Snub icosidodecadodecahedron explained

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₆. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices.^[1] As the name indicates, it belongs to the family of snub polyhedra.

Cartesian coordinates

Let

\rho ≈ 1.3247179572447454

be the real zero of the polynomial

x^3-x-1

. The number

\rho

is known as the plastic ratio. Denote by

\phi

the golden ratio. Let the point

be given by

p= \begin{pmatrix} \rho\\ \phi^2\rho^2-\phi^2\rho-1\\-\phi\rho^2+\phi^{2
\end{pmatrix}}

.Let the matrix

be given by

M= \begin{pmatrix} 1/2&-\phi/2&1/(2\phi)\\ \phi/2&1/(2\phi)&-1/2\\ 1/(2\phi)&1/2&\phi/2 \end{pmatrix}

is the rotation around the axis

(1,0,\phi)

by an angle of

2\pi/5

, counterclockwise. Let the linear transformations

T_0,\ldots,T₁₁

be the transformations which send a point

(x,y,z)

to the even permutations of

(\pmx,\pmy,\pmz)

with an even number of minus signs. The transformations

T_i

constitute the group of rotational symmetries of a regular tetrahedron.The transformations

T_iM^j

(i=0,\ldots,11

j=0,\ldots,4)

constitute the group of rotational symmetries of a regular icosahedron.Then the 60 points

T_iM^jp

are the vertices of a snub icosidodecadodecahedron. The edge length equals

2\sqrt{\phi^2\rho^2-2\phi-1}

, the circumradius equals

\sqrt{(\phi+2)\rho^{2+\rho-3\phi-1}}

, and the midradius equals

\sqrt{\rho^2+\rho-\phi}

For a snub icosidodecadodecahedron whose edge length is 1,the circumradius is

	12\sqrt{\rho
	^2+\rho+2}

≈ 1.126897912799939

Its midradius is

	12\sqrt{\rho
	^2+\rho+1}

≈ 1.0099004435452335

Related polyhedra

Medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

Notes and References

Web site: 46: snub icosidodecadodecahedron. Maeder. Roman. MathConsult.

Snub icosidodecadodecahedron explained

Cartesian coordinates

Related polyhedra

Medial hexagonal hexecontahedron

See also

Notes and References