geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a

nonconvex uniform polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

, indexed as . It has 92 faces (80

triangles A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...

and 12 pentagrams), 150 edges, and 60 vertices. It is given a

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

Cartesian coordinates

Let

\xi\approx -1.8934600671194555

be the smallest (most negative) zero of the polynomial

x^3+2x^2-\phi^

, where

\phi

is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

. Let the point

p

be given by :

p=
\begin
        \xi \\
        \phi^-\phi^\xi \\
        -\phi^+\phi^\xi+2\phi^\xi^2
\end

. Let the matrix

M

be given by :

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

M

is the rotation around the axis

(1, 0, \phi)

by an angle of

2\pi/5

, counterclockwise. Let the linear transformations

T_0, \ldots, T_

be the transformations which send a point

(x, y, z)

to the even permutations of

(\pm x, \pm y, \pm z)

with an even number of minus signs. The transformations

T_i

constitute the group of rotational symmetries of a

regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

. The transformations

T_i M^j

(i = 0,\ldots, 11

j = 0,\ldots, 4)

constitute the group of rotational symmetries of a

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

. Then the 60 points

T_i M^j p

are the vertices of a great snub icosahedron. The edge length equals

-2\xi\sqrt

, the circumradius equals

-\xi\sqrt

, and the midradius equals

-\xi

. For a great snub icosidodecahedron whose edge length is 1, the circumradius is :

R = \frac12\sqrt \approx 0.5800015046400155

Its midradius is :

r=\frac\sqrt \approx 0.2939417380786233

The four positive real roots of the sextic in ,

4096R^ - 27648R^ + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0

are the circumradii of the

snub dodecahedron In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex Isogonal figure, isogonal nonprismatic solids constructed by two or more types of regular polygon Face (geometry), faces. The snub dod ...

(U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

References

External links

* Uniform polyhedra {{Polyhedron-stub

Cartesian coordinates

See also

References

External links