Triakis icosahedron

As a Kleetope

The triakis icosahedron can be formed by gluing triangular pyramids to each face of a regular icosahedron. Depending on the height of these pyramids relative to their base, the result can be either convex or non-convex. This construction, of gluing pyramids to each face, is an instance of a general construction called the Kleetope; the triakis icosahedron is the Kleetope of the icosahedron. This interpretation is also expressed in the name, triakis, which is used for the Kleetopes of polyhedra with triangular faces.

When depicted in Leonardo's form, with equilateral triangle faces, it is an example of a non-convex deltahedron, one of the few known deltahedra that are isohedral (meaning that all faces are symmetric to each other).{{cite journal

Each edge of the triakis icosahedron has endpoints of total degree at least 13. By Kotzig's theorem, this is the most possible for any polyhedron. The same total degree is obtained from the Kleetope of any polyhedron with minimum degree five, but the triakis icosahedron is the simplest example of this construction.{{cite journal | hdl-access = free

As a Catalan solid

The triakis icosahedron is a Catalan solid, the dual polyhedron of the truncated dodecahedron. The truncated dodecahedron is an Archimedean solid, with faces that are regular decagons and equilateral triangles, and with all edges having unit length; its vertices lie on a common sphere, the circumsphere of the truncated decahedron. The polar reciprocation of this solid through this sphere is a convex form of the triakis icosahedron, with all faces tangent to the same sphere, now an inscribed sphere, with coordinates and dimensions that can be calculated as follows.

Let \varphi denote the golden ratio. The short edges of this form of the triakis icosahedron have length and the long edges have length Its faces are isosceles triangles with one obtuse angle of } and two acute angles of

As a Catalan solid, its dihedral angles are all equal, \cos^{-1}\left(\frac{6\varphi+3}{4\varphi+7}\right)\approx 160°36'45.188". One possible set of 32 Cartesian coordinates for the vertices of the triakis icosahedron centered at the origin (scaled differently than the one above) can be generated by combining the vertices of two appropriately scaled Platonic solids, the regular icosahedron and a regular dodecahedron:{{cite journal

• Twelve vertices of a regular icosahedron, scaled to have a unit circumradius, with the coordinates \frac{(0, \pm 1, \pm \varphi)}{\sqrt{\varphi^2 + 1}} , \frac{(\pm 1, \pm \varphi, 0)}{\sqrt{\varphi^2 + 1}} , \frac{(\pm \varphi, 0, \pm 1)}{\sqrt{\varphi^2 + 1}}.
• Twenty vertices of a regular dodecahedron, scaled to have circumradius \frac{2+\varphi}{3+2\varphi}\sqrt {\frac{3}{2-1/\varphi}}=\frac{1}{11}\sqrt {75 + 6\sqrt{5}}\approx 0.8548, with the coordinates (\pm 1, \pm 1,\pm 1)\frac{\sqrt {25 + 2\sqrt{5}}}{11} and (0, \pm \varphi, \pm \frac{1}{\varphi})\frac{\sqrt {25 + 2\sqrt{5}}}{11} , (\pm \frac{1}{\varphi}, 0 , \pm \varphi)\frac{\sqrt {25 + 2\sqrt{5}}}{11} , (\pm \varphi, \pm \frac{1}{\varphi},0)\frac{\sqrt {25 + 2\sqrt{5}}}{11}.

Symmetry

In any of its standard convex or non-convex forms, the triakis icosahedron has the same symmetries as a regular icosahedron. The three types of symmetry axes of the icosahedron, through two opposite vertices, edge midpoints, and face centroids, become respectively axes through opposite pairs of degree-ten vertices of the triakis icosahedron, through opposite midpoints of edges between degree-ten vertices, and through opposite pairs of degree-three vertices.

References

References

1. (2008). "The Symmetries of Things". AK Peters.
2. Wenninger, Magnus. (1974). "Polyhedron Models". Cambridge University Press.
3. "Triakis icosahedron".
4. {{The Geometrical Foundation of Natural Structure (book)