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:
by cells: pap tet
gap 20300
)
Confer
uniform relative:
ex  
External
links
hedrondude   wikipedia   WikiChoron     quickfur

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.


Incidence matrix

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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