- general polytopal classes:
- isogonal (for n = m only)
links
| Acronym | ... |
| Name | (n,m)-double antiprismatoid |
| Circumradius | ... |
| Face vector | 4nm, 20nm, 28nm+2n+2m, 12nm+2n+2m |
| Especially | gap (n = m = 5) padiap (n = m = 5/3) |
| Confer |
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The general building rule for (n,m)-double antiprismatoids is: construct (in 4D) 2 perpendicular rings of 2n m-gonal antiprisms, respectively of 2m n-gonal antiprisms. Then connect the triangles of the one ring to the vertices of the other (and vice versa), and further more connect the lacing edges of the antiprism of one to the nearest similar edges of the other. Combinatorically all this filling stuff would be tets. In fact, at every antiprismal triangle adjoins one tet(adjacent), as shown to the right. Two neighbouring such tet(adjacent) share a common lacing triangle, which extends the antiprismal base (n-gon or m-gon respectively). Thus the so far open surface topologically looks like a toroidal surface with attached tetragonal pyramids. Atop each still "visible" toroidal edge, i.e. the lacing ones of the first ring of antiprisms, now further tets can be introduced. These then are the tet(isolated). Their opposite edge then pairwise connects the tips of these "tetragonal pyramids", thus the new open surface is just the former in reverse, giving space for further topological "tetragonal pyramids", i.e. pairs of tet(adjacent). But the adjoining triangles now will be orthogonal to those of the former layer. And thus the so far constructed partial complex can be closed by a second orthogonal ring of antiprisms. – But for general n,m the total figure cannot be made unit edged only. (Additionally, a single vertex orbit for sure is possible only when n=m.) Uniform exceptions occur only for n=m=5 (gap) and n=m=5/3 (padiap).
Note that there is a crude mixture of gap and padiap too, which kind of is flattened somehow. In fact, gudap again uses a ring 10 paps and an orthogonal ring of 10 starps, but there the vertex set of both rings coincides. Accordingly the remaining space can be filled by a single type of 50 tets only.
When obtaining this polychoron as the vertex-alternation of the (2n,2m)-ditetragoltriate then besides of the snubbed cells each also the according vertex figures come in. Those there happen to be triangular bipyramids. These in here rather have to be divided equatorially into pairs of pyramids instead. This is how the faces, below marked by *), come in.
Note also that those tet(isolated) in fact feature as digonal antiprisms and their two base edges would connect to lacing edges of the respective antiptisms of either ring each.
2nm * | 2 4 2 2 0 0 | 1 6 2 8 4 1 0 0 | 2 6 4 2 0 * 2nm | 0 0 2 2 4 2 | 0 0 1 4 8 2 6 1 | 0 2 4 6 2 --------+-------------------------+-------------------------------+------------------ 2 0 | 2nm * * * * * | 1 2 1 0 0 0 0 0 | 2 2 0 0 0 edges of n-gons 2 0 | * 4nm * * * * | 0 2 0 2 0 0 0 0 | 1 2 1 0 0 lacings of n-aps 1 1 | * * 4nm * * * | 0 0 1 2 2 0 0 0 | 0 2 2 1 0 type 1, extending the n-gons 1 1 | * * * 4nm * * | 0 0 0 2 2 1 0 0 | 0 1 2 2 0 type 2, extending the m-gons 0 2 | * * * * 4nm * | 0 0 0 0 2 0 2 0 | 0 0 1 2 1 lacings of m-aps 0 2 | * * * * * 2nm | 0 0 0 0 0 1 2 1 | 0 0 0 2 2 edges of m-gons --------+-------------------------+-------------------------------+------------------ n 0 | n 0 0 0 0 0 | 2m * * * * * * * | 2 0 0 0 0 n-gon 3 0 | 1 2 0 0 0 0 | * 4nm * * * * * * | 1 1 0 0 0 2 1 | 1 0 2 0 0 0 | * * 2nm * * * * * | 0 2 0 0 0 *) 2 1 | 0 1 1 1 0 0 | * * * 8nm * * * * | 0 1 1 0 0 1 2 | 0 0 1 1 1 0 | * * * * 8nm * * * | 0 0 1 1 0 1 2 | 0 0 0 2 0 1 | * * * * * 2nm * * | 0 0 0 2 0 *) 0 3 | 0 0 0 0 2 1 | * * * * * * 4nm * | 0 0 0 1 1 0 m | 0 0 0 0 0 m | * * * * * * * 2n | 0 0 0 0 2 m-gon --------+-------------------------+-------------------------------+------------------ 2n 0 | 2n 2n 0 0 0 0 | 2 2n 0 0 0 0 0 0 | 2m * * * * n-ap 3 1 | 1 2 2 1 0 0 | 0 1 1 2 0 0 0 0 | * 4nm * * * pyramid upon lacing triangles of n-ap (adjacent tet variant) 2 2 | 0 1 2 2 1 0 | 0 0 0 2 2 0 0 0 | * * 4nm * * 2-ap (isolated tet variant) 1 3 | 0 0 1 2 2 1 | 0 0 0 0 2 1 1 0 | * * * 4nm * pyramid upon lacing triangles of m-ap (adjacent tet variant) 0 2m | 0 0 0 0 2m 2m | 0 0 0 0 0 0 2m 2 | * * * * 2n m-ap starting figure: (2n,2m)-ditetragoltriate
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