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
)
Dihedral angles
  • at {3} between tet and tet:   arccos[-(1+3 sqrt(5))/8] = 164.477512°
  • at {3} between pap and tet:   arccos[-sqrt(5/8)] = 142.238756°
  • at {5} between pap and pap:   144°
Confer
uniform relative:
ex  
related CRFs:
bicypdex  
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 antiprismoids 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. 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). 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 50 tets only.

Gap allows for an bi-cyclopenta-augmentation of any alternate paps. Thereby the introduced peppiess pairwise become corealmic with the remaining paps and thus will join into 10 tip-to-tip ikes. This bi-cyclopentaaugmented gap conversely can be considered as a bi-cyclopentadiminished ex (bicypdex).


Incidence matrix

50  * |  2   4   2   2   0  0 |  1   6  2   8   4  1   0  0 |  2   6   4   2  0  (vf) bitrapezoidal truncated ike
 * 50 |  0   0   2   2   4  2 |  0   0  1   4   8  2   6  1 |  0   2   4   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

colored: r=red, y=yellow, g=green, b=blue
(basically: coloring the antiprisms & implied coloring of tetrahedra)

50  * |  2  1  1  1  1  1  1  1  1  0  0  0  0  0 |  1  3  3  2  2  2  2  2  1  1  1  1  1  0  0  0 | 1 1  3  3  1  1  1  1  1  1 0 0
 * 50 |  0  0  0  0  0  1  1  1  1  1  1  1  1  2 |  0  0  0  1  1  1  1  1  2  2  2  2  2  3  3  1 | 0 0  1  1  1  1  1  1  3  3 1 1
------+-------------------------------------------+-------------------------------------------------+--------------------------------
 2  0 | 50  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0 | 1 1  1  1  0  0  0  0  0  0 0 0  ry 5gon edges
 2  0 |  * 25  *  *  *  *  *  *  *  *  *  *  *  * |  0  2  0  0  2  0  0  0  0  0  0  0  0  0  0  0 | 1 0  2  0  1  0  0  0  0  0 0 0  r/g lateral ap edges
 2  0 |  *  * 25  *  *  *  *  *  *  *  *  *  *  * |  0  2  0  0  0  2  0  0  0  0  0  0  0  0  0  0 | 1 0  2  0  0  1  0  0  0  0 0 0  r/b lateral ap edges
 2  0 |  *  *  * 25  *  *  *  *  *  *  *  *  *  * |  0  0  2  0  0  0  2  0  0  0  0  0  0  0  0  0 | 0 1  0  2  0  0  1  0  0  0 0 0  y/g lateral ap edges
 2  0 |  *  *  *  * 25  *  *  *  *  *  *  *  *  * |  0  0  2  0  0  0  0  2  0  0  0  0  0  0  0  0 | 0 1  0  2  0  0  0  1  0  0 0 0  y/b lateral ap edges
 1  1 |  *  *  *  *  * 50  *  *  *  *  *  *  *  * |  0  0  0  1  1  0  1  0  1  0  1  0  0  0  0  0 | 0 0  1  1  1  0  1  0  1  0 0 0  ry/g joining edges, extending ry 5gons
 1  1 |  *  *  *  *  *  * 50  *  *  *  *  *  *  * |  0  0  0  1  0  1  0  1  0  1  0  1  0  0  0  0 | 0 0  1  1  0  1  0  1  0  1 0 0  ry/b joining edges, extending ry 5gons
 1  1 |  *  *  *  *  *  *  * 50  *  *  *  *  *  * |  0  0  0  0  1  1  0  0  1  1  0  0  1  0  0  0 | 0 0  1  0  1  1  0  0  1  1 0 0  r/gb joining edges, extending gb 5gons
 1  1 |  *  *  *  *  *  *  *  * 50  *  *  *  *  * |  0  0  0  0  0  0  1  1  0  0  1  1  1  0  0  0 | 0 0  0  1  0  0  1  1  1  1 0 0  y/gb joining edges, extending gb 5gons
 0  2 |  *  *  *  *  *  *  *  *  * 25  *  *  *  * |  0  0  0  0  0  0  0  0  2  0  0  0  0  2  0  0 | 0 0  0  0  1  0  0  0  2  0 1 0  r/g lateral ap edges
 0  2 |  *  *  *  *  *  *  *  *  *  * 25  *  *  * |  0  0  0  0  0  0  0  0  0  2  0  0  0  0  2  0 | 0 0  0  0  0  1  0  0  0  2 0 1  r/b lateral ap edges
 0  2 |  *  *  *  *  *  *  *  *  *  *  * 25  *  * |  0  0  0  0  0  0  0  0  0  0  2  0  0  2  0  0 | 0 0  0  0  0  0  1  0  2  0 1 0  y/g lateral ap edges
 0  2 |  *  *  *  *  *  *  *  *  *  *  *  * 25  * |  0  0  0  0  0  0  0  0  0  0  0  2  0  0  2  0 | 0 0  0  0  0  0  0  1  0  2 0 1  y/b lateral ap edges
 0  2 |  *  *  *  *  *  *  *  *  *  *  *  *  * 50 |  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1 | 0 0  0  0  0  0  0  0  1  1 1 1  gb 5gon edges
------+-------------------------------------------+-------------------------------------------------+--------------------------------
 5  0 |  5  0  0  0  0  0  0  0  0  0  0  0  0  0 | 10  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1 1  0  0  0  0  0  0  0  0 0 0
 3  0 |  1  1  1  0  0  0  0  0  0  0  0  0  0  0 |  * 50  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1 0  1  0  0  0  0  0  0  0 0 0  r/gb
 3  0 |  1  0  0  1  1  0  0  0  0  0  0  0  0  0 |  *  * 50  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 1  0  1  0  0  0  0  0  0 0 0  y/gb
 2  1 |  1  0  0  0  0  1  1  0  0  0  0  0  0  0 |  *  *  * 50  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  1  1  0  0  0  0  0  0 0 0  extending ry 5gons
 2  1 |  0  1  0  0  0  1  0  1  0  0  0  0  0  0 |  *  *  *  * 50  *  *  *  *  *  *  *  *  *  *  * | 0 0  1  0  1  0  0  0  0  0 0 0  adjoined to r/g lateral ap edges
 2  1 |  0  0  1  0  0  0  1  1  0  0  0  0  0  0 |  *  *  *  *  * 50  *  *  *  *  *  *  *  *  *  * | 0 0  1  0  0  1  0  0  0  0 0 0  adjoined to r/b lateral ap edges
 2  1 |  0  0  0  1  0  1  0  0  1  0  0  0  0  0 |  *  *  *  *  *  * 50  *  *  *  *  *  *  *  *  * | 0 0  0  1  0  0  1  0  0  0 0 0  adjoined to y/g lateral ap edges
 2  1 |  0  0  0  0  1  0  1  0  1  0  0  0  0  0 |  *  *  *  *  *  *  * 50  *  *  *  *  *  *  *  * | 0 0  0  1  0  0  0  1  0  0 0 0  adjoined to y/b lateral ap edges
 1  2 |  0  0  0  0  0  1  0  1  0  1  0  0  0  0 |  *  *  *  *  *  *  *  * 50  *  *  *  *  *  *  * | 0 0  0  0  1  0  0  0  1  0 0 0  adjoined to r/g lateral ap edges
 1  2 |  0  0  0  0  0  0  1  1  0  0  1  0  0  0 |  *  *  *  *  *  *  *  *  * 50  *  *  *  *  *  * | 0 0  0  0  0  1  0  0  0  1 0 0  adjoined to r/b lateral ap edges
 1  2 |  0  0  0  0  0  1  0  0  1  0  0  1  0  0 |  *  *  *  *  *  *  *  *  *  * 50  *  *  *  *  * | 0 0  0  0  0  0  1  0  1  0 0 0  adjoined to y/g lateral ap edges
 1  2 |  0  0  0  0  0  0  1  0  1  0  0  0  1  0 |  *  *  *  *  *  *  *  *  *  *  * 50  *  *  *  * | 0 0  0  0  0  0  0  1  0  1 0 0  adjoined to y/b lateral ap edges
 1  2 |  0  0  0  0  0  0  0  1  1  0  0  0  0  1 |  *  *  *  *  *  *  *  *  *  *  *  * 50  *  *  * | 0 0  0  0  0  0  0  0  1  1 0 0  extending gb 5gons
 0  3 |  0  0  0  0  0  0  0  0  0  1  0  1  0  1 |  *  *  *  *  *  *  *  *  *  *  *  *  * 50  *  * | 0 0  0  0  0  0  0  0  1  0 1 0  ry/g
 0  3 |  0  0  0  0  0  0  0  0  0  0  1  0  1  1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 50  * | 0 0  0  0  0  0  0  0  0  1 0 1  ry/b
 0  5 |  0  0  0  0  0  0  0  0  0  0  0  0  0  5 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 10 | 0 0  0  0  0  0  0  0  0  0 1 1
------+-------------------------------------------+-------------------------------------------------+--------------------------------
10  0 | 10  5  5  0  0  0  0  0  0  0  0  0  0  0 |  2 10  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 5 *  *  *  *  *  *  *  *  * * *  r pap
10  0 | 10  0  0  5  5  0  0  0  0  0  0  0  0  0 |  2  0 10  0  0  0  0  0  0  0  0  0  0  0  0  0 | * 5  *  *  *  *  *  *  *  * * *  y pap
 3  1 |  1  1  1  0  0  1  1  1  0  0  0  0  0  0 |  0  1  0  1  1  1  0  0  0  0  0  0  0  0  0  0 | * * 50  *  *  *  *  *  *  * * *  r/gb tet(adj)
 3  1 |  1  0  0  1  1  1  1  0  1  0  0  0  0  0 |  0  0  1  1  0  0  1  1  0  0  0  0  0  0  0  0 | * *  * 50  *  *  *  *  *  * * *  y/gb tet(adj)
 2  2 |  0  1  0  0  0  2  0  2  0  1  0  0  0  0 |  0  0  0  0  2  0  0  0  2  0  0  0  0  0  0  0 | * *  *  * 25  *  *  *  *  * * *  rg tet(isol)
 2  2 |  0  0  1  0  0  0  2  2  0  0  1  0  0  0 |  0  0  0  0  0  2  0  0  0  2  0  0  0  0  0  0 | * *  *  *  * 25  *  *  *  * * *  rb tet(isol)
 2  2 |  0  0  0  1  0  2  0  0  2  0  0  1  0  0 |  0  0  0  0  0  0  2  0  0  0  2  0  0  0  0  0 | * *  *  *  *  * 25  *  *  * * *  yg tet(isol)
 2  2 |  0  0  0  0  1  0  2  0  2  0  0  0  1  0 |  0  0  0  0  0  0  0  2  0  0  0  2  0  0  0  0 | * *  *  *  *  *  * 25  *  * * *  yb tet(isol)
 1  3 |  0  0  0  0  0  1  0  1  1  1  0  1  0  1 |  0  0  0  0  0  0  0  0  1  0  1  0  1  1  0  0 | * *  *  *  *  *  *  * 50  * * *  ry/g tet(adj)
 1  3 |  0  0  0  0  0  0  1  1  1  0  1  0  1  1 |  0  0  0  0  0  0  0  0  0  1  0  1  1  0  1  0 | * *  *  *  *  *  *  *  * 50 * *  ry/b tet(adj)
 0 10 |  0  0  0  0  0  0  0  0  0  5  0  5  0 10 |  0  0  0  0  0  0  0  0  0  0  0  0  0 10  0  2 | * *  *  *  *  *  *  *  *  * 5 *  g pap
 0 10 |  0  0  0  0  0  0  0  0  0  0  5  0  5 10 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0 10  2 | * *  *  *  *  *  *  *  *  * * 5  b pap

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