Convex 108 Degree Rhombi Regular Polyhedra

This project is an attempt to enumerate all convex solids with regular faces that also contain 108 degree rhombi. The emphasis is on 108 degrees because this is the angle of a regular pentagon. Many of these were found simply by replacing pentagons with 108 degree rhombi and filling in the gaps with regular polygons. Some of them can be made by skewing faces of regular faced polyhedra and could be made with rhombi of a range of angles. These kinds of polyhedra could be termed Rhomboflexible and will be presented in a later study.

As of December 6th, 2006 there have been discovered 87 such solids. 28 of these are chiral. Interestingly, not one 108 Degree Rhombi Regular has been found to contain an Octagon. If you find a polyhedra that is not currently in this list please E-mail me at polyhedra@bigfoot.com and I will list it with proper credit.

The table below is sorted by the base figure that is being altered, starting with Platonic Solids, then Archimedean Solids, Prisms, Johnson Solids, and Miscellaneous. Each figure lists the number of faces of each type - Rhombi, Triangle, Square, Pentagon, Hexagon, Octagon and Decagon, as well as the total Face, Edge and Vertex counts. A (C) denotes that solid is chiral.

Alterations of the Cube		Alterations of the Icosahedron
108 Rhombohedron Prolate R 3 4 5 6 8 10 6 * * * * * * F=6       E=12       V=8 stel     wrl     switch	108 Rhombohedron Oblate R 3 4 5 6 8 10 6 * * * * * * F=6       E=12       V=8 stel     wrl     switch	108 Sheered Icosahedron R 3 4 5 6 8 10 3 10 * * * * * F=13       E=21       V=10 stel     wrl     switch	108 Sheered Pentagonal Anti-Prism R 3 4 5 6 8 10 3 5 * 1 * * * F=9       E=16       V=9 stel     wrl     switch
Alterations of the Dodecahedron
108 Dodecahedron Full R 3 4 5 6 8 10 10 10 * 2 * * * F=22       E=40       V=20 stel     wrl     switch	108 Dodecahedron Half R 3 4 5 6 8 10 5 5 * 7 * * * F=17       E=35       V=20 stel     wrl     switch	108 Dodecahedron Half+J2 1 R 3 4 5 6 8 10 5 10 * 6 * * * F=21       E=40       V=21 stel     wrl     switch	108 Dodecahedron Half+J2 2 - J.M. R 3 4 5 6 8 10 5 10 * 6 * * * F=21       E=40       V=21 stel     wrl     switch
More Dodecahedron	Cuboctahedron	Rhombicosidodecahedron	Rhombicuboctahedron
108 Dodecahedron Half+J2+J2 - J.M. R 3 4 5 6 8 10 5 15 * 5 * * * F=25       E=45       V=22 stel     wrl     switch	108 Cuboctahedron (C) R 3 4 5 6 8 10 6 8 * * * * * F=14       E=24       V=12 stel     wrl     switch	108 U27 R 3 4 5 6 8 10 5 15 20 7 5 * * F=52       E=105       V=55 stel     wrl     switch	108 Rhombicuboctahedron R 3 4 5 6 8 10 8 8 10 * * * * F=26       E=48       V=24 stel     wrl     switch
Alterations of the Icosidodecahedron
108 U24 1 R 3 4 5 6 8 10 3 17 3 9 * * * F=32       E=60       V=30 stel     wrl     switch	108 U24 2,1 (C) R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch	108 U24 2,2 R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch	108 U24 2,3 R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch
More Alterations of the Icosidodecahedron
108 U24 3,1 (C) R 3 4 5 6 8 10 9 11 9 3 * * * F=32       E=60       V=30 stel     wrl     switch	108 U24 3,2 R 3 4 5 6 8 10 9 11 9 3 * * * F=32       E=60       V=30 stel     wrl     switch	108 U24 4 (C) R 3 4 5 6 8 10 12 8 12 * * * * F=32       E=60       V=30 stel     wrl     switch
Alterations of the Triangular Prism
108 Slant Triangular Prism R 3 4 5 6 8 10 2 2 1 * * * * F=5       E=9       V=6 stel     wrl     switch	108 Slant Triangular Prism+J1 R 3 4 5 6 8 10 2 6 * * * * * F=8       E=13       V=7 stel     wrl     switch	108 Slant Tri-Prism+Tetrahedron R 3 4 5 6 8 10 2 4 1 * * * * F=7       E=12       V=7 stel     wrl     switch	108 Slant Tri-Prism+Triangular Prism R 3 4 5 6 8 10 2 4 2 * * * * F=8       E=14       V=8 stel     wrl     switch
More Alterations of the Triangular Prism
108 Slant Tri-Prism+(Self) 1 (C) R 3 4 5 6 8 10 4 4 * * * * * F=8       E=14       V=8 stel     wrl     switch	108 Slant Tri-Prism+(Self) 2,1 (C) R 3 4 5 6 8 10 2 4 2 * * * * F=8       E=14       V=8 stel     wrl     switch	108 Slant Tri-Prism+(Self) 2,2 (C) R 3 4 5 6 8 10 2 4 2 * * * * F=8       E=14       V=8 stel     wrl     switch	108 Ortho Slant Tri-Prism+(Self) (C) - J.M. R 3 4 5 6 8 10 4 4 * * * * * F=8       E=14       V=8 stel     wrl     switch
Diamond Prism			Hexagonal Prism
108 Diamond Prism R 3 4 5 6 8 10 2 * 4 * * * * F=6       E=12       V=8 stel     wrl     switch	108 Diamond Prism+J1 R 3 4 5 6 8 10 2 4 3 * * * * F=9       E=16       V=9 stel     wrl     switch	108 Diamond Prism+J1+J1 R 3 4 5 6 8 10 2 8 2 * * * * F=12       E=20       V=10 stel     wrl     switch	108 Slant Hex Prism R 3 4 5 6 8 10 4 * 2 * 2 * * F=8       E=18       V=12 stel     wrl     switch
More Alterations of the Hexagonal Prism
108 Slant Hex Prism+J1 R 3 4 5 6 8 10 4 4 1 * 2 * * F=11       E=22       V=13 stel     wrl     switch	108 Slant Hex Prism+J1+J1 R 3 4 5 6 8 10 4 8 * * 2 * * F=14       E=26       V=14 stel     wrl     switch	108 Slant Hex Prism+J3 R 3 4 5 6 8 10 4 4 5 * 1 * * F=14       E=27       V=15 stel     wrl     switch	108 Slant Hex Prism+J3+J3 R 3 4 5 6 8 10 4 8 8 * * * * F=20       E=36       V=18 stel     wrl     switch
Slant Square Prism
108 Slant Square Prism - J.M. R 3 4 5 6 8 10 4 * 2 * * * * F=6       E=12       V=8 stel     wrl     switch	108 Slant Square Prism+J1 - J.M. R 3 4 5 6 8 10 4 4 1 * * * * F=9       E=16       V=9 stel     wrl     switch	108 Slant Square Prism+J1+J1 - J.M. R 3 4 5 6 8 10 4 8 * * * * * F=12       E=20       V=10 stel     wrl     switch
Alterations of Johnson Solids - J6, J21, J25
108 J6 R 3 4 5 6 8 10 3 7 3 3 * * 1 F=17       E=35       V=20 stel     wrl     switch	108 J21 R 3 4 5 6 8 10 3 7 13 3 * * 1 F=27       E=55       V=30 stel     wrl     switch	108 J25 R 3 4 5 6 8 10 3 27 3 3 * * 1 F=37       E=65       V=30 stel     wrl     switch
Alterations of Johnson Solids - J27, J28, J29, J30
108 J27 R 3 4 5 6 8 10 6 8 * * * * * F=14       E=24       V=12 stel     wrl     switch	108 J28 R 3 4 5 6 8 10 8 8 2 * * * * F=18       E=32       V=16 stel     wrl     switch	108 J29 (C) R 3 4 5 6 8 10 8 8 2 * * * * F=18       E=32       V=16 stel     wrl     switch	108 J30 R 3 4 5 6 8 10 10 10 * 2 * * * F=22       E=40       V=20 stel     wrl     switch
Alterations of Johnson Solids - J31, J32, J33
108 J31 (C) R 3 4 5 6 8 10 10 10 * 2 * * * F=22       E=40       V=20 stel     wrl     switch	108 J32 R 3 4 5 6 8 10 3 12 8 4 * * * F=27       E=50       V=25 stel     wrl     switch	108 J33 R 3 4 5 6 8 10 3 12 8 4 * * * F=27       E=50       V=25 stel     wrl     switch
Alterations of Johnson Solids - J34
108 J34 1 R 3 4 5 6 8 10 3 17 3 9 * * * F=32       E=60       V=30 stel     wrl     switch	108 J34 2,1 R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch	108 J34 2,2 (C) R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch	108 J34 2,3 (C) R 3 4 5 6 8 10 6 14 6 6 * * * F=32       E=60       V=30 stel     wrl     switch
Alterations of Johnson Solids - J35, J36, J37, J38
108 J35 R 3 4 5 6 8 10 6 8 6 * * * * F=20       E=36       V=18 stel     wrl     switch	108 J36 (C) R 3 4 5 6 8 10 6 8 6 * * * * F=20       E=36       V=18 stel     wrl     switch	108 J37 (C) R 3 4 5 6 8 10 8 8 10 * * * * F=26       E=48       V=24 stel     wrl     switch	108 J38 R 3 4 5 6 8 10 10 10 10 2 * * * F=32       E=60       V=30 stel     wrl     switch
Alterations of Johnson Solids - J39, J40, J41
108 J39 (C) R 3 4 5 6 8 10 10 10 10 2 * * * F=32       E=60       V=30 stel     wrl     switch	108 J40 R 3 4 5 6 8 10 3 12 18 4 * * * F=37       E=70       V=35 stel     wrl     switch	108 J41 R 3 4 5 6 8 10 3 12 18 4 * * * F=37       E=70       V=35 stel     wrl     switch
Alterations of Johnson Solids - J42
108 J42 1 R 3 4 5 6 8 10 3 17 13 9 * * * F=42       E=80       V=40 stel     wrl     switch	108 J42 2,1 R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch	108 J42 2,2 (C) R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch	108 J42 2,3 (C) R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch
Alterations of Johnson Solids - J43
108 J43 1 R 3 4 5 6 8 10 3 17 13 9 * * * F=42       E=80       V=40 stel     wrl     switch	108 J43 2,1 (C) R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch	108 J43 2,2 (C) R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch	108 J43 2,3 R 3 4 5 6 8 10 6 14 16 6 * * * F=42       E=80       V=40 stel     wrl     switch
Alterations of Johnson Solids - J47, J48
108 J47 (C) R 3 4 5 6 8 10 3 32 8 4 * * * F=47       E=80       V=35 stel     wrl     switch	108 J48 1 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch	108 J48 2,1 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch	108 J48 2,2 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch
More Alterations of Johnson Solids - J48
108 J48 2,3 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch	108 J48 2,4 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch	108 J48 2,5 (C) R 3 4 5 6 8 10 3 37 3 9 * * * F=52       E=90       V=40 stel     wrl     switch
Alterations of Johnson Solids - J72, J74, J75, J76
108 J72 R 3 4 5 6 8 10 5 15 20 7 5 * * F=52       E=105       V=55 stel     wrl     switch	108 J74 R 3 4 5 6 8 10 5 15 20 7 5 * * F=52       E=105       V=55 stel     wrl     switch	108 J75 R 3 4 5 6 8 10 5 15 20 7 5 * * F=52       E=105       V=55 stel     wrl     switch	108 J76 R 3 4 5 6 8 10 5 10 15 6 5 * 1 F=42       E=90       V=50 stel     wrl     switch
Alterations of Johnson Solids - J78, J79, J80, J81
108 J78 R 3 4 5 6 8 10 5 10 15 6 5 * 1 F=42       E=90       V=50 stel     wrl     switch	108 J79 (C) R 3 4 5 6 8 10 5 10 15 6 5 * 1 F=42       E=90       V=50 stel     wrl     switch	108 J80 R 3 4 5 6 8 10 10 10 10 2 10 * * F=42       E=90       V=50 stel     wrl     switch	108 J81 R 3 4 5 6 8 10 5 5 10 5 5 * 2 F=32       E=75       V=45 stel     wrl     switch
Alterations of Johnson Solids - J92 Based
108 J92a R 3 4 5 6 8 10 3 10 6 * 1 * * F=20       E=36       V=18 stel     wrl     switch	108 J92b R 3 4 5 6 8 10 4 6 2 2 1 * * F=15       E=29       V=16 stel     wrl     switch	108 J92b+J2 R 3 4 5 6 8 10 4 11 2 1 1 * * F=19       E=34       V=17 stel     wrl     switch
Misc
108 Misc1 R 3 4 5 6 8 10 7 7 2 3 * * * F=19       E=36       V=19 stel     wrl     switch	108 Misc1+J2 (C) R 3 4 5 6 8 10 7 12 2 2 * * * F=23       E=41       V=20 stel     wrl     switch

Note: The polyhedra on this page were catalogued/discovered by Roger Kaufman except where identified by initials listed after the model's name. J.M. = Jim McNeill.

Generation of VRML models was expedited by the use of Robert Webb's Stella application, and the .Stel files are available above to Stella users. The Hedron application by Jim McNeill was used extensively for model creation and to generate switch files.

I'd also like to thank Alex Doskey for his spreadsheet method which made the construction of this page much easier. I also use JovoToys in polyhedra contruction.

History:

2007-01-05 Better premise.
2006-12-06 Jim McNeill found 108 Ortho Slant Tri-Prism+(Self).
2006-12-01 J18 Found to be duplicate, removed. Jim McNeill found 2 augmentations of the Dodecahedron and the Slant Square Prisms.
2006-11-29 Started using png files
2006-11-28 Added Hedron Switch Files, New Window Open
2006-11-27 Initial Release

Roger's Polyhedra, (c) 2006, Roger Kaufman