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
...
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
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.
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