Uniform polyhedron compound

A uniform polyhedron compound is a polyhedral compound whose constituents are identical 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 -gonal prisms UC₂₀ and UC₂₁ exist only when > 2, and when p and q are coprime. The prismatic compounds of -gonal antiprisms UC₂₂, UC₂₃, UC₂₄ and UC₂₅ exist only when >, and when p and q are coprime. Furthermore, when = 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	36	24	Rotational freedom	T_d	S₄
UC₀₂	dis		12	tetrahedra	48	72	48	Rotational freedom	O_h	S₄
UC₀₃	snu		6	tetrahedra	24	36	24		O_h	D_2d
UC₀₄	so		2	tetrahedra	8	12	8	Regular	O_h	T_d
UC₀₅	ki		5	tetrahedra	20	30	20	Regular	I	T
UC₀₆	e		10	tetrahedra	40	60	20	Regular 2 polyhedra per vertex	I_h	T
UC₀₇	risdoh		6	cubes		72	48	Rotational freedom	O_h	C_4h
UC₀₈	rah		3	cubes		36	24		O_h	D_4h
UC₀₉	rhom		5	cubes	30	60	20	Regular 2 polyhedra per vertex	I_h	T_h
UC₁₀	dissit		4	octahedra		48	24	Rotational freedom	T_h	S₆
UC₁₁	daso		8	octahedra		96	48	Rotational freedom	O_h	S₆
UC₁₂	sno		4	octahedra		48	24		O_h	D_3d
UC₁₃	addasi		20	octahedra		240	120	Rotational freedom	I_h	S₆
UC₁₄	dasi		20	octahedra		240	60	2 polyhedra per vertex	I_h	S₆
UC₁₅	gissi		10	octahedra		120	60		I_h	D_3d
UC₁₆	si		10	octahedra		120	60		I_h	D_3d
UC₁₇	se		5	octahedra	40	60	30	Regular	I_h	T_h
UC₁₈	hirki		5	tetrahemihexahedra	20 15	60	30		I	T
UC₁₉	sapisseri		20	tetrahemihexahedra	60	240	60	2 polyhedra per vertex	I	C₃
UC₂₀	-		2n	p/q-gonal prisms	4n{p/q} 2np	6np	4np	Rotational freedom	D_nph	C_ph
UC₂₁	-		n	p/q-gonal prisms	2n{p/q} np	3np	2np		D_nph	D_ph
UC₂₂	-		2n	p/q-gonal antiprisms	4n{p/q} 4np	8np	4np	Rotational freedom	D_npd D_nph	S_2p
UC₂₃	-		n	p/q-gonal antiprisms	2n{p/q} 2np	4np	2np		D_npd D_nph	D_pd
UC₂₄	-		2n	p/q-gonal antiprisms	4n{p/q} 4np	8np	4np	Rotational freedom	D_nph	C_ph
UC₂₅	-		n	p/q-gonal antiprisms	2n{p/q} 2np	4np	2np		D_nph	D_ph
UC₂₆	gadsid		12	pentagonal antiprisms	120 24	240	120	Rotational freedom	I_h	S₁₀
UC₂₇	gassid		6	pentagonal antiprisms	60 12	120	60		I_h	D_5d
UC₂₈	gidasid		12	pentagrammic crossed antiprisms	120 24	240	120	Rotational freedom	I_h	S₁₀
UC₂₉	gissed		6	pentagrammic crossed antiprisms	60 12	120	60		I_h	D_5d
UC₃₀	ro		4	triangular prisms	8 12	36	24		O	D₃
UC₃₁	dro		8	triangular prisms	16 24	72	48		O_h	D₃
UC₃₂	kri		10	triangular prisms	20 30	90	60		I	D₃
UC₃₃	dri		20	triangular prisms	40 60	180	60	2 polyhedra per vertex	I_h	D₃
UC₃₄	kred		6	pentagonal prisms	30 12	90	60		I	D₅
UC₃₅	dird		12	pentagonal prisms	60 24	180	60	2 polyhedra per vertex	I_h	D₅
UC₃₆	gikrid		6	pentagrammic prisms	30 12	90	60		I	D₅
UC₃₇	giddird		12	pentagrammic prisms	60 24	180	60	2 polyhedra per vertex	I_h	D₅
UC₃₈	griso		4	hexagonal prisms	24 8	72	48		O_h	D_3d
UC₃₉	rosi		10	hexagonal prisms	60 20	180	120		I_h	D_3d
UC₄₀	rassid		6	decagonal prisms	60 12	180	120		I_h	D_5d
UC₄₁	grassid		6	decagrammic prisms	60 12	180	120		I_h	T_h
UC₅₉	arie		5	cuboctahedra	40 30	120	60		I_h	T_h
UC₆₀	gari		5	cubohemioctahedra	30 20	120	60		I_h	T_h
UC₆₁	iddei		5	octahemioctahedra	40 20	120	60		I_h	T_h
UC₆₂	rasseri		5	rhombicuboctahedra	40	240	120		I_h	T_h
UC₆₃	rasher		5	small rhombihexahedra	60 30	240	120		I_h	T_h
UC₆₄	rahrie		5	small cubicuboctahedra	40 30 30	240	120		I_h	T_h
UC₆₅	raquahri		5	great cubicuboctahedra	40 30 30{8/34 Popular articles Javier Milei - Argentine libertarian economist, author, radio conductor and public speaker sympathetic to the Austrian School of economic thought. He became widely known for his regular ... Jimmy Carter - American politician, philanthropist, and former farmer who served as the 39th president of the United States from 1977 to 1981. A member of the Democratic Party, he previ... UEFA Euro 2024 - The 2024 UEFA European Football Championship , commonly referred to as UEFA Euro 2024 or simply Euro 2024 , will be the 17th edition of the UEFA European Championship, the quadrennial internationa... Argentina - country located mostly in the southern half of South America. Sharing the bulk of the Southern Cone with Chile to the west, the country is also b... Sam Altman - American entrepreneur, investor, programmer, and blogger. He is the former president of Y Combinator and now the CEO of OpenAI. Early life and education. ... 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