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).  The names of the majority of the Johnson Solids were ascribed by Johnson at this time, indeed in a communication dated December 2004, Dr Johnson states '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 of the Johnson solids.  All links are to HEDRON 'switchable' VRML files.

Polyhedra can also be generated which are isomorphs of the Johnson solids.  In cases where the solid contains a {5}, {8}, or {10}-gon then these can be replaced by {5/2}, {8/3} or {10/3}-gons respectively.  With vertices crossed and the symmetry of the original Jonhson Solid retained, then some of these polyhedra appear to pass the criteria for being prefixed "Great".  I am aware though that this prefix is normally reserved for Uniform Polyhedra.  Where an isomorph exists which does not appear to meet the 'Great' criteria I have tried to give as descriptive a name as possible.  All isomorph names are of my own invention.

Alex Doskey has also introduced a page to his website where he discusses 'Convex Diamond Regular Polyhedra', an extension to the Johnson Solids, Steve Waterman also has a similar page.  Alex and Steve catalogue 70 additional polyhedra which result from allowing one extra polygon, a rhombus formed from two coplanar triangles.

Thanks to Alex Doskey for providing the numbers in the second column.

 Number - Name Faces, Edges, Vertices Isomorphs and links to other pages on this site J01 - Square Pyramid 5,8,5 J02 - Pentagonal Pyramid 6,10,6 J03 - Triangular Cupola 8,15,9 J04 - Square Cupola 10,20,12 GJ04 -  "Great" Square Cupola, or 4/3-cupola (see also my Cupola page) J05 - Pentagonal Cupola 12,25,15 GJ05 -  "Great" Pentagonal Cupola, or 5/3-cupola (see also my Cupola page) J06 - Pentagonal Rotunda 17,35,20 GJ06 - "Great" Pentagonal Rotunda J07 - Elongated Triangular Pyramid 7,12,7 J08 - Elongated Square Pyramid 9,16,9 J09 - Elongated Pentagonal Pyramid 11,20,11 J10 - Gyroelongated Square Pyramid 13,20,9 J11 - Gyroelongated Pentagonal Pyramid 16,25,11 GJ11 - Diminished Great Icosahedron J12 - Triangular Dipyramid 6,9,5 J13 - Pentagonal Dipyramid 10,15,7 J14 - Elongated Triangular Dipyramid 9,15,8 J15 - Elongated Square Dipyramid 12,20,10 J16 - Elongated Pentagonal Dipyramid 15,25,12 J17 - Gyroenlongated Square Dipyramid 16,24,10 J18 - Elongated Triangular Cupola 14,27,15 J19 - Elongated Square Cupola 18,36,20 J20 - Elongated Pentagonal Cupola 22,45,25 J21 - Elongated Pentagonal Rotunda 27,55,30 J22 - Gyroelongated Triangular Cupola 20,33,15 J23 - Gyroelongated Square Cupola 26,44,20 J24 - Gyroelongated Pentagonal Cupola 32,55,25 J25 - Gyroelongated Pentagonal Rotunda 37,65,30 J26 - Gyrobifastigium 8,14,8 J27 - Triangular Orthobicupola 14,24,12 J28 - Square Orthobicupola 18,32,16 J29 - Square Gyrobicupola 18,32,16 J30 - Pentagonal Orthobicupola 22,40,20 GJ30 - "Great" Pentagonal Orthobicupola J31 - Pentagonal Gyrobicupola 22,40,20 GJ31i - "Great" Pentagonal Gyrobicupola (inverted) GJ31e - "Great" Pentagonal Gyrobicupola (everted) J32 - Pentagonal Orthocupolarotunda 27,50,25 GJ32i - "Great" Pentagonal Orthocupolarotunda (Cupola inverted) GJ32e - "Great" Pentagonal Orthocupolarotunda (Cupola everted) J33 - Pentagonal Gyrocupolarotunda 27,50,25 GJ33i - "Great" Pentagonal Gyrocupolarotunda (Cupola inverted) GJ33e - "Great" Pentagonal Gyrocupolarotunda (Cupola everted) J34 - Pentagonal Orthobirotunda 32,30,60 GJ34 - "Great" Pentagonal Orthobirotunda J35 - Elongated Triangular Orthobicupola 20,36,18 J36 - Elongated Triangular Gyrobicupola 20,36,18 J37 - Elongated Square Gyrobicupola (Miller's Solid) 26,48,24 GJ37 - "Great" Elogated Square Gyrobicupola J38 - Elongated Pentagonal Orthobicupola 32,60,30 J39 - Elongated Pentagonal Gyrobicupola 32,60,30 J40 - Elongated Pentagonal Orthocupolarotunda 37,70,35 J41 - Elongated Pentagonal Gyrocupolarotunda 37,70,35 J42 - Elongated Pentagonal Orthobirotunda 42,80,40 J43 - Elongated Pentagonal Gyrobirotunda 42,80,40 J44 - Gyroelongated Triangular Bicupola 26,42,18 J45 - Gyroelongated Square Bicupola 34,56,24 J46 - Gyroelongated Pentagonal Bicupola 42,70,30 J47 - Gyroelongated Pentagonal Cupolarotunda 47,80,35 J48 - Gyroelongated Pentagonal Birotunda 52,90,40 J49 - Augmented Triangular Prism 8,13,7 J50 - Biaugmented Triangular Prism 11,17,8 J51 - Triaugmented Triangular Prism 14,21,9 J51i - Inverted Triaugmented Triangular Prism J52 - Augmented Pentagonal Prism 10,19,11 J53 - Biaugmented Pentagonal Prism 13,23,12 J54 - Augmented Hexagonal Prism 11,22,13 J55 - Parabiaugmented Hexagonal Prism 14,26,14 J56 - Metabiaugmented Hexagonal Prism 14,26,14 J57 - Triaugmented Hexagonal Prism 17,30,15 J58 - Augmented Dodecahedron 16,21,35 GJ58 - Augmented Great Stellated Dodecahedron J59 - Parabiaugmented Dodecahedron 20,40,22 GJ59 - Parabiaugmented Great Stellated Dodecahedron J60 - Metabiaugmented Dodecahedron 20,42,22 GJ60 - Metabiaugmented Great Stellated Dodecahedron J61 - Triaugmented Dodecahedron 24,45,23 GJ61 - Triaugmented Great Stellated Dodecahedron J62 - Metabidiminished Icosahedron 12,20,10 GJ62 - Metabidiminished Great Icosahedron J63 - Tridiminished Icosahedron 8,15,9 GJ63 - Tridiminished Great Icosahedron J64 - Augmented Tridiminished Icosahedron 10,18,10 J65 - Augmented Truncated Tetrahedron 14,27,15 J66 - Augmented Truncated Cube 22,48,28 GJ66 - Augmented Truncated Stellated Hexahedron J67 - Biaugmented Truncated Cube 32,60,30 GJ67 - Biaugmented Truncated Stellated Hexahedron J68 - Augmented Truncated Dodecahedron 42,105,65 GJ68 - Augmented Great Truncated Stellated Dodecahedron J69 - Parabiaugmented Truncated Dodecahedron 52,120,70 GJ69 - Parabiaugmented Great Truncated Stellated Dodecahedron J70 - Metabiaugmented Truncated Dodecahedron 52,120,70 GJ70 - Metabiaugmented Great  Truncated Stellated Dodecahedron J71 - Triaugmented Truncated Dodecahedron 62,135,75 GJ71 - Triaugmented Great Truncated Stellated Dodecahedron J72 - Gyrate Rhombicosidodecahedron 62,120,60 GJ72 - Gyrate Great Rhombicosidodecahedron (see also my Great Rhombicosidodecahedron page) J73 - Parabigyrate Rhombicosidodecahedron 62,120,60 GJ73 - Parabigyrate Great Rhombicosidodecahedron J74 - Metabigyrate Rhombicosidodecahedron 62,120,60 GJ74 - Metabigyrate Great  Rhombicosidodecahedron J75 - Trigyrate Rhombicosidodecahedron 62,120,60 GJ75 - Trigyrate Great  Rhombicosidodecahedron J76 - Diminished Rhombicosidodecahedron 52,105,55 GJ76 - Diminished Great  Rhombicosidodecahedron J77 - Paragyrate Diminished Rhombicosidodecahedron 52,105,55 GJ77 - Paragyrate Diminished Great  Rhombicosidodecahedron J78 - Metagyrate Diminished Rhombicosidodecahedron 52,105,55 GJ78 - Metagyrate Diminished Great Rhombicosidodecahedron J79 - Bigyrate Diminished Rhombicosidodecahedron 52,105,55 GJ79 - Bigyrate Diminished Great Rhombicosidodecahedron J80 - Parabidiminished Rhombicosidodecahedron 42,90,50 GJ80 - Parabidiminished Great Rhombicosidodecahedron J81 - Metabidiminished Rhombicosidodecahedron 42,90,50 GJ81 - Metabidiminished Great Rhombicosidodecahedron J82 - Gyrate Bidiminished Rhombicosidodecahedron 42,90,50 GJ82 - Gyrate Bidiminished Great Rhombicosidodecahedron J83 - Tridiminished Rhombicosidodecahedron 32,75,45 GJ83 - Tridiminished Great Rhombicosidodecahedron J84 - Snub Disphenoid (Siamese Dodecahedron) 12,18,8 J85 - Snub Square Antiprism 26,40,16 GJ85 - "Great " Snub Square Antiprism (see also my Snub Anti-prisms page) J86 - Sphenocorona 14,22,10 GJ86 - "Great" Sphenocorona (see also my  Sphenocoronas page) J87 - Augmented Sphenocorona 17,26,11 J88 - Sphenomegacorona 18,28,12 (see also my Sphenomegacoronas page) J89 - Hebesphenomegacorona 21,33,14 GJ89 - "Great"  Hebesphenomegacorona (see also my  Hebesphenomegacoronas page) J90 - Disphenocingulum 24,38,16 GJ90 - "Great" Disphenocingulum (see also my Gyro-elongated Spheno-prisms page and this quasi-gyro-disphenocingulum) J91 - Bilunabirotunda 14,26,14 J91i - "Inverted" Bilunabirotunda GJ91 - "Great" Bilunabirotunda (combo model) GJ91i - "Great Inverted" Bilunabirotunda J92 - Triangular Hebesphenorotunda 20,36,18 GJ92 - "Great" Triangular Hebesphenorotunda