“Expanded cuboctahedron”的意思、由来-开放百科全书

词条

Expanded cuboctahedron

释义

Other names
Expansion
Honeycomb
Dissection
Related polyhedra
See also
References
External links

Expanded cuboctahedron

Schläfli symbol	rr = rrr{4,3}
Conway notation	edaC = aaaC
Faces	50: 8 {3} 6+24 {4} 12 rhombs
Edges	96
Vertices	48
Symmetry group	O_h, [4,3], (*432) order 48
Rotation group	O, [4,3]⁺, (432), order 24
Dual polyhedron	Deltoidal tetracontaoctahedron
Properties	convex
Net

The expanded cuboctahedron is a polyhedron, constructed as a expanded cuboctahedron. It has 50 faces: 8 triangles, 30 squares, and 12 rhombs. The 48 vertices exist at two sets of 24, with a slightly different distance from its center.

It can also be constructed as a rectified rhombicuboctahedron.

Other names

Expanded rhombic dodecahedron
Rectified rhombicuboctahedron
Rectified small rhombicuboctahedron
Rhombirhombicuboctahedron

Expansion

The expansion operation from the rhombic dodecahedron can be seen in this animation:

Honeycomb

The expanded cuboctahedron can fill space along with a cuboctahedron, octahedron, and triangular prism.

Dissection

Excavated expanded cuboctahedron
Faces	86: 8 {3} 6+24+48 {4}
Edges	168
Vertices	62
Euler characteristic	-20
genus	11
Symmetry group	O_h, [4,3], (*432) order 48

This polyhedron can be dissected into a central rhombic dodecahedron surrounded by: 12 rhombic prisms, 8 tetrahedra, 6 square pyramids, and 24 triangular prisms.

If the central rhombic dodecahedron and the 12 rhombic prisms are removed, you can create a toroidal polyhedron with all regular polygon faces.^[1] This toroid has 86 faces (8 triangles and 78 squares), 168 edges, and 62 vertices. 14 of the 62 vertices are on the interior, defining the removed central rhombic dodecahedron. With Euler characteristic χ = f + v - e = -20, its genus, g = (2-χ)/2 is 11.

Related polyhedra

Name	Cube	Cubocta- hedron	Rhombi- cuboctahedron	Expanded cuboctahedron
Coxeter^[2]	C	CO = rC	rCO = rrC	rrCO = rrrC
Conway	C	aC = aO	eC	eaC
Image
Conway	O = dC	jC	oC	oaC
Dual

References

1. ^A Dissection of the Expanded Rhombic Dodecahedron
2. ^http://mathworld.wolfram.com/UniformPolyhedron.html

Coxeter Regular Polytopes, Third edition, (1973), Dover edition, {{ISBN|0-486-61480-8}} (pp. 145–154 Chapter 8: Truncation)
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}}

External links

George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
Variations on a Rhombic Theme
www.software3d.com: Prism-Expanded Dissected Cuboctahedron

1 : Polyhedra

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