Uniform polyhedron compound

From HandWiki
Short description: Compound shape made of identical uniform polyhedra arranged uniformly

In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices.

The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering.

The prismatic compounds of {p/q}-gonal prisms (UC20 and UC21) exist only when p/q > 2, and when p and q are coprime. The prismatic compounds of {p/q}-gonal antiprisms (UC22, UC23, UC24 and UC25) exist only when p/q > 3/2, and when p and q are coprime. Furthermore, when p/q = 2, the antiprisms degenerate into tetrahedra with digonal bases.

Compound Bowers
acronym 		Picture Polyhedral
count 		Polyhedral type Faces Edges Vertices Notes Symmetry group Subgroup
restricting
to one
constituent
UC01 sis 50px 6 tetrahedra 24{3} 36 24 Rotational freedom Td S4
UC02 dis 50px 12 tetrahedra 48{3} 72 48 Rotational freedom Oh S4
UC03 snu 50px 6 tetrahedra 24{3} 36 24 Oh D2d
UC04 so 50px 2 tetrahedra 8{3} 12 8 Regular Oh Td
UC05 ki 50px 5 tetrahedra 20{3} 30 20 Regular I T
UC06 e 50px 10 tetrahedra 40{3} 60 20 Regular

2 polyhedra per vertex

Ih T
UC07 risdoh 50px 6 cubes (12+24){4} 72 48 Rotational freedom Oh C4h
UC08 rah 50px 3 cubes (6+12){4} 36 24 Oh D4h
UC09 rhom 50px 5 cubes 30{4} 60 20 Regular

2 polyhedra per vertex

Ih Th
UC10 dissit 50px 4 octahedra (8+24){3} 48 24 Rotational freedom Th S6
UC11 daso 50px 8 octahedra (16+48){3} 96 48 Rotational freedom Oh S6
UC12 sno 50px 4 octahedra (8+24){3} 48 24 Oh D3d
UC13 addasi 50px 20 octahedra (40+120){3} 240 120 Rotational freedom Ih S6
UC14 dasi 50px 20 octahedra (40+120){3} 240 60 2 polyhedra per vertex Ih S6
UC15 gissi 50px 10 octahedra (20+60){3} 120 60 Ih D3d
UC16 si 50px 10 octahedra (20+60){3} 120 60 Ih D3d
UC17 se 50px 5 octahedra 40{3} 60 30 Regular Ih Th
UC18 hirki 50px 5 tetrahemihexahedra 20{3}

15{4}

60 30 I T
UC19 sapisseri 50px 20 tetrahemihexahedra (20+60){3}

60{4}

240 60 2 polyhedra per vertex I C3
UC20 - 50px 2n

(2n ≥ 2)

p/q-gonal prisms 4n{p/q}

2np{4}

6np 4np Rotational freedom Dnph Cph
UC21 - 50px n

(n ≥ 2)

p/q-gonal prisms 2n{p/q}

np{4}

3np 2np Dnph Dph
UC22 - 50px 2n

(2n ≥ 2)

(q odd)

p/q-gonal antiprisms

(q odd)

4n{p/q} (if p/q ≠ 2)

4np{3}

8np 4np Rotational freedom Dnpd (if n odd)

Dnph (if n even)

S2p
UC23 - 50px n

(n ≥ 2)

p/q-gonal antiprisms

(q odd)

2n{p/q} (if p/q ≠ 2)

2np{3}

4np 2np Dnpd (if n odd)

Dnph (if n even)

Dpd
UC24 - 50px 2n

(2n ≥ 2)

p/q-gonal antiprisms

(q even)

4n{p/q} (if p/q ≠ 2)

4np{3}

8np 4np Rotational freedom Dnph Cph
UC25 - 50px n

(n ≥ 2)

p/q-gonal antiprisms

(q even)

2n{p/q} (if p/q ≠ 2)

2np{3}

4np 2np Dnph Dph
UC26 gadsid 50px 12 pentagonal antiprisms 120{3}

24{5}

240 120 Rotational freedom Ih S10
UC27 gassid 50px 6 pentagonal antiprisms 60{3}

12{5}

120 60 Ih D5d
UC28 gidasid 50px 12 pentagrammic crossed antiprisms 120{3}

24{5/2}

240 120 Rotational freedom Ih S10
UC29 gissed 50px 6 pentagrammic crossed antiprisms 60{3} 120 60 Ih D5d
UC30 ro 50px 4 triangular prisms 8{3}

12{4}

36 24 O D3
UC31 dro 50px 8 triangular prisms 16{3}

24{4}

72 48 Oh D3
UC32 kri 50px 10 triangular prisms 20{3}

30{4}

90 60 I D3
UC33 dri 50px 20 triangular prisms 40{3}

60{4}

180 60 2 polyhedra per vertex Ih D3
UC34 kred 50px 6 pentagonal prisms 30{4}

12{5}

90 60 I D5
UC35 dird 50px 12 pentagonal prisms 60{4}

24{5}

180 60 2 polyhedra per vertex Ih D5
UC36 gikrid 50px 6 pentagrammic prisms 30{4}

12{5/2}

90 60 I D5
UC37 giddird 50px 12 pentagrammic prisms 60{4}

24{5/2}

180 60 2 polyhedra per vertex Ih D5
UC38 griso 50px 4 hexagonal prisms 24{4}

8{6}

72 48 Oh D3d
UC39 rosi 50px 10 hexagonal prisms 60{4}

20{6}

180 120 Ih D3d
UC40 rassid 50px 6 decagonal prisms 60{4}

12{10}

180 120 Ih D5d
UC41 grassid 50px 6 decagrammic prisms 60{4}

12{10/3}

180 120 Ih D5d
UC42 gassic 50px 3 square antiprisms 24{3}

6{4}

48 24 O D4
UC43 gidsac 50px 6 square antiprisms 48{3}

12{4}

96 48 Oh D4
UC44 sassid 50px 6 pentagrammic antiprisms 60{3}

12{5/2}

120 60 I D5
UC45 sadsid 50px 12 pentagrammic antiprisms 120{3}

24{5/2}

240 120 Ih D5
UC46 siddo 50px 2 icosahedra (16+24){3} 60 24 Oh Th
UC47 sne 50px 5 icosahedra (40+60){3} 150 60 Ih Th
UC48 presipsido 50px 2 great dodecahedra 24{5} 60 24 Oh Th
UC49 presipsi 50px 5 great dodecahedra 60{5} 150 60 Ih Th
UC50 passipsido 50px 2 small stellated dodecahedra 24{5/2} 60 24 Oh Th
UC51 passipsi 50px 5 small stellated dodecahedra 60{5/2} 150 60 Ih Th
UC52 sirsido 50px 2 great icosahedra (16+24){3} 60 24 Oh Th
UC53 sirsei 50px 5 great icosahedra (40+60){3} 150 60 Ih Th
UC54 tisso 50px 2 truncated tetrahedra 8{3}

8{6}

36 24 Oh Td
UC55 taki 50px 5 truncated tetrahedra 20{3}

20{6}

90 60 I T
UC56 te 50px 10 truncated tetrahedra 40{3}

40{6}

180 120 Ih T
UC57 tar 50px 5 truncated cubes 40{3}

30{8}

180 120 Ih Th
UC58 quitar 50px 5 stellated truncated hexahedra 40{3}

30{8/3}

180 120 Ih Th
UC59 arie 50px 5 cuboctahedra 40{3}

30{4}

120 60 Ih Th
UC60 gari 50px 5 cubohemioctahedra 30{4}

20{6}

120 60 Ih Th
UC61 iddei 50px 5 octahemioctahedra 40{3}

20{6}

120 60 Ih Th
UC62 rasseri 50px 5 rhombicuboctahedra 40{3}

(30+60){4}

240 120 Ih Th
UC63 rasher 50px 5 small rhombihexahedra 60{4}

30{8}

240 120 Ih Th
UC64 rahrie 50px 5 small cubicuboctahedra 40{3}

30{4}

30{8}

240 120 Ih Th
UC65 raquahri 50px 5 great cubicuboctahedra 40{3}

30{4}

30{8/3}

240 120 Ih Th
UC66 rasquahr 50px 5 great rhombihexahedra 60{4}

30{8/3}

240 120 Ih Th
UC67 rosaqri 50px 5 nonconvex great rhombicuboctahedra 40{3}

(30+60){4}

240 120 Ih Th
UC68 disco 50px 2 snub cubes (16+48){3}

12{4}

120 48 Oh O
UC69 dissid 50px 2 snub dodecahedra (40+120){3}

24{5}

300 120 Ih I
UC70 giddasid 50px 2 great snub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC71 gidsid 50px 2 great inverted snub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC72 gidrissid 50px 2 great retrosnub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC73 disdid 50px 2 snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300 120 Ih I
UC74 idisdid 50px 2 inverted snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300 120 Ih I
UC75 desided 50px 2 snub icosidodecadodecahedra (40+120){3}

24{5}

24{5/2}

360 120 Ih I

References

  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society 79: 447–457, doi:10.1017/S0305004100052440 .

External links



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