Convex regular-faced polyhedra with conditional edges

I created this page to publicize the classification of convex regular-faced polyhedra with conditional edges, a very interesting result and including some very interesting polyhedra which deserve to be better-known among polyhedron enthusiasts.

These polyhedra are a natural generalization of the Johnson solids, relaxing the requirement of strict convexity so that adjacent regular faces can be co-planar, forming a single parquet face, but requiring that all vertices be proper (informally that all vertices of the regular faces are actually vertices of the parquet faces rather than interior to them). A conditional edge is an edge, of dihedral angle π, between co-planar faces.

Allowing conditional edges but requiring proper vertices, there are five varieties of parquet faces possible: the rhombus "3+3" composed of two equilateral triangles; the square and triangle "3+4"; the pentagon and triangle "3+5"; the square and two opposite triangles "3+4+3"; and the pentagon and two opposite triangles "3+5+3".

There are 78 convex regular-faced polyhedra with conditional edges and all proper vertices. The bulk of these polyhedra (the 71 containing a rhombus 3+3) were listed by Alex Doskey, Roger Kaufman, and Steve Waterman in 2006. The listing was independently produced and proven complete in 2010 by A. V. Timofeenko.

The further generalization, allowing conditional vertices, is known to admit of a finitary classification, and is ongoing work for Timofeenko and his students. Some examples of such polyhedra have been produced by Roger Kaufman and others.

Another page gives more exposition and references.

The List

The table below shows the 78 polyhedra.

The numbering Pn,k is from Timofeenko (2011; his 2009 paper does not consider the tripartite faces 3+4+3 and 3+5+3 and so the numbering is slightly different). The numbering Sn is from Gurin-Zalgaller 2008.

The names are generally taken from Timofeenko. Some names labeled (GZ) are more fanciful names given in Gurin-Zalgaller; others labeled (S) are my own invention. I have taken some liberties with translating Timofeenko's names; in particular I use "para-augmented" and "meta-augmented" for the two ways of augmenting a pentagonal rotunda; "gyrate augmented" for augmentations which are also found in a different orientation among the Johnson solids; and various modifiers to describe the 5 different ways of biaugmenting an orthobirotunda and the 3 different ways of triaugmenting one.

The viewers require Java. The first opening of a live viewer is likely to be very slow as your browser starts Java. After the first others will start more quickly.

Pn,k Sn Name Composition Image/Viewer V E F F3 F4 F5 F6 F8 F10 F3+3 Fetc
Q1 Q1 oblique hexagonal prism,
Ivanov solid Q1
noncomposite 12 18 8 2 2 4
Q2 Q2 hexarhombic dodecahedron (S),
Ivanov solid Q2
noncomposite 18 28 12 4 4 4
Q3 Q3 Ivanov solid Q3 noncomposite 15 29 16 9 2 3 2
Q4 Q4 Ivanov solid Q4 noncomposite 15 27 14 5 2 3 4
Q5 Q5 Ivanov solid Q5 noncomposite 22 42 22 10 4 2 2 4
Q6 Q6 Pryakhin solid Q6 noncomposite 18 33 17 7 3 3 1 3
P2,2 S3 rhombic prism,
bifastigium (S)
Π3 + Π3' 8 12 6 4 2
P2,3 S4 3+4-prism (S) Π3 + Π4 10 15 7 5 2
P2,4 S5 3+5-prism (S) Π3 + Π5 12 18 8 6 2
P2,22 S14 oblique triangular prism,
augmented square pyramid
M1 + M2 6 9 5 2 1 2
P2,25 S17 augmented triangular cupola M2 + M4 10 16 8 2 2 1 3
P2,29 S22 augmented bilunabirotunda M3 + M8 15 29 16 9 2 3 2
P2,30 S46 meta-augmented pentagonal rotunda M3 + M9 21 36 17 7 5 1 4
P2,31 S24 para-augmented pentagonal rotunda,
pentagonal helmet (GZ)
M3 + M9' 21 35 16 5 5 1 5
P2,33 S2 augmented triangular hebesphenorotunda M3 + M20,
M3 + Q6
19 38 21 12 3 2 1 3
P2,34 S1 metabiaugmented bilunabirotunda M3 + M8 + M3',
M3 + Q4
16 32 18 10 2 2 4
P2,38 S59 gyrate augmented truncated tetrahedron M4 + M10' 15 24 11 2 3 3 3
P2,42 S60 gyrate augmented truncated cube M5 + M11' 28 44 18 4 5 5 4
P2,48 S63 gyrate augmented truncated dodecahedron M6 + M12' 65 100 37 15 5 1 11 5
P3,1 S6 augmented rhombic prism,
augmented bifastigium (S)
Π3 + Π3' + M2 9 16 9 4 3 2
P3,2 S10 elongated gyrobifastigium (S) Π3 + Π4 + Π3 12 18 8 4 4
P3,3 S11 3+4+3-prism (S),
elongated bifastigium (S)
Π3 + Π4 + Π3' 12 18 8 6 2
P3,4 S9 elongated augmented triangular prism (S),
augmented 3+4-prism (S)
Π3 + Π4 + M2 11 19 10 4 4 2
P3,5 S12 3+5+3-prism (S) Π3 + Π5 + Π3 14 21 9 7 2
P3,6 S13 augmented 3+5-prism (S) Π3 + Π5 + M2 13 22 11 4 5 2
P3,22 S40 elongated para-augmented pentagonal rotunda Π10 + M9 + M3 31 55 26 5 10 5 1 5
P3,31 S41 gyroelongated para-augmented pentagonal rotunda A10 + M9 + M3 31 65 36 25 5 1 5
P3,33 S15 augmented octahedron,
trirhomb (GZ)
M1 + M2 + M2 7 12 7 4 3
P3,34 S18 augmented triangular orthobicupola M2 + M4 + M4 13 25 14 6 5 3
P3,35 S20 augmented cuboctahedron M2 + M4 + M4' 13 24 13 4 5 4
P3,36 S23 parabiaugmented bilunabirotunda M3 + M8 + M3 16 32 18 10 2 2 4
P3,37 S47 metabiaugmented pentagonal cupola M3 + M9 + M3 22 37 17 4 4 1 8
P3,38 S48 meta-augmented pentagonal gyrocupolarotunda M3 + M9 + M6 26 51 27 12 5 6 4
P3,39 S49 meta-augmented pentagonal orthocupolarotunda M3 + M9 + M6' 26 51 27 12 5 6 4
P3,40 S27 augmented icosidodecahedron M3 + M9 + M9 31 60 31 15 11 5
P3,41 S52 meta-augmented pentagonal orthobirotunda M3 + M9 + M9' 31 61 32 17 11 4
P3,42 S25 para-augmented pentagonal gyrocupolarotunda M3 + M9' + M6 26 50 26 10 5 6 5
P3,43 S26 para-augmented pentagonal orthocupolarotunda M3 + M9' + M6' 26 50 26 10 5 6 5
P3,44 S28 para-augmented pentagonal orthobirotunda M3 + M9' + M9 31 60 31 15 11 5
P3,48 S61 gyrate biaugmented truncated cube M5 + M11 + M5' 32 56 26 8 10 4 4
P3,49 S62 bigyrate biaugmented truncated cube M5 + M11' + M5 32 52 22 10 4 8
P3,51 S66 gyrate metabiaugmented truncated dodecahedron M6 + M12 + M6' 70 115 47 20 10 2 10 5
P3,53 S64 gyrate parabiaugmented truncated dodecahedron M6 + M12 + M6''' 70 115 47 20 10 2 10 5
P3,54 S67 bigyrate metabiaugmented truncated dodecahedron M6 + M12' + M6 70 110 42 10 10 2 10 10
P3,55 S65 bigyrate parabiaugmented truncated dodecahedron M6 + M12' + M6' 70 110 42 10 10 2 10 10
P4,1 S7 metabiaugmented rhombic prism Π3 + Π3' + M2 + M2 10 20 12 8 2 2
P4,2 S8 parabiaugmented rhombic prism Π3 + Π3' + M2 + M2' 10 20 12 8 2 2
P4,5 S31 elongated para-augmented pentagonal orthocupolarotunda Π10 + M6 + M9 + M3 36 70 36 10 15 6 5
P4,6 S32 elongated para-augmented pentagonal gyrocupolarotunda Π10 + M6 + M9' + M3 36 70 36 10 15 6 5
P4,7 S33 elongated para-augmented pentagonal orthobirotunda Π10 + M9 + M3 + M9 41 80 41 15 10 11 5
P4,8 S34 elongated para-augmented pentagonal gyrobirotunda Π10 + M9 + M3 + M9' 41 80 41 15 10 11 5
P4,9 S37 gyroelongated para-augmented pentagonal cupolarotunda A10 + M6 + M9 + M3 36 80 46 30 5 6 5
P4,10 S38 gyroelongated para-augmented pentagonal birotunda A10 + M9 + M3 + M9 41 90 51 35 11 5
P4,11 S16 biaugmented octahedron,
hexarhomb (GZ)
M1 + M2 + M2 + M1 8 12 6 6
P4,12 S19 biaugmented triangular orthobicupola M2 + M4 + M4 + M2 14 26 14 4 4 6
P4,13 S21 biaugmented cuboctahedron,
octarhombi expanded cuboctahedron (GZ)
M2 + M4 + M4' + M2 14 24 12 4 8
P4,14 S50 metabiaugmented pentagonal gyrocupolarotunda M3 + M9 + M3 + M6 27 52 27 9 5 5 8
P4,15 S51 metabiaugmented pentagonal orthocupolarotunda M3 + M9 + M3 + M6' 27 52 27 9 5 5 8
P4,16 S44 metabiaugmented icosidodecahedron M3 + M9 + M3 + M9 32 60 30 10 10 10
P4,17 S53 cismetabiaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' 32 62 32 14 10 8
P4,18 S29 parabiaugmented icosidodecahedron M3 + M9 + M9 + M3 32 60 30 10 10 10
P4,19 S57 orthobiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3 32 62 32 14 10 8
P4,20 S58 transmetabiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3' 32 62 32 14 10 8
P4,21 S42 metaparabiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3'' 32 61 31 12 10 9
P4,22 S30 parabiaugmented pentagonal orthobirotunda M3 + M9' + M9 + M3 32 60 30 10 10 10
P4,25 S68 gyrate triaugmented truncated dodecahedron M6 + M12 + M6 + M6' 75 130 57 25 15 3 9 5
P4,26 S69 bigyrate triaugmented truncated dodecahedron M6 + M12 + M6' + M6 75 125 52 15 15 3 9 10
P4,27 S70 trigyrate triaugmented truncated dodecahedron M6 + M12' + M6 + M6 75 120 47 5 15 3 9 15
P4,30 oblique square prism M1 + M2 + (M1 + M2) 8 12 6 2 4
P4,31 doubled augmented square pyramid,
doubled oblique triangular prism,
twist slant square prism (Alex Doskey)
M1 + M2 + (M1 + M2)' 8 14 8 4 2 2
P5,1 S35 elongated parabiaugmented pentagonal orthobirotunda Π10 + M9 + M3 + M9 + M3 42 80 40 10 10 10 10
P5,2 S36 elongated parabiaugmented pentagonal gyrobirotunda Π10 + M9 + M3 + M9' + M3 42 80 40 10 10 10 10
P5,3 S39 gyroelongated parabiaugmented pentagonal birotunda A10 + M9 + M3 + M9 + M3 42 90 50 30 10 10
P5,4 S45 triaugmented icosidodecahedron M3 + M9 + M3 + M9 + M3 33 60 29 5 9 15
P5,5 S55 orthotriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3 33 63 32 11 9 12
P5,6 S56 orthometatriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3' 33 63 32 11 9 12
P5,7 S43 metaparatriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3'' 33 62 31 9 9 13
P6,1 S54 4-augmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3 + M3 34 64 32 8 8 16

Software credits

Inspiration

I became interested in the classification of regular-faced polyhedra due to a toy called Magformers, which is much loved by all from ages 0 to adulthood. A couple of places to see or buy them are here and here. As can be seen some parquet faces are already provided!


Robert R Tupelo-Schneck / <schneck@gmail.com> / 2011-03-31