Uniform polyhedron compound

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 uniform 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

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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}

125

| 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 (3): 447–457, doi:10.1017/S0305004100052440 .

## External links

* http://www.interocitors.com/polyhedra/UCs/ShortNames.html \- Bowers style acronyms for uniform polyhedron compounds

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