Great Disdyakis Triacontahedron

In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U₆₈. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol and Coxeter-Dynkin diagram, .

Cartesian coordinates

Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of
\begin
\Bigl(& \pm\,\varphi,& \pm\,\varphi,& \pm \bigl -\frac\bigr&\Bigr),\\
\Bigl(& \pm\,2\varphi,& \pm\,\frac,& \pm\,\frac &\Bigl), \\
\Bigl(& \pm\,\varphi,& \pm\,\frac,& \pm \bigl +\frac\bigr&\Bigr), \\
\Bigl(& \pm\,\sqrt,& \pm\,2,& \pm\,\frac &\Bigr), \\
\Bigl(& \pm\,\frac,& \pm\,3,& \pm\,\frac &\Bigr),
\end

where $\varphi = \tfrac$ is the golden ratio.

Related polyhedra

Great disdyakis triacontahedron

The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.

Proportions

The triangles have one angle of $\arccos\left(\tfrac+\tfrac\sqrt\right) \approx 71.594\,636\,220\,88^$, one of $\arccos\left(\tfrac+\tfrac\sqrt\right) \approx 13.192\,999\,040\,74^$ and one of $\arccos\left(\tfrac-\tfrac\sqrt\right) \approx 95.212\,364\,738\,38^.$ The dihedral angle equals $\arccos\left(\tfrac\right) \approx 121.336\,250\,807\,39^.$ Part of each triangle lies within the solid, hence is invisible in solid models.

See also

* List of uniform polyhedra

References

* p. 96

External links

*
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Uniform polyhedra

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