Johnson Solids

The term Johnson Solid is used to describe a non-uniform convex polyhedron with regular faces. The table of all 92 such figures was first published by N.W. Johnson in "Convex Solids with Regular Faces", published in Canadian Journal of Mathematics, Vol 18 pp. 169–200 (1966). In 1969, Russian mathematician Viktor Abramovitch Zalgaller proved that the list was complete in Convex Polyhedra with Regular Faces (Выпуклые многогранники с правильными гранями).

The names of the majority of the Johnson Solids were ascribed by Johnson at this time. In a communication dated December 2004, Dr. Johnson stated:

"When I was investigating regular-faced solids in the early 1960s, there were no established names even for many of the obvious ones, except for pyramids and bipyramids. So I invented terms like 'cupola' and 'rotunda' and devised other terminology ('augmented,' 'diminished,' 'elongated,' etc.) that covered all but the eight or nine anomalous figures at the end of my list, which got their own descriptive names."

The table below lists all 92 Johnson solids. Links lead to 'switchable' VRML files featuring display options including framework, 'Leonardo', and molecular modes.

Alex Doskey discusses 'Convex Diamond Regular Polyhedra', an extension to the Johnson Solids; Steve Waterman and Roger Kaufman catalogue 71 additional polyhedra resulting from allowing one extra polygon: a rhombus formed from two coplanar triangles.

Roger Kaufman has also explored Convex Triamond regular polyhedra (three coplanar triangles), convex 108° Rhombi regular polyhedra, and convex regular polyhedra with two different coplanar faces.

Click on any image to load it into the 3D viewer where it can be rotated interactively.

J01 Square Pyramid
J01 — Square Pyramid
V: 5  |  E: 8  |  F: 5
Faces: 4{3} · 1{4}
Symmetry: C4v
J02 Pentagonal Pyramid
J02 — Pentagonal Pyramid
V: 6  |  E: 10  |  F: 6
Faces: 5{3} · 1{5}
Symmetry: C5v
J03 Triangular Cupola
J03 — Triangular Cupola
V: 9  |  E: 15  |  F: 8
Faces: 4{3} · 3{4} · 1{6}
Symmetry: C3v
J04 Square Cupola
J04 — Square Cupola
V: 12  |  E: 20  |  F: 10
Faces: 4{3} · 5{4} · 1{8}
Symmetry: C4v
J05 Pentagonal Cupola
J05 — Pentagonal Cupola
V: 15  |  E: 25  |  F: 12
Faces: 5{3} · 5{4} · 1{5} · 1{10}
Symmetry: C5v
J06 Pentagonal Rotunda
J06 — Pentagonal Rotunda
V: 20  |  E: 35  |  F: 17
Faces: 10{3} · 6{5} · 1{10}
Symmetry: C5v
J07 Elongated Triangular Pyramid
J07 — Elongated Triangular Pyramid
V: 7  |  E: 12  |  F: 7
Faces: 4{3} · 3{4}
Symmetry: C3v
J08 Elongated Square Pyramid
J08 — Elongated Square Pyramid
V: 9  |  E: 16  |  F: 9
Faces: 4{3} · 5{4}
Symmetry: C4v
J09 Elongated Pentagonal Pyramid
J09 — Elongated Pentagonal Pyramid
V: 11  |  E: 20  |  F: 11
Faces: 5{3} · 5{4} · 1{5}
Symmetry: C5v
J10 Gyroelongated Square Pyramid
J10 — Gyroelongated Square Pyramid
V: 9  |  E: 20  |  F: 13
Faces: 12{3} · 1{4}
Symmetry: C4v
J11 Gyroelongated Pentagonal Pyramid
J11 — Gyroelongated Pentagonal Pyramid
V: 11  |  E: 25  |  F: 16
Faces: 15{3} · 1{5}
Symmetry: C5v
J12 Triangular Dipyramid
J12 — Triangular Dipyramid
V: 5  |  E: 9  |  F: 6
Faces: 6{3}
Symmetry: D3h
J13 Pentagonal Dipyramid
J13 — Pentagonal Dipyramid
V: 7  |  E: 15  |  F: 10
Faces: 10{3}
Symmetry: D5h
J14 Elongated Triangular Dipyramid
J14 — Elongated Triangular Dipyramid
V: 8  |  E: 15  |  F: 9
Faces: 6{3} · 3{4}
Symmetry: D3h
J15 Elongated Square Dipyramid
J15 — Elongated Square Dipyramid
V: 10  |  E: 20  |  F: 12
Faces: 8{3} · 4{4}
Symmetry: D4h
J16 Elongated Pentagonal Dipyramid
J16 — Elongated Pentagonal Dipyramid
V: 12  |  E: 25  |  F: 15
Faces: 10{3} · 5{4}
Symmetry: D5h
J17 Gyroelongated Square Dipyramid
J17 — Gyroelongated Square Dipyramid
V: 10  |  E: 24  |  F: 16
Faces: 16{3}
Symmetry: D4d
J18 Elongated Triangular Cupola
J18 — Elongated Triangular Cupola
V: 15  |  E: 27  |  F: 14
Faces: 4{3} · 9{4} · 1{6}
Symmetry: C3v
J19 Elongated Square Cupola
J19 — Elongated Square Cupola
V: 20  |  E: 36  |  F: 18
Faces: 4{3} · 13{4} · 1{8}
Symmetry: C4v
J20 Elongated Pentagonal Cupola
J20 — Elongated Pentagonal Cupola
V: 25  |  E: 45  |  F: 22
Faces: 5{3} · 15{4} · 1{5} · 1{10}
Symmetry: C5v
J21 Elongated Pentagonal Rotunda
J21 — Elongated Pentagonal Rotunda
V: 30  |  E: 55  |  F: 27
Faces: 10{3} · 10{4} · 6{5} · 1{10}
Symmetry: C5v
J22 Gyroelongated Triangular Cupola
J22 — Gyroelongated Triangular Cupola
V: 15  |  E: 33  |  F: 20
Faces: 16{3} · 3{4} · 1{6}
Symmetry: C3v
J23 Gyroelongated Square Cupola
J23 — Gyroelongated Square Cupola
V: 20  |  E: 44  |  F: 26
Faces: 20{3} · 5{4} · 1{8}
Symmetry: C4v
J24 Gyroelongated Pentagonal Cupola
J24 — Gyroelongated Pentagonal Cupola
V: 25  |  E: 55  |  F: 32
Faces: 25{3} · 5{4} · 1{5} · 1{10}
Symmetry: C5v
J25 Gyroelongated Pentagonal Rotunda
J25 — Gyroelongated Pentagonal Rotunda
V: 30  |  E: 65  |  F: 37
Faces: 30{3} · 6{5} · 1{10}
Symmetry: C5v
J26 Gyrobifastigium
J26 — Gyrobifastigium
V: 8  |  E: 14  |  F: 8
Faces: 4{3} · 4{4}
Symmetry: D2d
J27 Triangular Orthobicupola
J27 — Triangular Orthobicupola
V: 12  |  E: 24  |  F: 14
Faces: 8{3} · 6{4}
Symmetry: D3h
J28 Square Orthobicupola
J28 — Square Orthobicupola
V: 16  |  E: 32  |  F: 18
Faces: 8{3} · 10{4}
Symmetry: D4h
J29 Square Gyrobicupola
J29 — Square Gyrobicupola
V: 16  |  E: 32  |  F: 18
Faces: 8{3} · 10{4}
Symmetry: D4d
J30 Pentagonal Orthobicupola
J30 — Pentagonal Orthobicupola
V: 20  |  E: 40  |  F: 22
Faces: 10{3} · 10{4} · 2{5}
Symmetry: D5h
J31 Pentagonal Gyrobicupola
J31 — Pentagonal Gyrobicupola
V: 20  |  E: 40  |  F: 22
Faces: 10{3} · 10{4} · 2{5}
Symmetry: D5d
J32 Pentagonal Orthocupolarotunda
J32 — Pentagonal Orthocupolarotunda
V: 25  |  E: 50  |  F: 27
Faces: 15{3} · 5{4} · 7{5}
Symmetry: C5v
J33 Pentagonal Gyrocupolarotunda
J33 — Pentagonal Gyrocupolarotunda
V: 25  |  E: 50  |  F: 27
Faces: 15{3} · 5{4} · 7{5}
Symmetry: C5v
J34 Pentagonal Orthobirotunda
J34 — Pentagonal Orthobirotunda
V: 30  |  E: 60  |  F: 32
Faces: 20{3} · 12{5}
Symmetry: D5h
J35 Elongated Triangular Orthobicupola
J35 — Elongated Triangular Orthobicupola
V: 18  |  E: 36  |  F: 20
Faces: 8{3} · 12{4}
Symmetry: D3h
J36 Elongated Triangular Gyrobicupola
J36 — Elongated Triangular Gyrobicupola
V: 18  |  E: 36  |  F: 20
Faces: 8{3} · 12{4}
Symmetry: D3d
J37 Elongated Square Gyrobicupola
J37 — Elongated Square Gyrobicupola
V: 24  |  E: 48  |  F: 26
Faces: 8{3} · 18{4}
Symmetry: D4d
aka: Miller's Solid
J38 Elongated Pentagonal Orthobicupola
J38 — Elongated Pentagonal Orthobicupola
V: 30  |  E: 60  |  F: 32
Faces: 10{3} · 20{4} · 2{5}
Symmetry: D5h
J39 Elongated Pentagonal Gyrobicupola
J39 — Elongated Pentagonal Gyrobicupola
V: 30  |  E: 60  |  F: 32
Faces: 10{3} · 20{4} · 2{5}
Symmetry: D5d
J40 Elongated Pentagonal Orthocupolarotunda
J40 — Elongated Pentagonal Orthocupolarotunda
V: 35  |  E: 70  |  F: 37
Faces: 15{3} · 15{4} · 7{5}
Symmetry: C5v
J41 Elongated Pentagonal Gyrocupolarotunda
J41 — Elongated Pentagonal Gyrocupolarotunda
V: 35  |  E: 70  |  F: 37
Faces: 15{3} · 15{4} · 7{5}
Symmetry: C5v
J42 Elongated Pentagonal Orthobirotunda
J42 — Elongated Pentagonal Orthobirotunda
V: 40  |  E: 80  |  F: 42
Faces: 20{3} · 10{4} · 12{5}
Symmetry: D5h
J43 Elongated Pentagonal Gyrobirotunda
J43 — Elongated Pentagonal Gyrobirotunda
V: 40  |  E: 80  |  F: 42
Faces: 20{3} · 10{4} · 12{5}
Symmetry: D5d
J44 Gyroelongated Triangular Bicupola
J44 — Gyroelongated Triangular Bicupola
V: 18  |  E: 42  |  F: 26
Faces: 20{3} · 6{4}
Symmetry: D3
J45 Gyroelongated Square Bicupola
J45 — Gyroelongated Square Bicupola
V: 24  |  E: 56  |  F: 34
Faces: 24{3} · 10{4}
Symmetry: D4
J46 Gyroelongated Pentagonal Bicupola
J46 — Gyroelongated Pentagonal Bicupola
V: 30  |  E: 70  |  F: 42
Faces: 30{3} · 10{4} · 2{5}
Symmetry: D5
J47 Gyroelongated Pentagonal Cupolarotunda
J47 — Gyroelongated Pentagonal Cupolarotunda
V: 35  |  E: 80  |  F: 47
Faces: 35{3} · 5{4} · 7{5}
Symmetry: C5
J48 Gyroelongated Pentagonal Birotunda
J48 — Gyroelongated Pentagonal Birotunda
V: 40  |  E: 90  |  F: 52
Faces: 40{3} · 12{5}
Symmetry: D5
J49 Augmented Triangular Prism
J49 — Augmented Triangular Prism
V: 7  |  E: 13  |  F: 8
Faces: 6{3} · 2{4}
Symmetry: C2v
J50 Biaugmented Triangular Prism
J50 — Biaugmented Triangular Prism
V: 8  |  E: 17  |  F: 11
Faces: 10{3} · 1{4}
Symmetry: C2v
J51 Triaugmented Triangular Prism
J51 — Triaugmented Triangular Prism
V: 9  |  E: 21  |  F: 14
Faces: 14{3}
Symmetry: D3h
J52 Augmented Pentagonal Prism
J52 — Augmented Pentagonal Prism
V: 11  |  E: 19  |  F: 10
Faces: 4{3} · 4{4} · 2{5}
Symmetry: C2v
J53 Biaugmented Pentagonal Prism
J53 — Biaugmented Pentagonal Prism
V: 12  |  E: 23  |  F: 13
Faces: 8{3} · 3{4} · 2{5}
Symmetry: C2v
J54 Augmented Hexagonal Prism
J54 — Augmented Hexagonal Prism
V: 13  |  E: 22  |  F: 11
Faces: 4{3} · 5{4} · 2{6}
Symmetry: C2v
J55 Parabiaugmented Hexagonal Prism
J55 — Parabiaugmented Hexagonal Prism
V: 14  |  E: 26  |  F: 14
Faces: 8{3} · 4{4} · 2{6}
Symmetry: D2h
J56 Metabiaugmented Hexagonal Prism
J56 — Metabiaugmented Hexagonal Prism
V: 14  |  E: 26  |  F: 14
Faces: 8{3} · 4{4} · 2{6}
Symmetry: C2v
J57 Triaugmented Hexagonal Prism
J57 — Triaugmented Hexagonal Prism
V: 15  |  E: 30  |  F: 17
Faces: 12{3} · 3{4} · 2{6}
Symmetry: D3h
J58 Augmented Dodecahedron
J58 — Augmented Dodecahedron
V: 21  |  E: 35  |  F: 16
Faces: 5{3} · 11{5}
Symmetry: C5v
J59 Parabiaugmented Dodecahedron
J59 — Parabiaugmented Dodecahedron
V: 22  |  E: 40  |  F: 20
Faces: 10{3} · 10{5}
Symmetry: D5d
J60 Metabiaugmented Dodecahedron
J60 — Metabiaugmented Dodecahedron
V: 22  |  E: 40  |  F: 20
Faces: 10{3} · 10{5}
Symmetry: C2v
J61 Triaugmented Dodecahedron
J61 — Triaugmented Dodecahedron
V: 23  |  E: 45  |  F: 24
Faces: 15{3} · 9{5}
Symmetry: C3v
J62 Metabidiminished Icosahedron
J62 — Metabidiminished Icosahedron
V: 10  |  E: 20  |  F: 12
Faces: 10{3} · 2{5}
Symmetry: C2v
J63 Tridiminished Icosahedron
J63 — Tridiminished Icosahedron
V: 9  |  E: 15  |  F: 8
Faces: 5{3} · 3{5}
Symmetry: C3v
J64 Augmented Tridiminished Icosahedron
J64 — Augmented Tridiminished Icosahedron
V: 10  |  E: 18  |  F: 10
Faces: 7{3} · 3{5}
Symmetry: C3v
J65 Augmented Truncated Tetrahedron
J65 — Augmented Truncated Tetrahedron
V: 15  |  E: 27  |  F: 14
Faces: 8{3} · 3{4} · 3{6}
Symmetry: C3v
J66 Augmented Truncated Cube
J66 — Augmented Truncated Cube
V: 28  |  E: 48  |  F: 22
Faces: 12{3} · 5{4} · 5{8}
Symmetry: C4v
J67 Biaugmented Truncated Cube
J67 — Biaugmented Truncated Cube
V: 32  |  E: 60  |  F: 30
Faces: 16{3} · 10{4} · 4{8}
Symmetry: D4h
J68 Augmented Truncated Dodecahedron
J68 — Augmented Truncated Dodecahedron
V: 65  |  E: 105  |  F: 42
Faces: 25{3} · 5{4} · 1{5} · 11{10}
Symmetry: C5v
J69 Parabiaugmented Truncated Dodecahedron
J69 — Parabiaugmented Truncated Dodecahedron
V: 70  |  E: 120  |  F: 52
Faces: 30{3} · 10{4} · 2{5} · 10{10}
Symmetry: D5d
J70 Metabiaugmented Truncated Dodecahedron
J70 — Metabiaugmented Truncated Dodecahedron
V: 70  |  E: 120  |  F: 52
Faces: 30{3} · 10{4} · 2{5} · 10{10}
Symmetry: C2v
J71 Triaugmented Truncated Dodecahedron
J71 — Triaugmented Truncated Dodecahedron
V: 75  |  E: 135  |  F: 62
Faces: 35{3} · 15{4} · 3{5} · 9{10}
Symmetry: C3v
J72 Gyrate Rhombicosidodecahedron
J72 — Gyrate Rhombicosidodecahedron
V: 60  |  E: 120  |  F: 62
Faces: 20{3} · 30{4} · 12{5}
Symmetry: C5v
J73 Parabigyrate Rhombicosidodecahedron
J73 — Parabigyrate Rhombicosidodecahedron
V: 60  |  E: 120  |  F: 62
Faces: 20{3} · 30{4} · 12{5}
Symmetry: D5d
J74 Metabigyrate Rhombicosidodecahedron
J74 — Metabigyrate Rhombicosidodecahedron
V: 60  |  E: 120  |  F: 62
Faces: 20{3} · 30{4} · 12{5}
Symmetry: C2v
J75 Trigyrate Rhombicosidodecahedron
J75 — Trigyrate Rhombicosidodecahedron
V: 60  |  E: 120  |  F: 62
Faces: 20{3} · 30{4} · 12{5}
Symmetry: C3v
J76 Diminished Rhombicosidodecahedron
J76 — Diminished Rhombicosidodecahedron
V: 55  |  E: 105  |  F: 52
Faces: 15{3} · 25{4} · 11{5} · 1{10}
Symmetry: C5v
J77 Paragyrate Diminished Rhombicosidodecahedron
J77 — Paragyrate Diminished Rhombicosidodecahedron
V: 55  |  E: 105  |  F: 52
Faces: 15{3} · 25{4} · 11{5} · 1{10}
Symmetry: Cs
J78 Metagyrate Diminished Rhombicosidodecahedron
J78 — Metagyrate Diminished Rhombicosidodecahedron
V: 55  |  E: 105  |  F: 52
Faces: 15{3} · 25{4} · 11{5} · 1{10}
Symmetry: Cs
J79 Bigyrate Diminished Rhombicosidodecahedron
J79 — Bigyrate Diminished Rhombicosidodecahedron
V: 55  |  E: 105  |  F: 52
Faces: 15{3} · 25{4} · 11{5} · 1{10}
Symmetry: Cs
J80 Parabidiminished Rhombicosidodecahedron
J80 — Parabidiminished Rhombicosidodecahedron
V: 50  |  E: 90  |  F: 42
Faces: 10{3} · 20{4} · 10{5} · 2{10}
Symmetry: D5d
J81 Metabidiminished Rhombicosidodecahedron
J81 — Metabidiminished Rhombicosidodecahedron
V: 50  |  E: 90  |  F: 42
Faces: 10{3} · 20{4} · 10{5} · 2{10}
Symmetry: C2v
J82 Gyrate Bidiminished Rhombicosidodecahedron
J82 — Gyrate Bidiminished Rhombicosidodecahedron
V: 50  |  E: 90  |  F: 42
Faces: 10{3} · 20{4} · 10{5} · 2{10}
Symmetry: Cs
J83 Tridiminished Rhombicosidodecahedron
J83 — Tridiminished Rhombicosidodecahedron
V: 45  |  E: 75  |  F: 32
Faces: 5{3} · 15{4} · 9{5} · 3{10}
Symmetry: C3v
J84 Snub Disphenoid
J84 — Snub Disphenoid
V: 8  |  E: 18  |  F: 12
Faces: 12{3}
Symmetry: D2d
aka: Siamese Dodecahedron
J85 Snub Square Antiprism
J85 — Snub Square Antiprism
V: 16  |  E: 40  |  F: 26
Faces: 24{3} · 2{4}
Symmetry: D4d
J86 Sphenocorona
J86 — Sphenocorona
V: 10  |  E: 22  |  F: 14
Faces: 12{3} · 2{4}
Symmetry: C2v
J87 Augmented Sphenocorona
J87 — Augmented Sphenocorona
V: 11  |  E: 26  |  F: 17
Faces: 16{3} · 1{4}
Symmetry: Cs
J88 Sphenomegacorona
J88 — Sphenomegacorona
V: 12  |  E: 28  |  F: 18
Faces: 16{3} · 2{4}
Symmetry: C2v
J89 Hebesphenomegacorona
J89 — Hebesphenomegacorona
V: 14  |  E: 33  |  F: 21
Faces: 18{3} · 3{4}
Symmetry: C2v
J90 Disphenocingulum
J90 — Disphenocingulum
V: 16  |  E: 38  |  F: 24
Faces: 20{3} · 4{4}
Symmetry: D2d
J91 Bilunabirotunda
J91 — Bilunabirotunda
V: 14  |  E: 26  |  F: 14
Faces: 8{3} · 2{4} · 4{5}
Symmetry: D2h
J92 Triangular Hebesphenorotunda
J92 — Triangular Hebesphenorotunda
V: 18  |  E: 36  |  F: 20
Faces: 13{3} · 3{4} · 3{5} · 1{6}
Symmetry: C3v
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