Johnson's polyhedra

A Johnson's polyhedron is convex with regular faces - thus all its edges have same length - which is neither regular or archimedean, nor a prism or an anti-prism. There are 92. You get most of them by modifying pyramids, regular or archimedean solids, prisms or anti-prisms:
prisms, cupolas and rotunda (J1-6), modified pyramids and dipyramids (J7-17), modified cupolas and rotundas (J18-48), augmented prisms ((J49-57), modified Platonic solids J58-64), modified Archimedes solids (J65-83), various (J84-92).
J37 is the only one whose vertices are superimposables (order 4, three squares and one triangle).
Only five are chiral (exist in two mirror image versions) : the gyroelongated polyhedra (modified by inserting an antiprism in the place of a facet) J44-48.

The table below shows their complete list in the order defined by Johnson, with the numbers of faces and vertices of each type (fn is a regular polygon with n sides, and p faces and p edges meet at a vertex vp).


name (with link to popup applet)
  Jnn faces - vertices  

image

face's colors: f3 f4 f5 f6 f8 f10

square pyramid
  J01 4f3 1f5 - 4v3 1v4  
J 01
pentagonal pyramid
  J02 5f3 1f5 - 5v3 1v5  
J 02
triangular cupola
  J03 4f3 3f4 1f6 - 6v3 3v4  
J 03
square cupola
  J04 4f3 5f4 1f8 - 8v3 4v4  
J 04
pentagonal cupola
  J05 5f3 5f4 1f5 1f10 - 10v3 5v4  
J 05
pentagonal rotunda
  J06 10f3 6f5 1f10 - 10v3 10v4  
J 06
elongated triangular pyramid
  J07 4f3 3f4 - 4v3 3v4  
J 07
elongated square pyramid
  J08 4f3 5f4 - 4v3 5v4  
J 08
elongated pentagonal pyramid
  J09 5f3 5f4 1f5 - 5v3 5v4 1v5  
J 09
gyroelongated square pyramid
  J10 12f3 1f4 - 5v4 4v5  
J 10
gyroelongated pentagonal pyramid
  J11 15f3 1f5 - 5v4 6v5  
J 11
triangular dipyramid
  J12 6f3 - 2v3 3v4  
J 12
pentagonal dipyramid
  J13 10f3 - 5v4 2v5  
J 13
elongated triangular dipyramid
  J14 6f3 3f4 - 2v3 6v4  
J 14
elongated square dipyramid
  J15 8f3 4f4 - 10v4  
J 15
elongated pentagonal dipyramid
  J16 10f3 5f4 - 10v4 2v5  
J 16
gyroelongated square dipyramid
  J17 16f3 - 2v4 8v5  
J 17
elongated triangular cupola
  J18 4f3 9f4 1f6 - 6v3 9v4  
J 18
elongated square cupola
  J19 4f3 13f4 1f8 - 8v3 12v4  
J 19
elongated pentagonal cupola
  J20 5f3 15f4 1f5 1f10 - 10v3 15v4  
J 20
elongated pentagonal rotunda
  J21 10f3 10f4 6f5 1f10 - 10v3 20v4  
J 21
gyroelongated triangular cupola
  J22 16f3 3f4 1f6 - 9v4 6v5  
J 22
gyroelongated square cupola
  J23 20f3 5f4 1f8 - 12v4 8v5  
J 23
gyroelongated pentagonal cupola
  J24 25f3 5f4 1f5 1f10 - 15v4 10v5  
J 24
gyroelongated pentagonal rotunda
  J25 25f3 5f4 1f5 1f10 - 15v4 10v5  
J 25
triangular gyrobiprism
  J26 4f3 4f4 - 4v3 4v4  
J 26
triangular orthobicupola
  J27 8f3 6f4 - 12v4  
J 27
square orthobicupola
  J28 8f3 10f4 - 16v4  
J 28
square gyrobicupola
  J29 8f3 10f4 - 16v4  
J 29
pentagonal orthobicupola
  J30 10f3 10f4 2f5 - 20v4  
J 30
pentagonal gyrobicupola
  J31 10f3 10f4 2f5 - 20v4  
J 31
pentagonal orthocupolarotunda
  J32 15f3 5f4 7f5 - 25v4  
J 32
pentagonal gyrocupolarotunda
  J33 15f3 5f4 7f5 - 25v4  
J 33
pentagonal orthobirotunda
  J34 20f3 12f5 - 30v4  
J 34
elongated triangular orthobicupola
  J35 8f3 12f4 - 18v4  
J 35
elongated triangular gyrobicupola
  J36 8f3 12f4 - 18v4  
J 36
elongated square gyrobicupola
  J37 8f3 18f4 - 24v4  
J 37
elongated pentagonal orthobicupola
  J38 10f3 20f4 2f5 - 30v4  
J 38
elongated pentagonal gyrobicupola
  J39 10f3 20f4 2f5 - 30v4  
J 39
elongated pentag. orthocupolarotunda
  J40 15f3 15f4 7f5 - 35v4  
J 40
elongated pentag. gyrocupolarotunda
  J41 15f3 15f4 7f5 - 35v4  
J 41
elongated pentagonal orthobirotunda
  J42 20f3 10f4 12f5 - 40v4  
J 42
elongated pentagonal gyrobirotunda
  J43 20f3 10f4 12f5 - 40v4  
J 43
gyroelongated triangular bicupola
  J44 20f3 6f4 - 6v4 12v5  
J 44
gyroelongated square bicupola
  J45 24f3 10f4 - 8v4 16v5  
J 45
gyroelongated pentagonal bicupola
  J46 30f3 10f4 2f5 - 10v4 20v5  
J 46
gyroelongated pentag. cupolarotunda
  J47 35f3 5f4 7f5 - 15v4 20v5  
J 47
gyroelongated pentag. birotunda
  J48 40f3 12f5 - 20v4 20v5  
J 48
augmented triangular prism
  J49 6f3 2f4 - 2v3 5v4  
J 49
biaugmented triangular prism
  J50 10f3 1f4 - 6v4 2v5  
J 50
triaugmented triangular prism
  J51 14f3 - 3v4 6v5  
J 51
augmented pentagonal prism
  J52 4f3 4f4 2f5 - 6v3 5v4  
J 52
biaugmented pentagonal prism
  J53 8f3 3f4 2f5 - 2v3 10v4  
J 53
augmented hexagonal prism
  J54 4f3 5f4 2f6 - 8v3 5v4  
J 54
parabiaugmented hexagonal prism
  J55 8f3 4f4 2f6 - 4v3 10v4  
J 55
metabiaugmented hexagonal prism
  J56 8f3 4f4 2f6 - 4v3 10v4  
J 56
triaugmented hexagonal prism
  J57 12f3 3f4 2f6 - 15v4  
J 57
augmented dodecahedron
  J58 5f3 11f5 - 15v3 5v4 1v5  
J 58
parabiaugmented dodecahedron
  J59 10f3 10f5 - 10v3 10v4 2v5  
J 59
metabiaugmented dodecahedron
  J60 10f3 10f5 - 10v3 10v4 2v5  
J 60
triaugmented dodecahedron
  J61 15f3 9f5 - 5v3 15v4 3v5  
J 61
metabidiminished icosahedron
  J62 10f3 2f5 - 2v3 6v4 2v5  
J 62
tridiminished icosahedron
  J63 5f3 3f5 - 6v3 3v4  
J 63
augmented tridiminished icosahedron
  J64 7f3 3f5 - 4v3 6v4  
J 64
augmented truncated tetrahedron
  J65 8f3 3f4 3f6 - 6v3 9v4  
J 65
augmented truncated cube
  J66 12f3 5f4 5f8 - 16v3 12v4  
J 66
biaugmented truncated cube
  J67 16f3 10f4 4f8 - 8v3 24v4  
J 67
augmented truncated dodecahedron
  J68 25f3 5f4 1f5 11f10 - 50v3 15v4  
J 68
parabiaugmented trunc. dodecahedron
  J69 30f3 10f4 2f5 10f10 - 40v3 30v4  
J 69
metabiaugmented trunc. dodecahedron
  J70 30f3 10f4 2f5 10f10 - 40v3 30v4  
J 70
triaugmented trunc. dodecahedron
  J71 35f3 15f4 3f5 9f10 - 30v3 45v4  
J 71
gyrate rhombicosidodecahedron
  J72 20f3 30f4 12f5 - 60v4  
J 72
parabigyrate rhombicosidodecahedron
  J73 20f3 30f4 12f5 - 60v4  
J 73
metabigyrate rhombicosidodec.
  J74 20f3 30f4 12f5 - 60v4  
J 74
trigyrate rhombicosidodecahedron
  J75 20f3 30f4 12f5 - 60v4  
J 75
diminished rhombicosidodecahedron
  J76 15f3 25f4 11f5 1f10 - 10v3 45v4  
J 76
paragyrate dimin. rhombicosidodec.
  J77 15f3 25f4 11f5 1f10 - 10v3 45v4  
J 77
metagyrate dimin. rhombicosidodec.
  J78 15f3 25f4 11f5 1f10 - 10v3 45v4  
J 78
bigyrate diminished rhombicosidodec.
  J79 15f3 25f4 11f5 1f10 - 10v3 45v4  
J 79
parabidiminished rhombicosidodec.
  J80 10f3 20f4 10f5 2f10 - 20v3 30v4  
J 80
metabidiminished rhombicosidodec.
  J81 10f3 20f4 10f5 2f10 - 20v3 30v4  
J 81
gyrate bidiminished rhombicosidodec.
  J82 10f3 20f4 10f5 2f10 - 20v3 30v4  
J 82
tridiminished rhombicosidodecahedron
  J83 5f3 15f4 9f5 3f10 - 30v3 15v4  
J 83
snub disphenoid
  J84 12f3 - 4v4 4v5  
J 84
snub square antiprism
  J85 24f3 2f4 - 16v5  
J 85
sphenocorona
  J86 12f3 2f4 - 6v4 4v5  
J 86
augmented sphenocorona
  J87 16f3 1f4 - 3v4 8v5  
J 87
sphenomegacorona
  J88 16f3 2f4 - 4v4 8v5  
J 88
hebesphenomegacorona
  J89 18f3 3f4 - 4v4 10v5  
J 89
disphenocingulum
  J90 20f3 4f4 - 4v4 12v5  
J 90
bilunabirotunda
  J91 8f3 2f4 4f5 - 4v3 10v4  
J 91
triangular hebesphenorotunda
  J92 13f3 3f4 3f5 1f6 - 18v4  
J 92

These images and the data for the applets have been produce using Hedron, the polyhedra software by Jim McNeill (label "Wales" on the link globe).

references: •  The Johnson Solids
•  Generalized Johnson solids (coplanar faces allowed)

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects November 1998
updated 24-08-2009