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Great Rhombicuboctahedron


U11GreatRhombicuboctNet
A03Proj1A03Proj2A03Proj3A03Proj4

polyhdron net The 26-faced Archimedean solid A_3 consisting of faces 12{4}+8{6}+6{8}. It is sometimes (improperly) called the truncated cuboctahedron (Ball and Coxeter 1987, p. 143), and is also more properly called the rhombitruncated cuboctahedron. It is uniform polyhedron U_(11) and Wenninger model W_(15). It has Schläfli symbol t{3; 4} and Wythoff symbol 234|.

The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It can be combined with cubes and truncated octahedra into a regular space-filling pattern.

The small cubicuboctahedron is a faceted version of the great rhombicuboctahedron.

Its dual is the disdyakis dodecahedron, also called the hexakis octahedron. The inradius r of the dual, midradius rho of the solid and dual, and circumradius R of the solid for a=1 are

r=3/(97)(14+sqrt(2))sqrt(13+6sqrt(2))
(1)
 approx 2.20974
(2)
rho=1/2sqrt(12+6sqrt(2))
(3)
 approx 2.26303
(4)
R=1/2sqrt(13+6sqrt(2))
(5)
 approx 2.31761.
(6)

Additional quantities are

t=tan(1/8pi)
(7)
=sqrt(2)-1
(8)
l=2t=2(sqrt(2)-1)
(9)
h=1+lsin(1/4pi)
(10)
=3-sqrt(2).
(11)

The distances between the solid center and centroids of the square and octagonal faces are

r_4=1/2(3+sqrt(2))
(12)
r_8=1/2(1+2sqrt(2)).
(13)

The surface area and volume are

S=12(2+sqrt(2)+sqrt(3))
(14)
V=22+14sqrt(2).
(15)

See also

Archimedean Solid, Equilateral Zonohedron, Great Truncated Cuboctahedron, Small Rhombicuboctahedron, Octahemioctahedron, Uniform Great Rhombicuboctahedron

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 138, 1987.Cundy, H. and Rollett, A. "Great Rhombicuboctahedron or Truncated Cuboctahedron. 4.6.8." §3.7.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 106, 1989.Geometry Technologies. "Rhombitruncated Cubeoctahedron." http://www.scienceu.com/geometry/facts/solids/rh_tr_cubeocta.html.Kasahara, K. "Two New Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 227, 1988.Wenninger, M. J. "The Rhombitruncated Cuboctahedron." Model 15 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 29, 1989.

Cite this as:

Weisstein, Eric W. "Great Rhombicuboctahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreatRhombicuboctahedron.html

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