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User:Tomruen/Convex regular-faced polyhedra with conditional edges

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In geometry there are 78 nonconcave polyhedra which made from regular polygon, besides the 91 Johnson solids, the 13 Archimedean solids, and 5 platonic solids, which include 180 degree dihedral angle edges, excluding vertices with 0 degree angle defect. [1]

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.

A conditional edge is an edge which can be removed which doesn't change the geometry of the polyhedron. Pairs of equilateral triangles for example merge into a rhombic face.

Examples

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Excluded examples with conditional vertices

List

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JE Pn,k Sn Name Composition V E F F3 F4 F5 F6 F8 F10 F3+3 Fetc
1 Q1 Q1 oblique hexagonal prism, Ivanov solid Q1 noncomposite 12 18 8 2 2 4
2 Q2 Q2 hexarhombic dodecahedron (S), Ivanov solid Q2 noncomposite 18 28 12 4 4 4
3 Q3 Q3 Ivanov solid Q3 noncomposite 15 29 16 9 2 3 2
4 Q4 Q4 Ivanov solid Q4 noncomposite 15 27 14 5 2 3 4
5 Q5 Q5 Ivanov solid Q5 noncomposite 22 42 22 10 4 2 2 4
6 Q6 Q6 Pryakhin solid Q6 noncomposite 18 33 17 7 3 3 1 3
7 P2,2 S03 rhombic prism, bifastigium (S) ?3 + ?3' 8 12 6 4 2
8 P2,3 S04 3+4-prism (S) ?3 + ?4 10 15 7 5 2
9 P2,4 S05 3+5-prism (S) ?3 + ?5 12 18 8 6 2
10 P2,22 S14 oblique triangular prism, augmented square pyramid M1 + M2 6 9 5 2 1 2
11 P2,25 S17 augmented triangular cupola M2 + M4 10 16 8 2 2 1 3
12 P2,29 S22 augmented bilunabirotunda M3 + M8 15 29 16 9 2 3 2
13 P2,30 S46 meta-augmented pentagonal rotunda M3 + M9 21 36 17 7 5 1 4
14 P2,31 S24 para-augmented pentagonal rotunda, pentagonal helmet (GZ) M3 + M9' 21 35 16 5 5 1 5
15 P2,33 S02 augmented triangular hebesphenorotunda M3 + M20, M3 + Q6 19 38 21 12 3 2 1 3
16 P2,34 S01 metabiaugmented bilunabirotunda M3 + M8 + M3', M3 + Q4 16 32 18 10 2 2 4
17 P2,38 S59 gyrate augmented truncated tetrahedron M4 + M10' 15 24 11 2 3 3 3
18 P2,42 S60 gyrate augmented truncated cube M5 + M11' 28 44 18 4 5 5 4
19 P2,48 S63 gyrate augmented truncated dodecahedron M6 + M12' 65 100 37 15 5 1 11 5
20 P3,1 S06 augmented rhombic prism, augmented bifastigium (S) ?3 + ?3' + M2 9 16 9 4 3 2
21 P3,2 S10 elongated gyrobifastigium (S) ?3 + ?4 + ?3 12 18 8 4 4
22 P3,3 S11 3+4+3-prism (S), elongated bifastigium (S) ?3 + ?4 + ?3' 12 18 8 6 2
23 P3,4 S09 elongated augmented triangular prism (S), augmented 3+4-prism (S) ?3 + ?4 + M2 11 19 10 4 4 2
24 P3,5 S12 3+5+3-prism (S) ?3 + ?5 + ?3 14 21 9 7 2
25 P3,6 S13 augmented 3+5-prism (S) ?3 + ?5 + M2 13 22 11 4 5 2
26 P3,22 S40 elongated para-augmented pentagonal rotunda ?10 + M9 + M3 31 55 26 5 10 5 1 5
27 P3,31 S41 gyroelongated para-augmented pentagonal rotunda A10 + M9 + M3 31 65 36 25 5 1 5
28 P3,33 S15 augmented octahedron, trirhomb (GZ) M1 + M2 + M2 7 12 7 4 3
29 P3,34 S18 augmented triangular orthobicupola M2 + M4 + M4 13 25 14 6 5 3
30 P3,35 S20 augmented cuboctahedron M2 + M4 + M4' 13 24 13 4 5 4
31 P3,36 S23 parabiaugmented bilunabirotunda M3 + M8 + M3 16 32 18 10 2 2 4
32 P3,37 S47 metabiaugmented pentagonal cupola M3 + M9 + M3 22 37 17 4 4 1 8
33 P3,38 S48 meta-augmented pentagonal gyrocupolarotunda M3 + M9 + M6 26 51 27 12 5 6 4
34 P3,39 S49 meta-augmented pentagonal orthocupolarotunda M3 + M9 + M6' 26 51 27 12 5 6 4
35 P3,40 S27 augmented icosidodecahedron M3 + M9 + M9 31 60 31 15 11 5
36 P3,41 S52 meta-augmented pentagonal orthobirotunda M3 + M9 + M9' 31 61 32 17 11 4
37 P3,42 S25 para-augmented pentagonal gyrocupolarotunda M3 + M9' + M6 26 50 26 10 5 6 5
38 P3,43 S26 para-augmented pentagonal orthocupolarotunda M3 + M9' + M6' 26 50 26 10 5 6 5
39 P3,44 S28 para-augmented pentagonal orthobirotunda M3 + M9' + M9 31 60 31 15 11 5
40 P3,48 S61 gyrate biaugmented truncated cube M5 + M11 + M5' 32 56 26 8 10 4 4
41 P3,49 S62 bigyrate biaugmented truncated cube M5 + M11' + M5 32 52 22 10 4 8
42 P3,51 S66 gyrate metabiaugmented truncated dodecahedron M6 + M12 + M6' 70 115 47 20 10 2 10 5
43 P3,53 S64 gyrate parabiaugmented truncated dodecahedron M6 + M12 + M6 70 115 47 20 10 2 10 5
44 P3,54 S67 bigyrate metabiaugmented truncated dodecahedron M6 + M12' + M6 70 110 42 10 10 2 10 10
45 P3,55 S65 bigyrate parabiaugmented truncated dodecahedron M6 + M12' + M6' 70 110 42 10 10 2 10 10
46 P4,1 S07 metabiaugmented rhombic prism ?3 + ?3' + M2 + M2 10 20 12 8 2 2
47 P4,2 S08 parabiaugmented rhombic prism ?3 + ?3' + M2 + M2' 10 20 12 8 2 2
48 P4,5 S31 elongated para-augmented pentagonal orthocupolarotunda ?10 + M6 + M9 + M3 36 70 36 10 15 6 5
49 P4,6 S32 elongated para-augmented pentagonal gyrocupolarotunda ?10 + M6 + M9' + M3 36 70 36 10 15 6 5
50 P4,7 S33 elongated para-augmented pentagonal orthobirotunda ?10 + M9 + M3 + M9 41 80 41 15 10 11 5
51 P4,8 S34 elongated para-augmented pentagonal gyrobirotunda ?10 + M9 + M3 + M9' 41 80 41 15 10 11 5
52 P4,9 S37 gyroelongated para-augmented pentagonal cupolarotunda A10 + M6 + M9 + M3 36 80 46 30 5 6 5
53 P4,10 S38 gyroelongated para-augmented pentagonal birotunda A10 + M9 + M3 + M9 41 90 51 35 11 5
54 P4,11 S16 biaugmented octahedron, hexarhomb (GZ) M1 + M2 + M2 + M1 8 12 6 6
55 P4,12 S19 biaugmented triangular orthobicupola M2 + M4 + M4 + M2 14 26 14 4 4 6
56 P4,13 S21 biaugmented cuboctahedron, octarhombi expanded cuboctahedron (GZ) M2 + M4 + M4' + M2 14 24 12 4 8
57 P4,14 S50 metabiaugmented pentagonal gyrocupolarotunda M3 + M9 + M3 + M6 27 52 27 9 5 5 8
58 P4,15 S51 metabiaugmented pentagonal orthocupolarotunda M3 + M9 + M3 + M6' 27 52 27 9 5 5 8
59 P4,16 S44 metabiaugmented icosidodecahedron M3 + M9 + M3 + M9 32 60 30 10 10 10
60 P4,17 S53 cismetabiaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' 32 62 32 14 10 8
61 P4,18 S29 parabiaugmented icosidodecahedron M3 + M9 + M9 + M3 32 60 30 10 10 10
62 P4,19 S57 orthobiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3 32 62 32 14 10 8
63 P4,20 S58 transmetabiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3' 32 62 32 14 10 8
64 P4,21 S42 metaparabiaugmented pentagonal orthobirotunda M3 + M9 + M9' + M3 32 61 31 12 10 9
65 P4,22 S30 parabiaugmented pentagonal orthobirotunda M3 + M9' + M9 + M3 32 60 30 10 10 10
66 P4,25 S68 gyrate triaugmented truncated dodecahedron M6 + M12 + M6 + M6' 75 130 57 25 15 3 9 5
67 P4,26 S69 bigyrate triaugmented truncated dodecahedron M6 + M12 + M6' + M6 75 125 52 15 15 3 9 10
68 P4,27 S70 trigyrate triaugmented truncated dodecahedron M6 + M12' + M6 + M6 75 120 47 5 15 3 9 15
69 P4,30 oblique square prism M1 + M2 + (M1 + M2) 8 12 6 2 4
70 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
71 P5,1 S35 elongated parabiaugmented pentagonal orthobirotunda ?10 + M9 + M3 + M9 + M3 42 80 40 10 10 10 10
72 P5,2 S36 elongated parabiaugmented pentagonal gyrobirotunda ?10 + M9 + M3 + M9' + M3 42 80 40 10 10 10 10
73 P5,3 S39 gyroelongated parabiaugmented pentagonal birotunda A10 + M9 + M3 + M9 + M3 42 90 50 30 10 10
74 P5,4 S45 triaugmented icosidodecahedron M3 + M9 + M3 + M9 + M3 33 60 29 5 9 15
75 P5,5 S55 orthotriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3 33 63 32 11 9 12
76 P5,6 S56 orthometatriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3' 33 63 32 11 9 12
77 P5,7 S43 metaparatriaugmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3 33 62 31 9 9 13
78 P6,1 S54 4-augmented pentagonal orthobirotunda M3 + M9 + M3 + M9' + M3 + M3 34 64 32 8 8 16

See also

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References

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  1. ^ Convex regular-faced polyhedra with conditional edges
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