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Echidnahedron


icos04

The "echidnahedron" is the term for the spiky fourth icosahedron stellation (in the enumeration of Maeder 1994) apparently first used in the Netlib polyhedron database.

It is implemented in the Wolfram Language as PolyhedronData["Echidnahedron"].

Echidnahedron vertex groups

It has 92 vertices, 270 edges, and 180 faces. Its vertices are arranged in three concentric groups of 20 (indicated in red in the above illustration), 12 (green), and 60 (blue).

Polyhedra determined by vertex groups

For an echidnahedron with edge lengths a, phia, phi^2a, and phi^2sqrt(2)a (where phi is the golden ratio), the vertices determine a regular dodecahedron, regular icosahedron, and (irregular) truncated icosahedron, with circumradii

R_(12)=sqrt(3/2(3+sqrt(5)))a
(1)
R_(20)=sqrt(1/2(25+11sqrt(5)))a
(2)
R_(60)=sqrt(1/2(97+43sqrt(5)))a,
(3)

respectively.

The surface area and volume are given by

S=1/(20)(13211+sqrt(174306161))a^2
(4)
V=(210+90sqrt(5))a^3.
(5)

A solid echidnahedron has moment of inertia tensor

 I=[1/(45)(299+131sqrt(5))Ma^2 0 0; 0 1/(45)(299+131sqrt(5))Ma^2 0; 0 0 1/(45)(299+131sqrt(5))Ma^2]
(6)

for uniform density solid of mass M.


See also

Icosahedron Stellations, Small Triambic Icosahedron, Spikey

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References

Maeder, R. E. "The Stellated Icosahedra." Mathematica in Education 3, 5-11, 1994. http://library.wolfram.com/infocenter/Articles/2519/.Sandia National Laboratories. "Polyhedron Database." http://netlib.sandia.gov/polyhedra/.Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, p. 65, 1971.

Cite this as:

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

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