Acronym padiap
Name prismatic double inverted antiprism,
pentagrammic double antiprismoid
Cross sections
 ©
Circumradius (sqrt(5)-1)/2 = 0.618034
Vertex figure is a bi-crossed-trapezoidal truncated great-icosahedron
General of army gap
Colonel of regiment (is itself not locally convex, but no other uniform polychoral member known:
by cells: starp tet
padiap 20300
)
External
links
hedrondude   WikiChoron  

The prismatic double inverted antiprism (padiap) is a faceting of the grand hexacosachoron. The antiprism cells (starp) form 2 rings of 10 each.

As abstract polychoron padiap is isomorphic to gap, thereby replacing the pentagrams by pentagons, resp. replacing starp by pap. Also the verftex figures turn from asymmetric facetings of gike into such facetings of ike.

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 starps and an orthogonal ring of 10 paps, 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) bi-crossed-trapezoidal truncated gike
 * 50 |  0   0   2   2   4  2 |  0   0  1   4   8  2   6  1 |  0   2   2   6  2  (vf) bi-crossed-trapezoidal truncated gike
------+-----------------------+-----------------------------+------------------
 2  0 | 50   *   *   *   *  * |  1   2  1   0   0  0   0  0 |  2   2   0   0  0  {5/2} 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 {5/2}s
 1  1 |  *   *   * 100   *  * |  0   0  0   2   2  1   0  0 |  0   1   2   2  0  joining edges, extending 2nd {5/2}s
 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  {5/2} 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 {5/2}s
 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 {5/2}s
 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   *   *   *  *  starp
 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  starp
or
100 |   2   4   4 |  1   6   3  12 |  2   8   4  (vf) bi-crossed-trapezoidal truncated gike
----+-------------+----------------+-----------
  2 | 100   *   * |  1   2   1   0 |  2   2   0  {5/2} 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 {5/2}
  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 {5/2})
  3 |   0   1   2 |  *   *   * 400 |  0   1   1  tet(adj)-tet(isol) trig
----+-------------+----------------+-----------
 10 |  10  10   0 |  2  10   0   0 | 20   *   *  5/3-ap (starp)
  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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