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.

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.

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 PolyhedraSmith@gmail.com and I will list it with proper credit.

Notes

• The model derived from J72 is the same as the one from U27

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 symmetry symbol, 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.

Links below display various models

• Clicking on pictures opens an Interactive OFF Viewer
• OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
• STEL files may be saved and opened with Stella
• VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL

Alterations of the Cube		Alterations of the Icosahedron
108 Rhombohedron Prolate S R 3 4 5 6 8 10 D3v 6 * * * * * * F=6       E=12       V=8 off     stel     wrl     switch	108 Rhombohedron Oblate S R 3 4 5 6 8 10 D3v 6 * * * * * * F=6       E=12       V=8 off     stel     wrl     switch	108 Sheered Icosahedron S R 3 4 5 6 8 10 C3v 3 10 * * * * * F=13       E=21       V=10 off     stel     wrl     switch	108 Sheered Pentagonal Anti-Prism S R 3 4 5 6 8 10 Cs 3 5 * 1 * * * F=9       E=16       V=9 off     stel     wrl     switch
Alterations of the Dodecahedron
108 Dodecahedron Full S R 3 4 5 6 8 10 D5v 10 10 * 2 * * * F=22       E=40       V=20 off     stel     wrl     switch	108 Dodecahedron Half S R 3 4 5 6 8 10 C5v 5 5 * 7 * * * F=17       E=35       V=20 off     stel     wrl     switch	108 Dodecahedron Half+J2 1 S R 3 4 5 6 8 10 C5v 5 10 * 6 * * * F=21       E=40       V=21 off     stel     wrl     switch	108 Dodecahedron Half +J2 2 - J.M. S R 3 4 5 6 8 10 Cs 5 10 * 6 * * * F=21       E=40       V=21 off     stel     wrl     switch
More Dodecahedron	Cuboctahedron	Rhombicosidodecahedron	Rhombicuboctahedron
108 Dodecahedron Half +J2+J2 - J.M. S R 3 4 5 6 8 10 Cs 5 15 * 5 * * * F=25       E=45       V=22 off     stel     wrl     switch	108 Cuboctahedron (C) S R 3 4 5 6 8 10 D3 6 8 * * * * * F=14       E=24       V=12 off     stel     wrl     switch	108 Rhombicuboctahedron S R 3 4 5 6 8 10 C1 8 8 10 * * * * F=26       E=48       V=24 off     stel     wrl     switch	108 U27 S R 3 4 5 6 8 10 C5v 5 15 20 7 5 * * F=52       E=105       V=55 off     stel     wrl     switch
Alterations of the Icosidodecahedron
108 U24 1 S R 3 4 5 6 8 10 C3v 3 17 3 9 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 U24 2,1 (C) S R 3 4 5 6 8 10 C2 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 U24 2,2 S R 3 4 5 6 8 10 C2v 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 U24 2,3 S R 3 4 5 6 8 10 D3v 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch
More Alterations of the Icosidodecahedron
108 U24 3,1 (C) S R 3 4 5 6 8 10 C3 9 11 9 3 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 U24 3,2 S R 3 4 5 6 8 10 Cs 9 11 9 3 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 U24 4 (C) S R 3 4 5 6 8 10 T 12 8 12 * * * * F=32       E=60       V=30 off     stel     wrl     switch
Alterations of the Triangular Prism
108 Slant Triangular Prism S R 3 4 5 6 8 10 Cs 2 2 1 * * * * F=5       E=9       V=6 off     stel     wrl     switch	108 Slant Triangular Prism+J1 S R 3 4 5 6 8 10 C2 2 6 * * * * * F=8       E=13       V=7 off     stel     wrl     switch	108 Slant Triangular Prism+ Slant Triangular Prism 1 (C) S R 3 4 5 6 8 10 C2 4 4 * * * * * F=8       E=14       V=8 off     stel     wrl     switch	108 Slant Triangular Prism+ Slant Triangular Prism 2,1 (C) S R 3 4 5 6 8 10 C2 2 4 2 * * * * F=8       E=14       V=8 off     stel     wrl     switch
More Alterations of the Triangular Prism
108 Slant Triangular Prism+ Slant Triangular Prism 2,2 (C) S R 3 4 5 6 8 10 Cs 2 4 2 * * * * F=8       E=14       V=8 off     stel     wrl     switch	108 Slant Triangular Prism+ Tetrahedron S R 3 4 5 6 8 10 Cs 2 4 1 * * * * F=7       E=12       V=7 off     stel     wrl     switch	108 Slant Triangular Prism+ Triangular Prism S R 3 4 5 6 8 10 Cs 2 4 2 * * * * F=8       E=14       V=8 off     stel     wrl     switch	108 Ortho Slant Triangular Prism+ Ortho Slant Triangular Prism (C) S R 3 4 5 6 8 10 D2 2 * 4 * * * * F=6       E=12       V=8 off     stel     wrl     switch
Diamond Prism			Hexagonal Prism
108 Diamond Prism S R 3 4 5 6 8 10 C2h 2 4 3 * * * * F=9       E=16       V=9 off     stel     wrl     switch	108 Diamond Prism+J1 S R 3 4 5 6 8 10 Cs 2 8 2 * * * * F=12       E=20       V=10 off     stel     wrl     switch	108 Diamond Prism+J1+J1 S R 3 4 5 6 8 10 D2h 4 * 2 * 2 * * F=8       E=18       V=12 off     stel     wrl     switch	108 Slant Hex Prism S R 3 4 5 6 8 10 Ci 4 4 1 * 2 * * F=11       E=22       V=13 off     stel     wrl     switch
More Alterations of the Hexagonal Prism
108 Slant Hex Prism+J1 S R 3 4 5 6 8 10 Cs 4 8 * * 2 * * F=14       E=26       V=14 off     stel     wrl     switch	108 Slant Hex Prism+J1+J1 S R 3 4 5 6 8 10 C2 4 4 5 * 1 * * F=14       E=27       V=15 off     stel     wrl     switch	108 Slant Hex Prism+J3 S R 3 4 5 6 8 10 C1 4 8 8 * * * * F=20       E=36       V=18 off     stel     wrl     switch	108 Slant Hex Prism +J3+J3 - J.M. S R 3 4 5 6 8 10 Ci 4 * 2 * * * * F=6       E=12       V=8 off     stel     wrl     switch
Slant Square Prism
108 Slant Square Prism - J.M. S R 3 4 5 6 8 10 C2h 4 4 1 * * * * F=9       E=16       V=9 off     stel     wrl     switch	108 Slant Square Prism+J1 - J.M. S R 3 4 5 6 8 10 Cs 4 8 * * * * * F=12       E=20       V=10 off     stel     wrl     switch	108 Slant Square Prism +J1+J1 - J.M. S R 3 4 5 6 8 10 C2h 4 4 * * * * * F=8       E=14       V=8 off     stel     wrl     switch
Alterations of Johnson Solids - J6, J21, J25
108 J6 S R 3 4 5 6 8 10 Cs 3 7 3 3 * * 1 F=17       E=35       V=20 off     stel     wrl     switch	108 J21 S R 3 4 5 6 8 10 Cs 3 7 13 3 * * 1 F=27       E=55       V=30 off     stel     wrl     switch	108 J25 S R 3 4 5 6 8 10 Cs 3 27 3 3 * * 1 F=37       E=65       V=30 off     stel     wrl     switch
Alterations of Johnson Solids - J27, J28, J29, J30
108 J27 S R 3 4 5 6 8 10 Cs 6 8 * * * * * F=14       E=24       V=12 off     stel     wrl     switch	108 J28 S R 3 4 5 6 8 10 C4h 8 8 2 * * * * F=18       E=32       V=16 off     stel     wrl     switch	108 J29 (C) S R 3 4 5 6 8 10 D2 8 8 2 * * * * F=18       E=32       V=16 off     stel     wrl     switch	108 J30 S R 3 4 5 6 8 10 C5h 10 10 * 2 * * * F=22       E=40       V=20 off     stel     wrl     switch
Alterations of Johnson Solids - J31, J32, J33
108 J31 (C) S R 3 4 5 6 8 10 C5 10 10 * 2 * * * F=22       E=40       V=20 off     stel     wrl     switch	108 J32 S R 3 4 5 6 8 10 Cs 3 12 8 4 * * * F=27       E=50       V=25 off     stel     wrl     switch	108 J33 S R 3 4 5 6 8 10 Cs 3 12 8 4 * * * F=27       E=50       V=25 off     stel     wrl     switch
Alterations of Johnson Solids - J34
108 J34 1 S R 3 4 5 6 8 10 Cs 3 17 3 9 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 J34 2,1 S R 3 4 5 6 8 10 C2v 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 J34 2,2 (C) S R 3 4 5 6 8 10 C2 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 J34 2,3 (C) S R 3 4 5 6 8 10 C2 6 14 6 6 * * * F=32       E=60       V=30 off     stel     wrl     switch
Alterations of Johnson Solids - J35, J36, J37, J38
108 J35 S R 3 4 5 6 8 10 Cs 6 8 6 * * * * F=20       E=36       V=18 off     stel     wrl     switch	108 J36 (C) S R 3 4 5 6 8 10 D3 6 8 6 * * * * F=20       E=36       V=18 off     stel     wrl     switch	108 J37 (C) S R 3 4 5 6 8 10 D4 8 8 10 * * * * F=26       E=48       V=24 off     stel     wrl     switch	108 J38 S R 3 4 5 6 8 10 C5h 10 10 10 2 * * * F=32       E=60       V=30 off     stel     wrl     switch
Alterations of Johnson Solids - J39, J40, J41
108 J39 (C) S R 3 4 5 6 8 10 D5 10 10 10 2 * * * F=32       E=60       V=30 off     stel     wrl     switch	108 J40 S R 3 4 5 6 8 10 Cs 3 12 18 4 * * * F=37       E=70       V=35 off     stel     wrl     switch	108 J41 S R 3 4 5 6 8 10 Cs 3 12 18 4 * * * F=37       E=70       V=35 off     stel     wrl     switch
Alterations of Johnson Solids - J42
108 J42 1 S R 3 4 5 6 8 10 Cs 3 17 13 9 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J42 2,1 S R 3 4 5 6 8 10 C2v 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J42 2,2 (C) S R 3 4 5 6 8 10 C2 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J42 2,3 (C) S R 3 4 5 6 8 10 C2 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch
Alterations of Johnson Solids - J43
108 J43 1 S R 3 4 5 6 8 10 Cs 3 17 13 9 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J43 2,1 (C) S R 3 4 5 6 8 10 C2 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J43 2,2 (C) S R 3 4 5 6 8 10 C2 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch	108 J43 2,3 S R 3 4 5 6 8 10 C2h 6 14 16 6 * * * F=42       E=80       V=40 off     stel     wrl     switch
Alterations of Johnson Solids - J47, J48
108 J47 (C) S R 3 4 5 6 8 10 C1 3 32 8 4 * * * F=47       E=80       V=35 off     stel     wrl     switch	108 J48 1 (C) S R 3 4 5 6 8 10 C1 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch	108 J48 2,1 (C) S R 3 4 5 6 8 10 C2 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch	108 J48 2,2 (C) S R 3 4 5 6 8 10 C2 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch
More Alterations of Johnson Solids - J48
108 J48 2,3 (C) S R 3 4 5 6 8 10 C2 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch	108 J48 2,4 (C) S R 3 4 5 6 8 10 C2 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch	108 J48 2,5 (C) S R 3 4 5 6 8 10 C2 6 34 6 6 * * * F=52       E=90       V=40 off     stel     wrl     switch
Alterations of Johnson Solids - J73, J74, J75, J76
108 J73 S R 3 4 5 6 8 10 C5v 5 15 20 7 5 * * F=52       E=105       V=55 off     stel     wrl     switch	108 J74 S R 3 4 5 6 8 10 Cs 5 15 20 7 5 * * F=52       E=105       V=55 off     stel     wrl     switch	108 J75 S R 3 4 5 6 8 10 Cs 5 15 20 7 5 * * F=52       E=105       V=55 off     stel     wrl     switch	108 J76 S R 3 4 5 6 8 10 C5v 5 10 15 6 5 * 1 F=42       E=90       V=50 off     stel     wrl     switch
Alterations of Johnson Solids - J78, J79, J80, J81
108 J78 S R 3 4 5 6 8 10 Cs 5 10 15 6 5 * 1 F=42       E=90       V=50 off     stel     wrl     switch	108 J79 (C) S R 3 4 5 6 8 10 C1 5 10 15 6 5 * 1 F=42       E=90       V=50 off     stel     wrl     switch	108 J80 S R 3 4 5 6 8 10 D5v 10 10 10 2 10 * * F=42       E=90       V=50 off     stel     wrl     switch	108 J81 S R 3 4 5 6 8 10 Cs 5 5 10 5 5 * 2 F=32       E=75       V=45 off     stel     wrl     switch
Alterations of Johnson Solids - J92 Based
108 J92a S R 3 4 5 6 8 10 C3v 3 10 6 * 1 * * F=20       E=36       V=18 off     stel     wrl     switch	108 J92b S R 3 4 5 6 8 10 Cs 4 6 2 2 1 * * F=15       E=29       V=16 off     stel     wrl     switch	108 J92b+J2 S R 3 4 5 6 8 10 Cs 4 11 2 1 1 * * F=19       E=34       V=17 off     stel     wrl     switch
Misc
108 Misc1 S R 3 4 5 6 8 10 Cs 7 7 2 3 * * * F=19       E=36       V=19 off     stel     wrl     switch	108 Misc1+J2 (C) S R 3 4 5 6 8 10 C1 7 12 2 2 * * * F=23       E=41       V=20 off     stel     wrl     switch

Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.

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:

2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-05 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-10 Put underlines in file names
2023-03-06 Added Symmetry Symbol
2023-03-04 Open Interactive Viewer from model pictures
2019-03-12 Changed email address from defunct bigfoot.com
2012-02-22 Changed to index.html
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) 2006-2024, Roger Kaufman