Uniform polyhedron compound

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		6	tetrahedra	24{3}	36	24	Rotational freedom	Td	S4
UC02	dis		12	tetrahedra	48{3}	72	48	Rotational freedom	Oh	S4
UC03	snu		6	tetrahedra	24{3}	36	24		Oh	D2d
UC04	so		2	tetrahedra	8{3}	12	8	Regular	Oh	Td
UC05	ki		5	tetrahedra	20{3}	30	20	Regular	I	T
UC06	e		10	tetrahedra	40{3}	60	20	Regular 2 polyhedra per vertex	Ih	T
UC07	risdoh		6	cubes	(12+24){4}	72	48	Rotational freedom	Oh	C4h
UC08	rah		3	cubes	(6+12){4}	36	24		Oh	D4h
UC09	rhom		5	cubes	30{4}	60	20	Regular 2 polyhedra per vertex	Ih	Th
UC10	dissit		4	octahedra	(8+24){3}	48	24	Rotational freedom	Th	S6
UC11	daso		8	octahedra	(16+48){3}	96	48	Rotational freedom	Oh	S6
UC12	sno		4	octahedra	(8+24){3}	48	24		Oh	D3d
UC13	addasi		20	octahedra	(40+120){3}	240	120	Rotational freedom	Ih	S6
UC14	dasi		20	octahedra	(40+120){3}	240	60	2 polyhedra per vertex	Ih	S6
UC15	gissi		10	octahedra	(20+60){3}	120	60		Ih	D3d
UC16	si		10	octahedra	(20+60){3}	120	60		Ih	D3d
UC17	se		5	octahedra	40{3}	60	30	Regular	Ih	Th
UC18	hirki		5	tetrahemihexahedra	20{3} 15{4}	60	30		I	T
UC19	sapisseri		20	tetrahemihexahedra	(20+60){3} 60{4}	240	60	2 polyhedra per vertex	I	C3
UC20	-		2n (2n ≥ 2)	p/q-gonal prisms	4n{p/q} 2np{4}	6np	4np	Rotational freedom	Dnph	Cph
UC21	-		n (n ≥ 2)	p/q-gonal prisms	2n{p/q} np{4}	3np	2np		Dnph	Dph
UC22	-		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	-		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	-		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	-		n (n ≥ 2)	p/q-gonal antiprisms (q even)	2n{p/q} (if ⁠p/q⁠ ≠ 2) 2np{3}	4np	2np		Dnph	Dph
UC26	gadsid		12	pentagonal antiprisms	120{3} 24{5}	240	120	Rotational freedom	Ih	S10
UC27	gassid		6	pentagonal antiprisms	60{3} 12{5}	120	60		Ih	D5d
UC28	gidasid		12	pentagrammic crossed antiprisms	120{3} 24{5/2}	240	120	Rotational freedom	Ih	S10
UC29	gissed		6	pentagrammic crossed antiprisms	60{3} 125	120	60		Ih	D5d
UC30	ro		4	triangular prisms	8{3} 12{4}	36	24		O	D3
UC31	dro		8	triangular prisms	16{3} 24{4}	72	48		Oh	D3
UC32	kri		10	triangular prisms	20{3} 30{4}	90	60		I	D3
UC33	dri		20	triangular prisms	40{3} 60{4}	180	60	2 polyhedra per vertex	Ih	D3
UC34	kred		6	pentagonal prisms	30{4} 12{5}	90	60		I	D5
UC35	dird		12	pentagonal prisms	60{4} 24{5}	180	60	2 polyhedra per vertex	Ih	D5
UC36	gikrid		6	pentagrammic prisms	30{4} 12{5/2}	90	60		I	D5
UC37	giddird		12	pentagrammic prisms	60{4} 24{5/2}	180	60	2 polyhedra per vertex	Ih	D5
UC38	griso		4	hexagonal prisms	24{4} 8{6}	72	48		Oh	D3d
UC39	rosi		10	hexagonal prisms	60{4} 20{6}	180	120		Ih	D3d
UC40	rassid		6	decagonal prisms	60{4} 12{10}	180	120		Ih	D5d
UC41	grassid		6	decagrammic prisms	60{4} 12{10/3}	180	120		Ih	D5d
UC42	gassic		3	square antiprisms	24{3} 6{4}	48	24		O	D4
UC43	gidsac		6	square antiprisms	48{3} 12{4}	96	48		Oh	D4
UC44	sassid		6	pentagrammic antiprisms	60{3} 12{5/2}	120	60		I	D5
UC45	sadsid		12	pentagrammic antiprisms	120{3} 24{5/2}	240	120		Ih	D5
UC46	siddo		2	icosahedra	(16+24){3}	60	24		Oh	Th
UC47	sne		5	icosahedra	(40+60){3}	150	60		Ih	Th
UC48	presipsido		2	great dodecahedra	24{5}	60	24		Oh	Th
UC49	presipsi		5	great dodecahedra	60{5}	150	60		Ih	Th
UC50	passipsido		2	small stellated dodecahedra	24{5/2}	60	24		Oh	Th
UC51	passipsi		5	small stellated dodecahedra	60{5/2}	150	60		Ih	Th
UC52	sirsido		2	great icosahedra	(16+24){3}	60	24		Oh	Th
UC53	sirsei		5	great icosahedra	(40+60){3}	150	60		Ih	Th
UC54	tisso		2	truncated tetrahedra	8{3} 8{6}	36	24		Oh	Td
UC55	taki		5	truncated tetrahedra	20{3} 20{6}	90	60		I	T
UC56	te		10	truncated tetrahedra	40{3} 40{6}	180	120		Ih	T
UC57	tar		5	truncated cubes	40{3} 30{8}	180	120		Ih	Th
UC58	quitar		5	stellated truncated hexahedra	40{3} 30{8/3}	180	120		Ih	Th
UC59	arie		5	cuboctahedra	40{3} 30{4}	120	60		Ih	Th
UC60	gari		5	cubohemioctahedra	30{4} 20{6}	120	60		Ih	Th
UC61	iddei		5	octahemioctahedra	40{3} 20{6}	120	60		Ih	Th
UC62	rasseri		5	rhombicuboctahedra	40{3} (30+60){4}	240	120		Ih	Th
UC63	rasher		5	small rhombihexahedra	60{4} 30{8}	240	120		Ih	Th
UC64	rahrie		5	small cubicuboctahedra	40{3} 30{4} 30{8}	240	120		Ih	Th
UC65	raquahri		5	great cubicuboctahedra	40{3} 30{4} 30{8/3}	240	120		Ih	Th
UC66	rasquahr		5	great rhombihexahedra	60{4} 30{8/3}	240	120		Ih	Th
UC67	rosaqri		5	nonconvex great rhombicuboctahedra	40{3} (30+60){4}	240	120		Ih	Th
UC68	disco		2	snub cubes	(16+48){3} 12{4}	120	48		Oh	O
UC69	dissid		2	snub dodecahedra	(40+120){3} 24{5}	300	120		Ih	I
UC70	giddasid		2	great snub icosidodecahedra	(40+120){3} 24{5/2}	300	120		Ih	I
UC71	gidsid		2	great inverted snub icosidodecahedra	(40+120){3} 24{5/2}	300	120		Ih	I
UC72	gidrissid		2	great retrosnub icosidodecahedra	(40+120){3} 24{5/2}	300	120		Ih	I
UC73	disdid		2	snub dodecadodecahedra	120{3} 24{5} 24{5/2}	300	120		Ih	I
UC74	idisdid		2	inverted snub dodecadodecahedra	120{3} 24{5} 24{5/2}	300	120		Ih	I
UC75	desided		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, MR 0397554.

External links

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