Wythoff constructions with alternations produce vertex-transitive figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures is scaliform polytopes. [1] This category allows a subset of Johnson solids as cells, for example triangular cupola.
Each vertex configuration within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex.
The symmetry order of a vertex-transitive polytope is the number of vertices times the symmetry order of the vertex figure.
The scaliforms and some related non-Wythoffian 4-polytopes are give below:
Name |
16-cell (Snub cubic hosochoron) |
Runcic snub cubic hosochoron [2] [3] [4] |
Snub 24-cell (24-diminished 600-cell) |
Runcic snub 24-cell [5] [6] |
120-diminished rectified 600-cell [7] [8] |
Grand antiprism (20-diminished 600-cell) |
bi-24-diminished 600-cell [9] [10] [11] |
---|---|---|---|---|---|---|---|
Classification | Regular | Scaliform | Semiregular | Scaliform | Scaliform | Uniform | Scaliform |
Coxeter | s{2,4,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
s3{2,4,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
s{3,4,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
s3{3,4,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Non-Wythoffian | ||
Symmetry | [2+,4,3], order 48 ±1/12[T×T].2 |
[3+,4,3], order 576 ±[T×T].2 |
(?) order 1200 (?) ±[I×D10] |
[[10,2+,10]], order 400 ? |
[2[(6,2+)[2]]], order 144 ? | ||
BSA | hex | tutcup | sadi | prissi | spidrox | gap | bidex |
Discoverer | Schlaefli, 1850 | Richard Klitzing, 2000 | Gosset, 1900 | Richard Klitzing, 2005 | George Olshevsky, 2002? | Conway, 1965 | Andrew Weimholt, 2004 |
Relation | 8 of 16 vertices of cubic prism, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 of 48 vertices of rhombicuboctahedral prism, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
96 of 120 vertices of 600-cell, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
288 of 576 vertices of runcitruncated 24-cell, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
600 of 720 vertices of rectified 600-cell, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
100 of 120 vertices of 600-cell, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
72 of 120 vertices of 600-cell, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Net |
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Images |
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Vertices | 8 | 24 | 96 | 288 | 600 | 100 | 72 |
Cells | 4+4:
![]() 6: ![]() 2: ![]() |
4+4:
![]() 6: ![]() 2: ![]() |
24:
![]() 96: ![]() 24 ![]() |
24:
![]() 96: ![]() 24: ![]() 96: ![]() |
600:
![]() 120: ![]() 120: ![]() |
100+200
![]() 20 ![]() |
48:
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Vertex figure |
![]() (8) 3.3.3 |
![]() (1) 3.4.3.4 (2) 3.4.6 (1) 3.3.3 (1) 3.6.6 |
![]() Tridiminished icosahedron (5) 3.3.3 (3) 3.3.3.3.3 |
![]() (1) 3.4.3.4 (2) 3.4.6 (2) 3.4.4 (1) 3.6.6 (1) 3.3.3.3.3 |
![]() Bidiminished pentagonal prism (1) 3.3.3.3 (4) 3.3.4 (2) 4.4.5 (2) 3.3.3.5 |
![]() Bidiminished icosahedron (12) 3.3.3 (2) 3.3.3.5 |
![]() Bitridiminished icosahedron (2) 3.3.3.5 (4) 3.5.5 |