Likewise, o3o3m5m and m3o3o5m both have 2400 faces.


Wendy and username5243, I see.
The problem of designing duals of truncates, rectates, and bitruncates is that they are all formed by apiculation of cells with varying heights. Height plays an important role, not just the vertex layout. The height, for example may be not enough (apiculation), exactly where the convex hull forms bipyramidal cells (surtegmation), or higher than that (dual bitruncation). Due to having no obvious way of generalizing the vertex layout of other dual uniforms, other than these, we end up having no official names for dual cantellation, runcination, etc (in 4D and above). You'd end up having names that even then, won't have patterns for higher dimensions (unless you find one) and it quickly gets complicated.

The problem of different dual uniforms with congruent cell numbers is handled by the fact that they have different cell types for each one of them. For example, both the dual sirco and dual snic have 24 faces, yet they are called deltoidal icositetrahedron and pentagonal icositetrahedron, which follows my dual naming convention. In 3D there are names for dual cantellates such as ortho, but it simply doesn't extend to runcination.

For the prism-sums and tegum-sums, we'd have to find a formula for what polytopes to use in order to get the dual uniform of choice. Finding compounds that are inscribed within dual uniforms is a hard process, let alone fully name those individual polytopes.

So it still stands. Having figured this out, I think that the naming convention used by the dual snic and the dual sirco fits the 4D catalans the best,

For Wendy's examples,
m3o4m3o's has isosceles-triangular bipyramidal cells and is called, in my own terms, an isotribipyramidal 288-cell.
o3m4m3o has tetragonal disphenoid cells and was described earlier.

Eventually my naming convention fails for the dual cantitruncates and dual runcitruncates, but I have a solution. Take their duals and see from which regular is derived from. Add that, and voila!

m3m3m5o and o3m3m5m have sphenoids (Cs-symmetric tetrahedron) for cells, but can be termed as sphenoidal hexacosichoric 7200-cell and sphenoidal hecatonicosachoric 7200-cell. One is derived from the 600-cell, and the other, from the 120-cell.

For this example both the runcicantellate and runcitruncate of one of either the 120- or 600-cell cover both x3x3o5x and x3o5x3x. Generally, for this convention, runcitruncation is preferred. Runcicantellation is the same as the runcitruncated dual only in 4D.

m3m3o5m and m3o5m3m have kite pyramids for cells, but can be termed as kite-pyramidal hexacosichoric 7200-cell and kite-pyramidal hecatonicosachoric 7200-cell.

Or try this. I could invent dutruncates, ducantellates, or duruncitruncates for duals of truncates, cantellates, or runcitruncates. That works, and is simple as well! :D
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Re: Names for the duals of the uniform polychora

Postby polychoronlover » Wed Jun 28, 2017 5:37 am

As for the names of uniform polychoron duals, I could find the following in the OBSA list:
  • tibbid - triangular-bipyramidal decachoron (dual of rap)
  • tibbit - triangular-bipyramidal triacontidichoron (dual of rit)
  • tabene - trigonal-bipyramidal enneacontahexachoron (dual of rico)
  • pibhaki - pentagonal-bipyramidal heptacosiicosachoron (dual of rox)
  • tibbic - triangular-bipyramidal chilliadiacosichoron (dual of rahi)

I don't know if there are acronyms for any others.
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