## Mathematical Model – Great Dodecahedron

Accession Number: 2016.mat.26

A large cardboard model of a great dodecahedron coloured in several shades of blue. The model is badly damaged at its vertices.

Primary Materials: Cardboard

Dimensions (cm): Longest dimension = 24 cm

Function: Pedagogical mathematical model.

Poor: The model is badly ripped in several places at its vertices.

Manufacturer: Unknown

Date of Manufacture: c. 1960

“It is one of the star polyhedra, which means that like the platonic solids, it has polygons for faces, and a number of polygons come together at the vertex. It from the Platonic solids, in that the faces cross one another, so that you have the edges that count as edges, and the outermost edges and also forced edges where polygons cut through one another. There are 30 real edges and the other ones come because one pentagon is in a different plane from another, and they cut through one another along a diagonal. So you see how this has pentagons for faces, pentagrams for the vertex figure. The vertex figure is what you get if you cut off the corner. For example, a vertex figure of a cube is a triangle. So the vertex figure of this is a pentagram. So you have a pair of numbers, the first number is the number of sides to the face, and the other number is the number of faces to the vertex. Then modified here, you see 5/2, because the vertex figure is a star pentagon and not an ordinary pentagon, a pentagon of density two. So this has simple 5:5/2. 5 for the pentagon, and 5/2 for the vertex figure. And until recently, it was believed that this was only one way. About 1810 it was first noticed by a man called Poinsot, in France, and it is in the mosaic of the floor of a cathedral somewhere in Italy. Two of those historical ones are such that you have the Kepler-Poinsot solids: 5/2:5, 5/2:3, 5:5/2 and 3:5/2, and this one, which is 5:5/2, is called the great dodecahedron, and it’s reciprocal 5/2:5 is the one that has the corners of the star sticking out. “ H.S.M. Coxeter, March 10, 2000