Uniform polyhedron compound explained

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 prisms (UC₂₀ and UC₂₁) exist only when, and when and are coprime. The prismatic compounds of antiprisms (UC₂₂, UC₂₃, UC₂₄ and UC₂₅) exist only when, and when and are coprime. Furthermore, when, 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	36	24	Rotational freedom	Td	S4
UC02	dis	50px	12	tetrahedra	48	72	48	Rotational freedom	Oh	S4
UC03	snu	50px	6	tetrahedra	24	36	24		Oh	D2d
UC04	so	50px	2	tetrahedra	8	12	8	Regular	Oh	Td
UC05	ki	50px	5	tetrahedra	20	30	20	Regular	I	T
UC06	e	50px	10	tetrahedra	40	60	20	Regular 2 polyhedra per vertex	Ih	T
UC07	risdoh	50px	6	cubes	(12+24)	72	48	Rotational freedom	Oh	C4h
UC08	rah	50px	3	cubes	(6+12)	36	24		Oh	D4h
UC09	rhom	50px	5	cubes	30	60	20	Regular 2 polyhedra per vertex	Ih	Th
UC10	dissit	50px	4	octahedra	(8+24)	48	24	Rotational freedom	Th	S6
UC11	daso	50px	8	octahedra	(16+48)	96	48	Rotational freedom	Oh	S6
UC12	sno	50px	4	octahedra	(8+24)	48	24		Oh	D3d
UC13	addasi	50px	20	octahedra	(40+120)	240	120	Rotational freedom	Ih	S6
UC14	dasi	50px	20	octahedra	(40+120)	240	60	2 polyhedra per vertex	Ih	S6
UC15	gissi	50px	10	octahedra	(20+60)	120	60		Ih	D3d
UC16	si	50px	10	octahedra	(20+60)	120	60		Ih	D3d
UC17	se	50px	5	octahedra	40	60	30	Regular	Ih	Th
UC18	hirki	50px	5	tetrahemihexahedra	20 15	60	30		I	T
UC19	sapisseri	50px	20	tetrahemihexahedra	(20+60) 60	240	60	2 polyhedra per vertex	I	C3
UC20	-	50px	2n (2n ≥ 2)	p/q-gonal prisms	4n 2np	6np	4np	Rotational freedom	Dnph	Cph
UC21	-	50px	n (n ≥ 2)	p/q-gonal prisms	2n np	3np	2np		Dnph	Dph
UC22	-	50px	2n (2n ≥ 2) (q odd)	p/q-gonal antiprisms (q odd)	4n (if ≠ 2) 4np	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 (if ≠ 2) 2np	4np	2np		Dnpd (if n odd) Dnph (if n even)	Dpd
UC24	-	50px	2n (2n ≥ 2)	p/q-gonal antiprisms (q even)	4n (if ≠ 2) 4np	8np	4np	Rotational freedom	Dnph	Cph
UC25	-	50px	n (n ≥ 2)	p/q-gonal antiprisms (q even)	2n (if ≠ 2) 2np	4np	2np		Dnph	Dph
UC26	gadsid	50px	12	pentagonal antiprisms	120 24	240	120	Rotational freedom	Ih	S10
UC27	gassid	50px	6	pentagonal antiprisms	60 12	120	60		Ih	D5d
UC28	gidasid	50px	12	pentagrammic crossed antiprisms	120 24	240	120	Rotational freedom	Ih	S10
UC29	gissed	50px	6	pentagrammic crossed antiprisms	60 12	120	60		Ih	D5d
UC30	ro	50px	4	triangular prisms	8 12	36	24		O	D3
UC31	dro	50px	8	triangular prisms	16 24	72	48		Oh	D3
UC32	kri	50px	10	triangular prisms	20 30	90	60		I	D3
UC33	dri	50px	20	triangular prisms	40 60	180	60	2 polyhedra per vertex	Ih	D3
UC34	kred	50px	6	pentagonal prisms	30 12	90	60		I	D5
UC35	dird	50px	12	pentagonal prisms	60 24	180	60	2 polyhedra per vertex	Ih	D5
UC36	gikrid	50px	6	pentagrammic prisms	30 12	90	60		I	D5
UC37	giddird	50px	12	pentagrammic prisms	60 24	180	60	2 polyhedra per vertex	Ih	D5
UC38	griso	50px	4	hexagonal prisms	24 8	72	48		Oh	D3d
UC39	rosi	50px	10	hexagonal prisms	60 20	180	120		Ih	D3d
UC40	rassid	50px	6	decagonal prisms	60 12	180	120		Ih	D5d
UC41	grassid	50px	6	decagrammic prisms	60 12	180	120		Ih	D5d
UC42	gassic	50px	3	square antiprisms	24 6	48	24		O	D4
UC43	gidsac	50px	6	square antiprisms	48 12	96	48		Oh	D4
UC44	sassid	50px	6	pentagrammic antiprisms	60 12	120	60		I	D5
UC45	sadsid	50px	12	pentagrammic antiprisms	120 24	240	120		Ih	D5
UC46	siddo	50px	2	icosahedra	(16+24)	60	24		Oh	Th
UC47	sne	50px	5	icosahedra	(40+60)	150	60		Ih	Th
UC48	presipsido	50px	2	great dodecahedra	24	60	24		Oh	Th
UC49	presipsi	50px	5	great dodecahedra	60	150	60		Ih	Th
UC50	passipsido	50px	2	small stellated dodecahedra	24	60	24		Oh	Th
UC51	passipsi	50px	5	small stellated dodecahedra	60	150	60		Ih	Th
UC52	sirsido	50px	2	great icosahedra	(16+24)	60	24		Oh	Th
UC53	sirsei	50px	5	great icosahedra	(40+60)	150	60		Ih	Th
UC54	tisso	50px	2	truncated tetrahedra	8 8	36	24		Oh	Td
UC55	taki	50px	5	truncated tetrahedra	20 20	90	60		I	T
UC56	te	50px	10	truncated tetrahedra	40 40	180	120		Ih	T
UC57	tar	50px	5	truncated cubes	40 30	180	120		Ih	Th
UC58	quitar	50px	5	stellated truncated hexahedra	40 30	180	120		Ih	Th
UC59	arie	50px	5	cuboctahedra	40 30	120	60		Ih	Th
UC60	gari	50px	5	cubohemioctahedra	30 20	120	60		Ih	Th
UC61	iddei	50px	5	octahemioctahedra	40 20	120	60		Ih	Th
UC62	rasseri	50px	5	rhombicuboctahedra	40 (30+60)	240	120		Ih	Th
UC63	rasher	50px	5	small rhombihexahedra	60 30	240	120		Ih	Th
UC64	rahrie	50px	5	small cubicuboctahedra	40 30 30	240	120		Ih	Th
UC65	raquahri	50px	5	great cubicuboctahedra	40 30 30	240	120		Ih	Th
UC66	rasquahr	50px	5	great rhombihexahedra	60 30	240	120		Ih	Th
UC67	rosaqri	50px	5	nonconvex great rhombicuboctahedra	40 (30+60)	240	120		Ih	Th
UC68	disco	50px	2	snub cubes	(16+48) 12	120	48		Oh	O
UC69	dissid	50px	2	snub dodecahedra	(40+120) 24	300	120		Ih	I
UC70	giddasid	50px	2	great snub icosidodecahedra	(40+120) 24	300	120		Ih	I
UC71	gidsid	50px	2	great inverted snub icosidodecahedra	(40+120) 24	300	120		Ih	I
UC72	gidrissid	50px	2	great retrosnub icosidodecahedra	(40+120) 24	300	120		Ih	I
UC73	disdid	50px	2	snub dodecadodecahedra	120 24 24	300	120		Ih	I
UC74	idisdid	50px	2	inverted snub dodecadodecahedra	120 24 24	300	120		Ih	I
UC75	desided	50px	2	snub icosidodecadodecahedra	(40+120) 24 24	360	120		Ih	I

References

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External links

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