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 derrived by simple augmentations of regular-faced polyhedra.

    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.

    Augmentations of the Octahedron - S3Augmentation of the Truncated Tetrahedron - T3
    (S3)+Y3[TT] - B.M.S.
    D3456810
    34*****
    F=7       E=12       V=7
    .stel           .wrl
    (S3)+Y3+Y3[TT,BB] - B.M.S.
    D3456810
    6******
    F=6       E=12       V=8
    .stel           .wrl
    (T3)+J3[TT] - B.M.S.
    D3456810
    323*3**
    F=11       E=24       V=15
    .stel           .wrl
    		  
    Augmentations of the Cuboctahedron - B4Augmentations of the Truncated Cube - T4
    (B4)+Y4[TT] - B.M.S.
    D3456810
    445****
    F=13       E=24       V=13
    .stel           .wrl
    (B4)+Y4+Y4[TT,BB] - B.M.S.
    D3456810
    8*4****
    F=12       E=24       V=14
    .stel           .wrl
    (T4)+J4[TT] - B.M.S.
    D3456810
    445**5*
    F=18       E=44       V=28
    .stel           .wrl
    (T4)+J4+J4[TT,BB] - B.M.S.
    D3456810
    8*10**4*
    F=22       E=52       V=32
    .stel           .wrl
    Augmentations of the Icosidodecahedron - B5
    (B5)+Y5[TT] - B.M.S.
    D3456810
    515*11***
    F=31       E=60       V=31
    .stel           .wrl
    (B5)+Y5+Y5[TT,BB] - B.M.S.
    D3456810
    1010*10***
    F=30       E=60       V=32
    .stel           .wrl
    (B5)+Y5+Y5[TT,B1] - B.M.S.
    D3456810
    1010*10***
    F=30       E=60       V=32
    .stel           .wrl
    (B5)+Y5+Y5+Y5[TT,B1,B3] - B.M.S.
    D3456810
    155*9***
    F=29       E=60       V=33
    .stel           .wrl
    Augmentations of the Truncated Dodecahedron - T5
    (T5)+J5[TT] - B.M.S.
    D3456810
    51551**11
    F=37       E=100       V=65
    .stel           .wrl
    (T5)+J5+J5[TT,BB] - B.M.S.
    D3456810
    1010102**10
    F=42       E=110       V=70
    .stel           .wrl
    (T5)+J5+J5[TT,B1] - B.M.S.
    D3456810
    1010102**10
    F=42       E=110       V=70
    .stel           .wrl
    (T5)+J5+J5+J5[TT,B1,B3] - B.M.S.
    D3456810
    155153**9
    F=47       E=120       V=75
    .stel           .wrl
    Diamond Prisms and Their Augmentations - P3+P3
    (P3)+P3[S1] - B.M.S.
    D3456810
    2*4****
    F=6       E=12       V=8
    .stel           .wrl
    (P3+P3)+Y4[T1] - R.K.
    D3456810
    243****
    F=9       E=16       V=9
    .stel           .wrl
    (P3+P3)+Y4+Y4[T1,B1] - R.K.
    D3456810
    282****
    F=12       E=20       V=10
    .stel           .wrl
    (P3+P3)+Y4+Y4[T1,B2] - R.K.
    D3456810
    282****
    F=12       E=20       V=10
    .stel           .wrl
    Augmentation of the Square Pyramid - J1Augmentation of the Triangular Cupola - J3
    (J1)+Y3[T1] - B.M.S.
    D3456810
    221****
    F=5       E=9       V=6
    .stel           .wrl
    		  
    (J3)+Y4[T1] - A.D.
    D3456810
    322*1**
    F=8       E=16       V=10
    .stel           .wrl
    		  
    Augmentations of the Pentagonal Rotunda - J6
    (J6)+Y5[TT] - B.M.S.
    D3456810
    55*5**1
    F=16       E=35       V=21
    .stel           .wrl
    (J6)+Y5[T1] - A.D.
    D3456810
    47*5**1
    F=17       E=36       V=21
    .stel           .wrl
    (J6)+Y5+Y5[T1,T3] - A.D.
    D3456810
    84*4**1
    F=17       E=37       V=22
    .stel           .wrl
    Augmentations of the (Gyro)Elongated Rotundas - J21 & J25Augmentations of the Triangular Orthobicupola - J27
    (J21)+Y5[TT] - S.W.
    D3456810
    55105**1
    F=26       E=55       V=31
    .stel           .wrl
    (J25)+Y5[TT] - S.W.
    D3456810
    525*5**1
    F=36       E=65       V=31
    .stel           .wrl
    (J27)+Y4[T1] - A.D.
    D3456810
    365****
    F=14       E=25       V=13
    .stel           .wrl
    (J27)+Y4+Y4[T1,B2] - A.D.
    D3456810
    644****
    F=14       E=26       V=14
    .stel           .wrl
    Augmentations of the Pentagonal Orthocupolarotunda - J32
    (J32)+Y5[TT] - B.M.S.
    D3456810
    51056***
    F=26       E=50       V=26
    .stel           .wrl
    (J32)+Y5[T1] - A.D.
    D3456810
    41256***
    F=27       E=51       V=26
    .stel           .wrl
    (J32)+Y5[T1,T3] - A.D.
    D3456810
    8955***
    F=27       E=52       V=27
    .stel           .wrl
    Augmentations of the Pentagonal Gyrocupolarotunda - J33
    (J33)+Y5[TT] - S.W.
    D3456810
    51056***
    F=26       E=50       V=26
    .stel           .wrl
    (J33)+Y5[T1] - A.D.
    D3456810
    41256***
    F=27       E=51       V=26
    .stel           .wrl
    (J33)+Y5[T1,T3] - A.D.
    D3456810
    8955***
    F=27       E=52       V=27
    .stel           .wrl
    Augmentations of the Pentagonal Orthobirotunda - J34
    (J34)+Y5[TT] - B.M.S.
    D3456810
    515*11***
    F=31       E=60       V=31
    .stel           .wrl
    (J34)+Y5+Y5[TT,BB] - B.M.S.
    D3456810
    1010*10***
    F=30       E=60       V=32
    .stel           .wrl
    (J34)+Y5+Y5[TT,B1] - A.D.
    D3456810
    912*10***
    F=31       E=61       V=32
    .stel           .wrl
    (J34)+Y5+Y5+Y5[TT,B1,B3] - A.D.
    D3456810
    139*9***
    F=31       E=62       V=33
    .stel           .wrl
    More Augmentations of the Pentagonal Orthobirotunda - J34
    (J34)+Y5[T1] - A.D.
    D3456810
    417*11***
    F=32       E=61       V=31
    .stel           .wrl
    (J34)+Y5+Y5[T1,T3] - A.D.
    D3456810
    814*10***
    F=32       E=62       V=32
    .stel           .wrl
    (J34)+Y5+Y5[T1,B2] - A.D.
    D3456810
    814*10***
    F=32       E=62       V=32
    .stel           .wrl
    (J34)+Y5+Y5[T1,B3] - A.D.
    D3456810
    814*10***
    F=32       E=62       V=32
    .stel           .wrl
    Still More Augmentations of the Pentagonal Orthobirotunda - J34
    (J34)+Y5+Y5+Y5[T1,T3,B2] - A.D.
    D3456810
    1211*9***
    F=32       E=63       V=33
    .stel           .wrl
    (J34)+Y5+Y5+Y5[T1,T3,B4] - A.D.
    D3456810
    1211*9***
    F=32       E=63       V=33
    .stel           .wrl
    (J34)+Y5+Y5+Y5+Y5[T1,T3,B2,B4] - A.D.
    D3456810
    168*8***
    F=32       E=64       V=34
    .stel           .wrl
    Augmentations of the Elongated Cupolarotundas - J40 & 41Augmentations of the Elongated Orthobirotundas - J42
    (J40)+Y5[TT] - S.W.
    D3456810
    510156***
    F=36       E=70       V=36
    .stel           .wrl
    (J41)+Y5[TT] - S.W.
    D3456810
    510156***
    F=36       E=70       V=36
    .stel           .wrl
    (J42)+Y5[TT] - S.W.
    D3456810
    5151011***
    F=41       E=80       V=41
    .stel           .wrl
    (J42)+Y5[TT,BB] - S.W.
    D3456810
    10101010***
    F=40       E=80       V=42
    .stel           .wrl
    Augmentations of the Elongated Gyrobirotunda - J43Augmentation of the Gyroelongated Cupolarotunda - J47
    (J43)+Y5[TT] - S.W.
    D3456810
    5151011***
    F=41       E=80       V=41
    .stel           .wrl
    (J43)+Y5[TT,BB] - S.W.
    D3456810
    10101010***
    F=40       E=80       V=42
    .stel           .wrl
    (J47)+Y5[TT] - S.W.
    D3456810
    53056***
    F=46       E=80       V=36
    .stel           .wrl
    Augmentations of the Gyroelongated Birotunda - J48Augmentation of the Augmented Truncated Cube - J66
    (J48)+Y5[TT] - S.W.
    D3456810
    535*11***
    F=51       E=90       V=41
    .stel           .wrl
    (J48)+Y5[TT,BB] - S.W.
    D3456810
    1030*10***
    F=50       E=90       V=42
    .stel           .wrl
    (J66)+J4[BB] - A.D.
    D3456810
    4810**4*
    F=26       E=56       V=32
    .stel           .wrl
    Augmentations of the Augmented and Biaugmented Truncated Dodecahedron - J68 and J70
    (J68)+J5[BB] - A.D.
    D3456810
    520102**10
    F=47       E=115       V=70
    .stel           .wrl
    (J68)+J5[B1] - A.D.
    D3456810
    520102**10
    F=47       E=115       V=70
    .stel           .wrl
    (J68)+J5+J5[B1,B3] - A.D.
    D3456810
    1015153**9
    F=52       E=125       V=75
    .stel           .wrl
    (J70)+J5[B3] - A.D.
    D3456810
    525153**9
    F=57       E=130       V=75
    .stel           .wrl
    Augmentations of the Bilunabirotunda - J91
    (J91)+Y5[T1] - A.D.
    D3456810
    2923***
    F=16       E=29       V=15
    .stel           .wrl
    (J91)+Y5+Y5[T1,T2] - A.D.
    D3456810
    41022***
    F=18       E=32       V=16
    .stel           .wrl
    (J91+Y5+Y5[T1,T2])-Y5[side] - A.D.
    D3456810
    4523***
    F=14       E=27       V=15
    .stel           .wrl
    (J91)+Y5+Y5[T1,B2] - A.D.
    D3456810
    41022***
    F=18       E=32       V=16
    .stel           .wrl
    Augmentations of the Triangular Hebesphenorotunda - J92
    (J92)+Y5[T1] - S.W.
    D3456810
    312321**
    F=21       E=38       V=19
    .stel           .wrl
    (J92+Y5)-Y5[T1] - A.D.
    D3456810
    37331**
    F=17       E=33       V=18
    .stel           .wrl
    Miscellaneus Conglomerations
    Slant Square Prism
    (Y4+Y3)+Y3+Y4 - A.D.
    D3456810
    4*2****
    F=6       E=12       V=8
    .stel           .wrl
    Slant Hex Prism
    (J3+Y4+Y4)+Y3+Y3+Y4 - A.D.
    D3456810
    4*2*2**
    F=8       E=18       V=12
    .stel           .wrl
    Twist Slant Square Prism
    (Y4+Y3)+Y4+Y3 - A.D.
    D3456810
    242****
    F=8       E=14       V=8
    .stel           .wrl
    More Miscellaneus Conglomerations
    Quad Hex.
    (T3)+J3+J3+Y4+Y4+Y3 - S.W.
    D3456810
    4*4*4**
    F=12       E=28       V=18
    .stel           .wrl
    J91 With a TWist
    (StewartX)+J62 - A.D.
    D3456810
    2923***
    F=16       E=29       V=15
    .stel           .wrl

    Note: The polyhedra on this page were discovered by B.M. Stewart, Steve Waterman, Roger Kaufman and Alex Doskey, as identified by their initials listed after the model's name, and many of them were modeled using my Jovo polyhedron toys. 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.

    For those interested in exploring for new Convex Diamond-Regular polyhedra, please note that all of the models discovered so far 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).

    If anyone finds a convex diamond-regular polyhedron that is not related to one of these, please let me know.


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    Link to this page as http://Polyhedra.Doskey.com/DiamondRegular/DiamondRegular.html
    Alexander's Polyhedra, (c) 1998-2006, Alex Doskey