Acronym | gap | ||||||
Name |
great antiprism, grand antiprism, pentagonal double antiprismoid (but not: great duoantiprism) | ||||||
Cross sections |
© | ||||||
Circumradius | (1+sqrt(5))/2 = 1.618034 | ||||||
Vertex figure | is a bitrapezoidal truncated icosahedron | ||||||
Lace city in approx. ASCII-art |
o5x x5o x5o f5o o5f o5f o5x o5x f5o f5o o5f o5f x5o x5o f5o f5o o5f o5x o5x x5o | ||||||
General of army | (is itself convex) | ||||||
Colonel of regiment |
(is itself locally convex
– uniform polychoral members:
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Confer | |||||||
External links |
The great antiprism (gap) is a faceting of the hexacosachoron. The antiprism cells (pap) form 2 rings of 10 each.
As abstract polychoron gap is isomorphic to padiap, thereby replacing the pentagons by pentagrams, resp. replacing pap by starp. Also the vertex figures turn from asymmetric facetings of ike into such facetings of gike.
The general building rule for double antiprimoids would be: 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. 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 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 50 tets only.
50 * | 2 4 2 2 0 0 | 1 6 2 8 4 1 0 0 | 2 6 2 2 0 (vf) bitrapezoidal truncated ike * 50 | 0 0 2 2 4 2 | 0 0 1 4 8 2 6 1 | 0 2 2 6 2 (vf) bitrapezoidal truncated ike ------+-----------------------+-----------------------------+------------------ 2 0 | 50 * * * * * | 1 2 1 0 0 0 0 0 | 2 2 0 0 0 5gon edges 2 0 | * 100 * * * * | 0 2 0 2 0 0 0 0 | 1 2 1 0 0 lateral ap edges 1 1 | * * 100 * * * | 0 0 1 2 2 0 0 0 | 0 2 2 1 0 joining edges, extending 1st 5gons 1 1 | * * * 100 * * | 0 0 0 2 2 1 0 0 | 0 1 2 2 0 joining edges, extending 2nd 5gons 0 2 | * * * * 100 * | 0 0 0 0 2 0 2 0 | 0 0 1 2 1 lateral ap edges 0 2 | * * * * * 50 | 0 0 0 0 0 1 2 1 | 0 0 0 2 2 5gon edges ------+-----------------------+-----------------------------+------------------ 5 0 | 5 0 0 0 0 0 | 10 * * * * * * * | 2 0 0 0 0 3 0 | 1 2 0 0 0 0 | * 100 * * * * * * | 1 1 0 0 0 2 1 | 1 0 2 0 0 0 | * * 50 * * * * * | 0 2 0 0 0 extending 1st 5gons 2 1 | 0 1 1 1 0 0 | * * * 200 * * * * | 0 1 1 0 0 adjoined to lateral ap edges 1 2 | 0 0 1 1 1 0 | * * * * 200 * * * | 0 0 1 1 0 adjoined to lateral ap edges 1 2 | 0 0 0 2 0 1 | * * * * * 50 * * | 0 0 0 2 0 extending 2nd 5gons 0 3 | 0 0 0 0 2 1 | * * * * * * 100 * | 0 0 0 1 1 0 5 | 0 0 0 0 0 5 | * * * * * * * 10 | 0 0 0 0 2 ------+-----------------------+-----------------------------+------------------ 10 0 | 10 10 0 0 0 0 | 2 10 0 0 0 0 0 0 | 10 * * * * pap 3 1 | 1 2 2 1 0 0 | 0 1 1 2 0 0 0 0 | * 100 * * * tet(adjacent) 2 2 | 0 1 2 2 1 0 | 0 0 0 2 2 0 0 0 | * * 100 * * tet(isolated) 1 3 | 0 0 1 2 2 1 | 0 0 0 0 2 1 1 0 | * * * 100 * tet(adjacent) 0 10 | 0 0 0 0 10 10 | 0 0 0 0 0 0 10 2 | * * * * 10 pap or: 100 | 2 4 4 | 1 6 3 12 | 2 8 4 (vf) bitrapezoidal truncated ike ----+-------------+----------------+----------- 2 | 100 * * | 1 2 1 0 | 2 2 0 5gon edges 2 | * 200 * | 0 2 0 2 | 1 2 1 lateral ap edges 2 | * * 200 | 0 0 1 4 | 0 3 2 edges joining the ap rings ----+-------------+----------------+----------- 5 | 5 0 0 | 20 * * * | 2 0 0 ap-ap 5gon 3 | 1 2 0 | * 200 * * | 1 1 0 ap-tet(adj) trig 3 | 1 0 2 | * * 100 * | 0 2 0 tet(adj)-tet(adj) trig (extending the 5gon) 3 | 0 1 2 | * * * 400 | 0 1 1 tet(adj)-tet(isol) trig ----+-------------+----------------+----------- 10 | 10 10 0 | 2 10 0 0 | 20 * * 5ap (pap) 4 | 1 2 3 | 0 1 1 2 | * 200 * tet(adjacent) = ap-tet(adj) 3pyr 4 | 0 2 4 | 0 0 0 4 | * * 100 tet(isolated) = lateral-ap-edges 2ap
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