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User:Tomruen/Scaliforms

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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:

Convex equilateral nonwythoffian vertex-transitive 4-polytopes
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 tuta sadi prissi spidrox gap bidex
Discoverer Schläfli, 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
Images





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:
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
  1. ^ http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform
  2. ^ http://bendwavy.org/klitzing/incmats/tut=invtut.htm
  3. ^ Category S1: Simple Scaliforms tutcup
  4. ^ 4.55 truncated tetrahedron || inverse truncated tetrahedron
  5. ^ http://bendwavy.org/klitzing/incmats/prissi.htm
  6. ^ Category S3: Special Scaliforms prissi
  7. ^ http://bendwavy.org/klitzing/incmats/spidrox.htm
  8. ^ Category S4: Scaliform Swirlprisms spidrox
  9. ^ http://bendwavy.org/klitzing/incmats/bidex.htm
  10. ^ Category S3: Special Scaliforms bidex
  11. ^ The Bi-icositetradiminished 600-cell
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