Convex Diamond-Regular Polyhedra

Diamonds are rhombuses formed by two coplanar equilateral triangles, sharing an edge. The polyhedra on this page are convex, and are composed only of regular polygons or diamonds, and each figure has at least one diamond. Most of these can be derived by simple augmentations of regular-faced polyhedra.

As of September 14th, 2006 the last of 71 such solids was added to the list. 9 of these are chiral. You won't find any more such solids but you can E-mail me at PolyhedraSmith@gmail.com anyway.

Note: The polyhedra on this page were co/re-discovered by B.M. Stewart, Steve Waterman, Roger Kaufman and Alex Doskey, as identified by their initials listed after the model's name.

Since the project was started, historical information was discovered pertaining to the discovery of these models.

• This page includes these 71 models and 7 others if composite faces include squares and pentagons (S4,S5,S9-13)
  • The 7 other models in the paper above are shown on Convex Regular Polyhedra with Coplanar Squares and Pentagons
  • The Timofeenko model and Zalgaller name are next to the Face, Edge, Vertex counts
• The list of 28 non-composite strictly convex regular-faced polyhedra
• Steve Waterman of the 71 solids co-compiled a listing on this page

(Notes from Alex Doskey) For those interested in exploring, please note that all of the models discovered have been based on one of three polyhedral relationships or their variants:

• 1. you can create a prism with a diamond base, by separating two diamonds with a band of squares. Just as a diamond is formed from two triangles, a diamond prism is formed from two triangular prisms.
• 2. square pyramid plus triangular pyramid (commonly seen as the octahedron-tetrahedron relationship)
  • 2b. Triangular cupola plus square pyramid (note that a cupola can be decomposed into square and triangular pyramids)
  • 2c. truncated tet plus triangular cupola (which are Stott expansions of octahedron and tetrahedron)
  • 2d. truncated cube plus square cupola (which are Stott expansions of cuboctahedron and square pyramid (see 2b.))
• 3. pentagonal rotunda plus pentagonal pyramid (seen in icosidodecahedron plus pyramid)
  • 3b. truncated dodecahedron plus pentagonal cupola (which are Stott expansions of icosidodecahedron and pent pyr.)

Even J91 - bilunabirotunda, and J92 - hebesphenorotunda are able to work because they have some of the same faces and dihedral angles as the pentagonal rotunda (or icosidodecahedron - however you want to look at it).

The 71st and last model added to this list, the Ivanov Q5 is attributed to B.A. Ivanov. Excavating this model by J92 on the two hexagons, yields J91 as seen in this model. Alternate views (1) and (2).

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

The naming conventions use B.M Stewart's abbreviations which can be found here

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

Augmentations of the Octahedron - S3		Augmentation of the Truncated Tetrahedron - T3
(S3)+Y3[TT] - B.M.S. SYM D 3 4 5 6 8 10 C3v 3 4 * * * * * S15 F=7   E=12   V=7 off     stel     wrl     switch	(S3)+Y3+Y3[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 D3v 6 * * * * * * S16 F=6   E=12   V=8 off     stel     wrl     switch	(T3)+J3[TT] - B.M.S. SYM D 3 4 5 6 8 10 C3v 3 2 3 * 3 * * S59 F=11   E=24   V=15 off     stel     wrl     switch
Augmentations of the Cuboctahedron - B4		Augmentations of the Truncated Cube - T4
(B4)+Y4[TT] - B.M.S. SYM D 3 4 5 6 8 10 C4v 4 4 5 * * * * S20 F=13   E=24   V=13 off     stel     wrl     switch	(B4)+Y4+Y4[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 D4h 8 * 4 * * * * S21 F=12   E=24   V=14 off     stel     wrl     switch	(T4)+J4[TT] - B.M.S. SYM D 3 4 5 6 8 10 C4v 4 4 5 * * 5 * S60 F=18   E=44   V=28 off     stel     wrl     switch	(T4)+J4+J4[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 D4h 8 * 10 * * 4 * S62 F=22   E=52   V=32 off     stel     wrl     switch
Augmentations of the Icosidodecahedron - B5
(B5)+Y5[TT] - B.M.S. SYM D 3 4 5 6 8 10 C5v 5 15 * 11 * * * S27 F=31   E=60   V=31 off     stel     wrl     switch	(B5)+Y5+Y5[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 D5v 10 10 * 10 * * * S29 F=30   E=60   V=32 off     stel     wrl     switch	(B5)+Y5+Y5[TT,B1] - B.M.S. SYM D 3 4 5 6 8 10 C2v 10 10 * 10 * * * S44 F=30   E=60   V=32 off     stel     wrl     switch	(B5)+Y5+Y5+Y5[TT,B1,B3] - B.M.S. SYM D 3 4 5 6 8 10 C3v 15 5 * 9 * * * S45 F=29   E=60   V=33 off     stel     wrl     switch
Augmentations of the Truncated Dodecahedron - T5
(T5)+J5[TT] - B.M.S. SYM D 3 4 5 6 8 10 C5v 5 15 5 1 * * 11 S63 F=37   E=100   V=65 off     stel     wrl     switch	(T5)+J5+J5[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 D5v 10 10 10 2 * * 10 S65 F=42   E=110   V=70 off     stel     wrl     switch	(T5)+J5+J5[TT,B1] - B.M.S. SYM D 3 4 5 6 8 10 C2v 10 10 10 2 * * 10 S67 F=42   E=110   V=70 off     stel     wrl     switch	(T5)+J5+J5+J5[TT,B1,B3] - B.M.S. SYM D 3 4 5 6 8 10 C3v 15 5 15 3 * * 9 S70 F=47   E=120   V=75 off     stel     wrl     switch
Diamond Prisms and Their Augmentations - P3+P3
(P3)+P3[S1] - B.M.S. SYM D 3 4 5 6 8 10 D2h 2 * 4 * * * * S3 F=6   E=12   V=8 off     stel     wrl     switch	(P3+P3)+Y4[T1] - R.K. SYM D 3 4 5 6 8 10 Cs 2 4 3 * * * * S6 F=9   E=16   V=9 off     stel     wrl     switch	(P3+P3)+Y4+Y4[T1,B1] - R.K. SYM D 3 4 5 6 8 10 C2v 2 8 2 * * * * S7 F=12   E=20   V=10 off     stel     wrl     switch	(P3+P3)+Y4+Y4[T1,B2] - R.K. SYM D 3 4 5 6 8 10 C2h 2 8 2 * * * * S8 F=12   E=20   V=10 off     stel     wrl     switch
Augmentation of the Square Pyramid - J1		Augmentation of the Triangular Cupola - J3
(J1)+Y3[T1] - B.M.S. SYM D 3 4 5 6 8 10 Cs 2 2 1 * * * * S14 F=5   E=9   V=6 off     stel     wrl     switch		(J3)+Y4[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 3 2 2 * 1 * * S17 F=8   E=16   V=10 off     stel     wrl     switch
Augmentations of the Pentagonal Rotunda - J6
(J6)+Y5[TT] - B.M.S. SYM D 3 4 5 6 8 10 C5v 5 5 * 5 * * 1 S24 F=16   E=35   V=21 off     stel     wrl     switch	(J6)+Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 4 7 * 5 * * 1 S46 F=17   E=36   V=21 off     stel     wrl     switch	(J6)+Y5+Y5[T1,T3] - A.D. SYM D 3 4 5 6 8 10 Cs 8 4 * 4 * * 1 S47 F=17   E=37   V=22 off     stel     wrl     switch
Augmentations of the (Gyro)Elongated Rotundas - J21 & J25		Augmentations of the Triangular Orthobicupola - J27
(J21)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 5 10 5 * * 1 S40 F=26   E=55   V=31 off     stel     wrl     switch	(J25)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 25 * 5 * * 1 S41 F=36   E=65   V=31 off     stel     wrl     switch	(J27)+Y4[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 3 6 5 * * * * S18 F=14   E=25   V=13 off     stel     wrl     switch	(J27)+Y4+Y4[T1,B2] - A.D. (C) SYM D 3 4 5 6 8 10 C2 6 4 4 * * * * S19 F=14   E=26   V=14 off     stel     wrl     switch
Augmentations of the Pentagonal Orthocupolarotunda - J32
(J32)+Y5[TT] - B.M.S. SYM D 3 4 5 6 8 10 C5v 5 10 5 6 * * * S25 F=26   E=50   V=26 off     stel     wrl     switch	(J32)+Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 4 12 5 6 * * * S49 F=27   E=51   V=26 off     stel     wrl     switch	(J32)+Y5[T1,T3] - A.D. SYM D 3 4 5 6 8 10 Cs 8 9 5 5 * * * S50 F=27   E=52   V=27 off     stel     wrl     switch
Augmentations of the Pentagonal Gyrocupolarotunda - J33
(J33)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 10 5 6 * * * S26 F=26   E=50   V=26 off     stel     wrl     switch	(J33)+Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 4 12 5 6 * * * S48 F=27   E=51   V=26 off     stel     wrl     switch	(J33)+Y5[T1,T3] - A.D. SYM D 3 4 5 6 8 10 Cs 8 9 5 5 * * * S51 F=27   E=52   V=27 off     stel     wrl     switch
Augmentations of the Pentagonal Orthobirotunda - J34
(J34)+Y5[TT] - B.M.S. SYM D 3 4 5 6 8 10 C5v 5 15 * 11 * * * S28 F=31   E=60   V=31 off     stel     wrl     switch	(J34)+Y5+Y5[TT,BB] - B.M.S. SYM D 3 4 5 6 8 10 C5h 10 10 * 10 * * * S30 F=30   E=60   V=32 off     stel     wrl     switch	(J34)+Y5+Y5[TT,B1] - A.D. SYM D 3 4 5 6 8 10 Cs 9 12 * 10 * * * S42 F=31   E=61   V=32 off     stel     wrl     switch	(J34)+Y5+Y5+Y5[TT,B1,B3] - A.D. SYM D 3 4 5 6 8 10 Cs 13 9 * 9 * * * S43 F=31   E=62   V=33 off     stel     wrl     switch
More Augmentations of the Pentagonal Orthobirotunda - J34
(J34)+Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 4 17 * 11 * * * S52 F=32   E=61   V=31 off     stel     wrl     switch	(J34)+Y5+Y5[T1,T3] - A.D. SYM D 3 4 5 6 8 10 Cs 8 14 * 10 * * * S53 F=32   E=62   V=32 off     stel     wrl     switch	(J34)+Y5+Y5[T1,B2] - A.D. (C) SYM D 3 4 5 6 8 10 C2 8 14 * 10 * * * S57 F=32   E=62   V=32 off     stel     wrl     switch	(J34)+Y5+Y5[T1,B3] - A.D. (C) SYM D 3 4 5 6 8 10 C2 8 14 * 10 * * * S58 F=32   E=62   V=32 off     stel     wrl     switch
Still More Augmentations of the Pentagonal Orthobirotunda - J34
(J34)+Y5+Y5+Y5[T1,T3,B2] - A.D. SYM D 3 4 5 6 8 10 Cs 12 11 * 9 * * * S55 F=32   E=63   V=33 off     stel     wrl     switch	(J34)+Y5+Y5+Y5[T1,T3,B4] - A.D. (C) SYM D 3 4 5 6 8 10 C1 12 11 * 9 * * * S56 F=32   E=63   V=33 off     stel     wrl     switch	(J34)+Y5+Y5+Y5+Y5[T1,T3,B2,B4] - A.D. (C) SYM D 3 4 5 6 8 10 C2 16 8 * 8 * * * S54 F=32   E=64   V=34 off     stel     wrl     switch
Augmentations of the Elongated Cupolarotundas - J40 & 41		Augmentations of the Elongated Orthobirotundas - J42
(J40)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 10 15 6 * * * S32 F=36   E=70   V=36 off     stel     wrl     switch	(J41)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 10 15 6 * * * S31 F=36   E=70   V=36 off     stel     wrl     switch	(J42)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 15 10 11 * * * S33 F=41   E=80   V=41 off     stel     wrl     switch	(J42)+Y5[TT,BB] - S.W. SYM D 3 4 5 6 8 10 D5h 10 10 10 10 * * * S35 F=40   E=80   V=42 off     stel     wrl     switch
Augmentations of the Elongated Gyrobirotunda - J43		Augmentation of the Gyroelongated Cupolarotunda - J47
(J43)+Y5[TT] - S.W. SYM D 3 4 5 6 8 10 C5v 5 15 10 11 * * * S34 F=41   E=80   V=41 off     stel     wrl     switch	(J43)+Y5[TT,BB] - S.W. SYM D 3 4 5 6 8 10 C5h 10 10 10 10 * * * S36 F=40   E=80   V=42 off     stel     wrl     switch	(J47)+Y5[TT] - S.W. (C) SYM D 3 4 5 6 8 10 C5 5 30 5 6 * * * S37 F=46   E=80   V=36 off     stel     wrl     switch
Augmentations of the Gyroelongated Birotunda - J48		Augmentation of the Augmented Truncated Cube - J66
(J48)+Y5[TT] - S.W. (C) SYM D 3 4 5 6 8 10 C5 5 35 * 11 * * * S38 F=51   E=90   V=41 off     stel     wrl     switch	(J48)+Y5[TT,BB] - S.W. (C) SYM D 3 4 5 6 8 10 D5 10 30 * 10 * * * S39 F=50   E=90   V=42 off     stel     wrl     switch	(J66)+J4[BB] - A.D. SYM D 3 4 5 6 8 10 C4v 4 8 10 * * 4 * S61 F=26   E=56   V=32 off     stel     wrl     switch
Augmentations of the Augmented and Biaugmented Truncated Dodecahedron - J68 and J70
(J68)+J5[BB] - A.D. SYM D 3 4 5 6 8 10 C5v 5 20 10 2 * * 10 S64 F=47   E=115   V=70 off     stel     wrl     switch	(J68)+J5[B1] - A.D. SYM D 3 4 5 6 8 10 Cs 5 20 10 2 * * 10 S66 F=47   E=115   V=70 off     stel     wrl     switch	(J68)+J5+J5[B1,B3] - A.D. SYM D 3 4 5 6 8 10 Cs 10 15 15 3 * * 9 S69 F=52   E=125   V=75 off     stel     wrl     switch	(J70)+J5[B3] - A.D. SYM D 3 4 5 6 8 10 Cs 5 25 15 3 * * 9 S68 F=57   E=130   V=75 off     stel     wrl     switch
Augmentations of the Bilunabirotunda - J91
(J91)+Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 2 9 2 3 * * * S22 F=16   E=29   V=15 off     stel     wrl     switch	(J91)+Y5+Y5[T1,T2] - A.D. SYM D 3 4 5 6 8 10 C2v 4 10 2 2 * * * S1 F=18   E=32   V=16 off     stel     wrl     switch	(J91+Y5+Y5[T1,T2])-Y5[side] - A.D. SYM D 3 4 5 6 8 10 Cs 4 5 2 3 * * * Q4 F=14   E=27   V=15 off     stel     wrl     switch	(J91)+Y5+Y5[T1,B2] - A.D. SYM D 3 4 5 6 8 10 C2h 4 10 2 2 * * * S23 F=18   E=32   V=16 off     stel     wrl     switch
Augmentations of the Triangular Hebesphenorotunda - J92
(J92)+Y5[T1] - S.W. SYM D 3 4 5 6 8 10 Cs 3 12 3 2 1 * * S2 F=21   E=38   V=19 off     stel     wrl     switch	(J92+Y5)-Y5[T1] - A.D. SYM D 3 4 5 6 8 10 Cs 3 7 3 3 1 * * Q6 F=17   E=33   V=18 off     stel     wrl     switch
Miscellaneus Conglomerations
(Y4+Y3)+Y3+Y4 - A.D. SYM D 3 4 5 6 8 10 C2h 4 * 2 * * * * XX F=6   E=12   V=8 off     stel     wrl     switch	(J3+Y4+Y4)+Y3+Y3+Y4 - A.D. SYM D 3 4 5 6 8 10 C2h 4 * 2 * 2 * * Q1 F=8   E=18   V=12 off     stel     wrl     switch	(Y4+Y3)+Y4+Y3 - A.D. (C) SYM D 3 4 5 6 8 10 C2 2 4 2 * * * * XX F=8   E=14   V=8 off     stel     wrl     switch
More Miscellaneus Conglomerations
(T3)+J3+J3+Y4+Y4+Y3 - S.W. SYM D 3 4 5 6 8 10 D2h 4 * 4 * 4 * * Q2 F=12   E=28   V=18 off     stel     wrl     switch	(StewartX)+J62 - A.D. SYM D 3 4 5 6 8 10 Cs 2 9 2 3 * * * Q3 F=16   E=29   V=15 off     stel     wrl     switch	Ivanov_Q5 - B.A.I. SYM D 3 4 5 6 8 10 C2h 4 10 4 2 2 * * Q5 F=22   E=42   V=22 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
2023-09-10 Added Timofeenko models and Zalgaller names
2023-09-06 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-14 Initial Release

Roger's Polyhedra, (c) 2006-2024, Roger Kaufman