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Snub Cube-Pentagonal Icositetrahedron Compound


SnubCubeDualCompound

The compound of the snub cube and its dual, the pentagonal icositetrahedron.

Surprisingly, the tribonacci constant t is intimately related to the metric properties of the snub cube, its dual, and their compound.

The compound can be constructed from the snub cube with unit edge length by heights h_3 and h_4, given by

h_3=(3456x^6-864x^4+216x^2-1)_2
(1)
=1/2sqrt((2-t)/(3(t+1)))
(2)
=0.06868...
(3)
h_4=(128x^6+96x^4+16x^2-1)_2
(4)
=1/2sqrt((2-t)/(t-1))
(5)
=0.218797...,
(6)

where t is the tribonacci constant.

The corresponding solid has edge lengths

s_1=(128x^6-64x^4+16x^2-1)_2
(7)
=1/21/(sqrt(t+1))
(8)
=0.296733...
(9)
s_2=1/2
(10)
s_3=(128x^6-6x^3-1)_2
(11)
s_4=1/2sqrt(2).
(12)

The circumradius is given by

R=(32x^6-80x^4+44x^2-7)_2
(13)
=1/2sqrt((t-3)/(t-2))
(14)
=1.34371...,
(15)

the surface area by the root of

 1028869776+35418062592S-45028405440S^2+22712607360S^3-5396081328S^4+463818960S^5+35732664S^6-7379424S^7+23652S^8+29160S^9-576S^(10)-36S^(11)+S^(12),
(16)

and volume by

V=(128x^6-8864x^4+19152x^2-10609)_2
(17)
=1/2sqrt((583t-694)/(97t-177))
(18)
=8.18758....
(19)

See also

Pentagonal Icositetrahedron, Polyhedron Compound, Snub Cube, Tribonacci Constant

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Cite this as:

Weisstein, Eric W. "Snub Cube-Pentagonal Icositetrahedron Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SnubCube-PentagonalIcositetrahedronCompound.html

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