# 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
pentagonal pyramid
 J02 5f3 1f5 - 5v3 1v5
triangular cupola
 J03 4f3 3f4 1f6 - 6v3 3v4
square cupola
 J04 4f3 5f4 1f8 - 8v3 4v4
pentagonal cupola
 J05 5f3 5f4 1f5 1f10 - 10v3 5v4
pentagonal rotunda
 J06 10f3 6f5 1f10 - 10v3 10v4
elongated triangular pyramid
 J07 4f3 3f4 - 4v3 3v4
elongated square pyramid
 J08 4f3 5f4 - 4v3 5v4
elongated pentagonal pyramid
 J09 5f3 5f4 1f5 - 5v3 5v4 1v5
gyroelongated square pyramid
 J10 12f3 1f4 - 5v4 4v5
gyroelongated pentagonal pyramid
 J11 15f3 1f5 - 5v4 6v5
triangular dipyramid
 J12 6f3 - 2v3 3v4
pentagonal dipyramid
 J13 10f3 - 5v4 2v5
elongated triangular dipyramid
 J14 6f3 3f4 - 2v3 6v4
elongated square dipyramid
 J15 8f3 4f4 - 10v4
elongated pentagonal dipyramid
J16 10f3 5f4 - 10v4 2v5
gyroelongated square dipyramid
J17 16f3 - 2v4 8v5
elongated triangular cupola
 J18 4f3 9f4 1f6 - 6v3 9v4
elongated square cupola
 J19 4f3 13f4 1f8 - 8v3 12v4
elongated pentagonal cupola
 J20 5f3 15f4 1f5 1f10 - 10v3 15v4
elongated pentagonal rotunda
 J21 10f3 10f4 6f5 1f10 - 10v3 20v4
gyroelongated triangular cupola
 J22 16f3 3f4 1f6 - 9v4 6v5
gyroelongated square cupola
 J23 20f3 5f4 1f8 - 12v4 8v5
gyroelongated pentagonal cupola
 J24 25f3 5f4 1f5 1f10 - 15v4 10v5
gyroelongated pentagonal rotunda
J25 25f3 5f4 1f5 1f10 - 15v4 10v5
triangular gyrobiprism
J26 4f3 4f4 - 4v3 4v4
triangular orthobicupola
 J27 8f3 6f4 - 12v4
square orthobicupola
 J28 8f3 10f4 - 16v4
square gyrobicupola
 J29 8f3 10f4 - 16v4
pentagonal orthobicupola
 J30 10f3 10f4 2f5 - 20v4
pentagonal gyrobicupola
 J31 10f3 10f4 2f5 - 20v4
pentagonal orthocupolarotunda
 J32 15f3 5f4 7f5 - 25v4
pentagonal gyrocupolarotunda
J33 15f3 5f4 7f5 - 25v4
pentagonal orthobirotunda
 J34 20f3 12f5 - 30v4
elongated triangular orthobicupola
 J35 8f3 12f4 - 18v4
elongated triangular gyrobicupola
J36 8f3 12f4 - 18v4
elongated square gyrobicupola
 J37 8f3 18f4 - 24v4
elongated pentagonal orthobicupola
 J38 10f3 20f4 2f5 - 30v4
elongated pentagonal gyrobicupola
 J39 10f3 20f4 2f5 - 30v4
elongated pentag. orthocupolarotunda
 J40 15f3 15f4 7f5 - 35v4
elongated pentag. gyrocupolarotunda
 J41 15f3 15f4 7f5 - 35v4
elongated pentagonal orthobirotunda
 J42 20f3 10f4 12f5 - 40v4
elongated pentagonal gyrobirotunda
 J43 20f3 10f4 12f5 - 40v4
gyroelongated triangular bicupola
 J44 20f3 6f4 - 6v4 12v5
gyroelongated square bicupola
 J45 24f3 10f4 - 8v4 16v5
gyroelongated pentagonal bicupola
 J46 30f3 10f4 2f5 - 10v4 20v5
gyroelongated pentag. cupolarotunda
 J47 35f3 5f4 7f5 - 15v4 20v5
gyroelongated pentag. birotunda
 J48 40f3 12f5 - 20v4 20v5
augmented triangular prism
 J49 6f3 2f4 - 2v3 5v4
biaugmented triangular prism
 J50 10f3 1f4 - 6v4 2v5
triaugmented triangular prism
 J51 14f3 - 3v4 6v5
augmented pentagonal prism
 J52 4f3 4f4 2f5 - 6v3 5v4
biaugmented pentagonal prism
 J53 8f3 3f4 2f5 - 2v3 10v4
augmented hexagonal prism
 J54 4f3 5f4 2f6 - 8v3 5v4
parabiaugmented hexagonal prism
 J55 8f3 4f4 2f6 - 4v3 10v4
metabiaugmented hexagonal prism
 J56 8f3 4f4 2f6 - 4v3 10v4
triaugmented hexagonal prism
 J57 12f3 3f4 2f6 - 15v4
augmented dodecahedron
 J58 5f3 11f5 - 15v3 5v4 1v5
parabiaugmented dodecahedron
 J59 10f3 10f5 - 10v3 10v4 2v5
metabiaugmented dodecahedron
 J60 10f3 10f5 - 10v3 10v4 2v5
triaugmented dodecahedron
 J61 15f3 9f5 - 5v3 15v4 3v5
metabidiminished icosahedron
 J62 10f3 2f5 - 2v3 6v4 2v5
tridiminished icosahedron
 J63 5f3 3f5 - 6v3 3v4
augmented tridiminished icosahedron
 J64 7f3 3f5 - 4v3 6v4
augmented truncated tetrahedron
 J65 8f3 3f4 3f6 - 6v3 9v4
augmented truncated cube
 J66 12f3 5f4 5f8 - 16v3 12v4
biaugmented truncated cube
 J67 16f3 10f4 4f8 - 8v3 24v4
augmented truncated dodecahedron
 J68 25f3 5f4 1f5 11f10 - 50v3 15v4
parabiaugmented trunc. dodecahedron
 J69 30f3 10f4 2f5 10f10 - 40v3 30v4
metabiaugmented trunc. dodecahedron
 J70 30f3 10f4 2f5 10f10 - 40v3 30v4
triaugmented trunc. dodecahedron
 J71 35f3 15f4 3f5 9f10 - 30v3 45v4
gyrate rhombicosidodecahedron
 J72 20f3 30f4 12f5 - 60v4
parabigyrate rhombicosidodecahedron
 J73 20f3 30f4 12f5 - 60v4
metabigyrate rhombicosidodec.
 J74 20f3 30f4 12f5 - 60v4
trigyrate rhombicosidodecahedron
 J75 20f3 30f4 12f5 - 60v4
diminished rhombicosidodecahedron
 J76 15f3 25f4 11f5 1f10 - 10v3 45v4
paragyrate dimin. rhombicosidodec.
 J77 15f3 25f4 11f5 1f10 - 10v3 45v4
metagyrate dimin. rhombicosidodec.
 J78 15f3 25f4 11f5 1f10 - 10v3 45v4
bigyrate diminished rhombicosidodec.
 J79 15f3 25f4 11f5 1f10 - 10v3 45v4
parabidiminished rhombicosidodec.
 J80 10f3 20f4 10f5 2f10 - 20v3 30v4
metabidiminished rhombicosidodec.
 J81 10f3 20f4 10f5 2f10 - 20v3 30v4
gyrate bidiminished rhombicosidodec.
 J82 10f3 20f4 10f5 2f10 - 20v3 30v4
tridiminished rhombicosidodecahedron
 J83 5f3 15f4 9f5 3f10 - 30v3 15v4
snub disphenoid
 J84 12f3 - 4v4 4v5
snub square antiprism
 J85 24f3 2f4 - 16v5
sphenocorona
 J86 12f3 2f4 - 6v4 4v5
augmented sphenocorona
 J87 16f3 1f4 - 3v4 8v5
sphenomegacorona
 J88 16f3 2f4 - 4v4 8v5
hebesphenomegacorona
 J89 18f3 3f4 - 4v4 10v5
disphenocingulum
 J90 20f3 4f4 - 4v4 12v5
bilunabirotunda
 J91 8f3 2f4 4f5 - 4v3 10v4
triangular hebesphenorotunda
 J92 13f3 3f4 3f5 1f6 - 18v4

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)

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