Great truncated icosidodecahedron

Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of

$$\backslash begin\{array\}\{ccclc\}$$

\Bigl(& \pm\,\varphi,& \pm\,\varphi,& \pm \bigl[3-\frac{1}{\varphi}\bigr] &\Bigr),\\

\Bigl(& \pm\,2\varphi,& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^3} &\Bigl), \\

\Bigl(& \pm\,\varphi,& \pm\,\frac{1}{\varphi^2},& \pm \bigl[1+\frac{3}{\varphi}\bigr] &\Bigr), \\

\Bigl(& \pm\,\sqrt{5},& \pm\,2,& \pm\,\frac{\sqrt{5}}{\varphi} &\Bigr), \\

\Bigl(& \pm\,\frac{1}{\varphi},& \pm\,3,& \pm\,\frac{2}{\varphi} &\Bigr),

\end{array}

where $\backslash varphi\; =\; \backslash tfrac\{1\; +\; \backslash sqrt\; 5\}\{2\}$ is the golden ratio.

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{{Uniform polyhedra db|Uniform dual polyhedron stat table|gtID}}

File:Great disdyakis triacontahedron.stl

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.

The triangles have one angle of $\backslash arccos\backslash left(\backslash tfrac\{1\}\{6\}+\backslash tfrac\{1\}\{15\}\backslash sqrt\{5\}\backslash right)\; \backslash approx\; 71.594\backslash ,636\backslash ,220\backslash ,88^\{\backslash circ\}$, one of $\backslash arccos\backslash left(\backslash tfrac\{3\}\{4\}+\backslash tfrac\{1\}\{10\}\backslash sqrt\{5\}\backslash right)\; \backslash approx\; 13.192\backslash ,999\backslash ,040\backslash ,74^\{\backslash circ\}$ and one of $\backslash arccos\backslash left(\backslash tfrac\{3\}\{8\}-\backslash tfrac\{5\}\{24\}\backslash sqrt\{5\}\backslash right)\; \backslash approx\; 95.212\backslash ,364\backslash ,738\backslash ,38^\{\backslash circ\}.$ The dihedral angle equals $\backslash arccos\backslash left(\backslash tfrac\{-179+24\backslash sqrt\{5\}\}\{241\}\backslash right)\; \backslash approx\; 121.336\backslash ,250\backslash ,807\backslash ,39^\{\backslash circ\}.$ Part of each triangle lies within the solid, hence is invisible in solid models.

• List of uniform polyhedra

{{Reflist}}

• {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}} p. 96

• {{mathworld | urlname = GreatTruncatedIcosidodecahedron| title = Great truncated icosidodecahedron}}
• {{mathworld | urlname = GreatDisdyakisTriacontahedron| title =Great disdyakis triacontahedron}}

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Category:Uniform polyhedra

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