Great truncated icosidodecahedron

Great truncated icosidodecahedron

Type	Uniform star polyhedron
Elements	F = 62, E = 180 V = 120 (χ = 2)
Faces by sides	30{4}+20{6}+12{10/3}
Coxeter diagram
Wythoff symbol	2 3 5/3 \|
Symmetry group	I_h, [5,3], *532
Index references	U₆₈, C₈₇, W₁₀₈
Dual polyhedron	Great disdyakis triacontahedron
Vertex figure	4.6.10/3
Bowers acronym	Gaquatid

3D model of a great truncated icosidodecahedron

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.^[1] It is given a Schläfli symbol $t0,1,2{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}5⁄3,3},$ 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{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 $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio.

Related polyhedra

Great disdyakis triacontahedron

Great disdyakis triacontahedron

Type	Star polyhedron
Face
Elements	F = 120, E = 180 V = 62 (χ = 2)
Symmetry group	I_h, [5,3], *532
Index references	DU₆₈
dual polyhedron	Great truncated icosidodecahedron

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

References

↑ Maeder, Roman. "68: great truncated icosidodecahedron". MathConsult.

Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 p. 96

External links

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Maeder, Roman. "68: great truncated icosidodecahedron". MathConsult.

[1]

Star-polyhedra navigator
Kepler-Poinsot polyhedra (nonconvex regular polyhedra)	small stellated dodecahedron great dodecahedron great stellated dodecahedron great icosahedron
Uniform truncations of Kepler-Poinsot polyhedra	dodecadodecahedron truncated great dodecahedron rhombidodecadodecahedron truncated dodecadodecahedron snub dodecadodecahedron great icosidodecahedron truncated great icosahedron nonconvex great rhombicosidodecahedron great truncated icosidodecahedron
Nonconvex uniform hemipolyhedra	tetrahemihexahedron cubohemioctahedron octahemioctahedron small dodecahemidodecahedron small icosihemidodecahedron great dodecahemidodecahedron great icosihemidodecahedron great dodecahemicosahedron small dodecahemicosahedron
Duals of nonconvex uniform polyhedra	medial rhombic triacontahedron small stellapentakis dodecahedron medial deltoidal hexecontahedron small rhombidodecacron medial pentagonal hexecontahedron medial disdyakis triacontahedron great rhombic triacontahedron great stellapentakis dodecahedron great deltoidal hexecontahedron great disdyakis triacontahedron great pentagonal hexecontahedron
Duals of nonconvex uniform polyhedra with infinite stellations	tetrahemihexacron hexahemioctacron octahemioctacron small dodecahemidodecacron small icosihemidodecacron great dodecahemidodecacron great icosihemidodecacron great dodecahemicosacron small dodecahemicosacron