Triakis icosahedron

3D view of a triacis icosahedron ( animation )

The Triakis icosahedron is a convex polyhedron , which is composed of 60 isosceles triangles and belongs to the Catalan solids . It is dual to the truncated dodecahedron and has 32 vertices and 90 edges.

Emergence

If pyramids with the flank length are placed on the 20 boundary surfaces of an icosahedron (edge ​​length ) , a triacis icosahedron is created, provided the following condition is met: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\frac {a}{3}}{\sqrt {3}}<b<{\frac {a}{4}}{\sqrt {10-2{\sqrt {5}}}}}$

• For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the icosahedron with the edge length remains. ${\displaystyle b}$ ${\displaystyle a}$
• The special Triakis icosahedron with equal surface angles arises when is. ${\displaystyle b={\frac {a}{22}}\,(15-{\sqrt {5}})}$
• Takes the above At the maximum value, the triacisicosahedron degenerates into a rhombic triacontahedron with the length of the edge . ${\displaystyle b}$ ${\displaystyle b}$
• If the maximum value is exceeded , the polyhedron is no longer convex and finally degenerates into an icosahedral star . ${\displaystyle b}$ ${\displaystyle b={\frac {a}{2}}\left(1+{\sqrt {5}}\right)}$

Formulas

General

Sizes of a triacis kosahedron with edge lengths a , b
volume	${\displaystyle V={\frac {5}{12}}a^{2}\left(a(3+{\sqrt {5}})+4{\sqrt {3b^{2}-a^{2}}}\right)}$
Surface area	${\displaystyle A_{O}=15a{\sqrt {4b^{2}-a^{2}}}}$
Pyramid height	${\displaystyle k={\frac {1}{3}}{\sqrt {9b^{2}-3a^{2}}}}$
Inc sphere radius	${\displaystyle \rho ={\frac {a}{4}}{\sqrt {{\frac {10a+4b}{a+2b}}+2{\sqrt {5}}}}}$
Dihedral angle  (over edge a )	${\displaystyle \cos \,\alpha _{1}={\frac {(12b^{2}-5a^{2}){\sqrt {5}}-8a{\sqrt {3b^{2}-a^{2}}}}{9(4b^{2}-a^{2})}}}$
Dihedral angle  (over edge b )	${\displaystyle \cos \,\alpha _{2}={\frac {2b^{2}-a^{2}}{4b^{2}-a^{2}}}}$

Special

Edge sphere in the special triacis icosahedron: the spherical caps clearly stand out on the individual triangular surfaces. The incircles are also intersections of the triangles with the edge sphere.

Sizes of a triacis kosahedron with edge length a
volume	${\displaystyle V={\frac {5}{44}}a^{3}(5+7{\sqrt {5}})}$
Surface area	${\displaystyle A_{O}={\frac {15}{11}}a^{2}{\sqrt {109-30{\sqrt {5}}}}}$
2. Side length	${\displaystyle b={\frac {a}{22}}\,(15-{\sqrt {5}})}$
Pyramid height	${\displaystyle k={\frac {a}{66}}(5{\sqrt {5}}-9){\sqrt {3}}}$
Inc sphere radius	${\displaystyle \rho ={\frac {a}{4}}{\sqrt {\frac {10(33+13{\sqrt {5}})}{61}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{4}}\left(1+{\sqrt {5}}\right)}$
Face angle  ≈ 160 ° 36 ′ 45 ″	${\displaystyle \cos \,\alpha =-{\frac {1}{61}}(24+15{\sqrt {5}})}$

Remarks

1. ↑ ${\displaystyle {\frac {a}{3}}{\sqrt {3}}<b<{\frac {a}{4}}{\sqrt {10-2{\sqrt {5}}}}}$
2. ↑ ${\displaystyle b={\frac {a}{22}}\,(15-{\sqrt {5}})}$

Web links

Commons : Triakisikosahedra  - collection of images, videos and audio files

• Eric W. Weisstein : Triakisikosahedron . In: MathWorld (English).