Uniform polyhedron compound

Last updated December 09, 2024

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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠p/q⁠ > 2$ , 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	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	T_d	S₄
UC₀₂	dis	12	tetrahedra	48{3}	72	48	Rotational freedom	O_h	S₄
UC₀₃	snu	6	tetrahedra	24{3}	36	24		O_h	D_2d
UC₀₄	so	2	tetrahedra	8{3}	12	8	Regular	O_h	T_d
UC₀₅	ki	5	tetrahedra	20{3}	30	20	Regular	I	T
UC₀₆	e	10	tetrahedra	40{3}	60	20	Regular 2 polyhedra per vertex	I_h	T
UC₀₇	risdoh	6	cubes	(12+24){4}	72	48	Rotational freedom	O_h	C_4h
UC₀₈	rah	3	cubes	(6+12){4}	36	24		O_h	D_4h
UC₀₉	rhom	5	cubes	30{4}	60	20	Regular 2 polyhedra per vertex	I_h	T_h
UC₁₀	dissit	4	octahedra	(8+24){3}	48	24	Rotational freedom	T_h	S₆
UC₁₁	daso	8	octahedra	(16+48){3}	96	48	Rotational freedom	O_h	S₆
UC₁₂	sno	4	octahedra	(8+24){3}	48	24		O_h	D_3d
UC₁₃	addasi	20	octahedra	(40+120){3}	240	120	Rotational freedom	I_h	S₆
UC₁₄	dasi	20	octahedra	(40+120){3}	240	60	2 polyhedra per vertex	I_h	S₆
UC₁₅	gissi	10	octahedra	(20+60){3}	120	60		I_h	D_3d
UC₁₆	si	10	octahedra	(20+60){3}	120	60		I_h	D_3d
UC₁₇	se	5	octahedra	40{3}	60	30	Regular	I_h	T_h
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	D_nph	C_ph
UC₂₁	-	n (n ≥ 2)	p/q-gonal prisms	2n{p/q} np{4}	3np	2np		D_nph	D_ph
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	D_npd (if n odd) D_nph (if n even)	S_2p
UC₂₃	-	n (n ≥ 2)	p/q-gonal antiprisms (q odd)	2n{p/q} (if ⁠p/q⁠ ≠ 2) 2np{3}	4np	2np		D_npd (if n odd) D_nph (if n even)	D_pd
UC₂₄	-	2n (2n ≥ 2)	p/q-gonal antiprisms (q even)	4n{p/q} (if ⁠p/q⁠ ≠ 2) 4np{3}	8np	4np	Rotational freedom	D_nph	C_ph
UC₂₅	-	n (n ≥ 2)	p/q-gonal antiprisms (q even)	2n{p/q} (if ⁠p/q⁠ ≠ 2) 2np{3}	4np	2np		D_nph	D_ph
UC₂₆	gadsid	12	pentagonal antiprisms	120{3} 24{5}	240	120	Rotational freedom	I_h	S₁₀
UC₂₇	gassid	6	pentagonal antiprisms	60{3} 12{5}	120	60		I_h	D_5d
UC₂₈	gidasid	12	pentagrammic crossed antiprisms	120{3} 24{5/2}	240	120	Rotational freedom	I_h	S₁₀
UC₂₉	gissed	6	pentagrammic crossed antiprisms	60{3} 125	120	60		I_h	D_5d
UC₃₀	ro	4	triangular prisms	8{3} 12{4}	36	24		O	D₃
UC₃₁	dro	8	triangular prisms	16{3} 24{4}	72	48		O_h	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	I_h	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	I_h	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	I_h	D₅
UC₃₈	griso	4	hexagonal prisms	24{4} 8{6}	72	48		O_h	D_3d
UC₃₉	rosi	10	hexagonal prisms	60{4} 20{6}	180	120		I_h	D_3d
UC₄₀	rassid	6	decagonal prisms	60{4} 12{10}	180	120		I_h	D_5d
UC₄₁	grassid	6	decagrammic prisms	60{4} 12{10/3}	180	120		I_h	D_5d
UC₄₂	gassic	3	square antiprisms	24{3} 6{4}	48	24		O	D₄
UC₄₃	gidsac	6	square antiprisms	48{3} 12{4}	96	48		O_h	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		I_h	D₅
UC₄₆	siddo	2	icosahedra	(16+24){3}	60	24		O_h	T_h
UC₄₇	sne	5	icosahedra	(40+60){3}	150	60		I_h	T_h
UC₄₈	presipsido	2	great dodecahedra	24{5}	60	24		O_h	T_h
UC₄₉	presipsi	5	great dodecahedra	60{5}	150	60		I_h	T_h
UC₅₀	passipsido	2	small stellated dodecahedra	24{5/2}	60	24		O_h	T_h
UC₅₁	passipsi	5	small stellated dodecahedra	60{5/2}	150	60		I_h	T_h
UC₅₂	sirsido	2	great icosahedra	(16+24){3}	60	24		O_h	T_h
UC₅₃	sirsei	5	great icosahedra	(40+60){3}	150	60		I_h	T_h
UC₅₄	tisso	2	truncated tetrahedra	8{3} 8{6}	36	24		O_h	T_d
UC₅₅	taki	5	truncated tetrahedra	20{3} 20{6}	90	60		I	T
UC₅₆	te	10	truncated tetrahedra	40{3} 40{6}	180	120		I_h	T
UC₅₇	tar	5	truncated cubes	40{3} 30{8}	180	120		I_h	T_h
UC₅₈	quitar	5	stellated truncated hexahedra	40{3} 30{8/3}	180	120		I_h	T_h
UC₅₉	arie	5	cuboctahedra	40{3} 30{4}	120	60		I_h	T_h
UC₆₀	gari	5	cubohemioctahedra	30{4} 20{6}	120	60		I_h	T_h
UC₆₁	iddei	5	octahemioctahedra	40{3} 20{6}	120	60		I_h	T_h
UC₆₂	rasseri	5	rhombicuboctahedra	40{3} (30+60){4}	240	120		I_h	T_h
UC₆₃	rasher	5	small rhombihexahedra	60{4} 30{8}	240	120		I_h	T_h
UC₆₄	rahrie	5	small cubicuboctahedra	40{3} 30{4} 30{8}	240	120		I_h	T_h
UC₆₅	raquahri	5	great cubicuboctahedra	40{3} 30{4} 30{8/3}	240	120		I_h	T_h
UC₆₆	rasquahr	5	great rhombihexahedra	60{4} 30{8/3}	240	120		I_h	T_h
UC₆₇	rosaqri	5	nonconvex great rhombicuboctahedra	40{3} (30+60){4}	240	120		I_h	T_h
UC₆₈	disco	2	snub cubes	(16+48){3} 12{4}	120	48		O_h	O
UC₆₉	dissid	2	snub dodecahedra	(40+120){3} 24{5}	300	120		I_h	I
UC₇₀	giddasid	2	great snub icosidodecahedra	(40+120){3} 24{5/2}	300	120		I_h	I
UC₇₁	gidsid	2	great inverted snub icosidodecahedra	(40+120){3} 24{5/2}	300	120		I_h	I
UC₇₂	gidrissid	2	great retrosnub icosidodecahedra	(40+120){3} 24{5/2}	300	120		I_h	I
UC₇₃	disdid	2	snub dodecadodecahedra	120{3} 24{5} 24{5/2}	300	120		I_h	I
UC₇₄	idisdid	2	inverted snub dodecadodecahedra	120{3} 24{5} 24{5/2}	300	120		I_h	I
UC₇₅	desided	2	snub icosidodecadodecahedra	(40+120){3} 24{5} 24{5/2}	360	120		I_h	I

Related Research Articles

In geometry, an $n$ -gonal antiprism or $n$ -antiprism is a polyhedron composed of two parallel direct copies of an $n$ -sided polygon, connected by an alternating band of $2 n$ triangles. They are represented by the Conway notation $A n$ .

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In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an $n$ -polytope and an $m$ -polytope is an $(n + m)$ -polytope, where $n$ and $m$ are dimensions of 2 (polygon) or higher.

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces.

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

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Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of prisms sharing a common axis of rotational symmetry.

In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.

Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry. It arises from superimposing two copies of the corresponding prismatic compound of antiprisms, and rotating each copy by an equal and opposite angle.

Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of prisms sharing a common axis of rotational symmetry. It arises from superimposing two copies of the corresponding prismatic compound of prisms, and rotating each copy by an equal and opposite angle.

In geometry, a compound of three tetrahedra can be constructed by three tetrahedra rotated by 60 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D_3d, order 12. It is a uniform prismatic compound of antiprisms, UC23.

<span class="mw-page-title-main">Compound of four tetrahedra</span> Polyhedral compound

In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.

In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.

<span class="mw-page-title-main">Hendecagrammic prism</span>

In geometry, a hendecagrammic prism is a star polyhedron made from two identical regular hendecagrams connected by squares. The related hendecagrammic antiprisms are made from two identical regular hendecagrams connected by equilateral triangles.

References

Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society , 79 (3): 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

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