12 a. Truncated icosidodecahedron 12/10 20/6 30/4 from truncation of icosahedron 12/5 20/6

(=Great Rhombicosidodecahedron)

Data

The distance from the centre of the icosahedron to the centre of the triangle is EP×0.76

The distance from the centre of the great rhombicosidodecahedron to the hexagon (red) is EA×3.67.

These two distances are equal, thus EA = EP×0.21.

The circum-circle radius of icosahedron triangle ( EP 0.58) minus the in-circle radius

of the truncated icosidodecahedron hexagon (red)( EA 0.87) is the

distance between icosahedron triangle corner and the mid-edge of the hexagon, i.e.,

EP×0.40.

The in-circle radius of the icosahedron triangle ( EP 0.29) minus the in-circle radius of

of the hexagon (red)(EA×0.87) is the distance between the icosahedron triangle

mid-point and the truncated icosadodecahedron hexagon mid-point,i.e., EP×0.11.

Construction

Twenty hexagons (red) are applied on the icosahedron triangles. The twelve corners are

cut off, resulting in twelve decagons (green-blue) . The thirty edges (between the hexagons)

are cut off, resulting in thirty squares (green-blue)

If the icosidodecahedron is truncated, rectangles are formed with edges E_pF . Kepler gave this polyhedron the name truncated icosidodecahedron, but he knew that it was not possible to perform truncation to form the great rhombicosidodecahedron.

Line 1: Icosahedron, net and solid

Line 2: Truncated icosidodecahedron, net and solid.

12b. Truncated icosidodecahedron from truncation of dodecahedron.

Data

The distance from the centre of the dodecahedron to the centre of pentagon is EP×1.11.

The distance from the centre of the truncated icosidodecahedron to the centre of the

decagon (red) is EA×3.44. These two distances are equal, thus, EA = EP×0.32

The distance between the pentagon mid-edge and the decagon mid-edge is the in-circle

radius of the pentagon (EP×0.69) and the in-circle radius of the decagon(red)(EA×1.54),

i.e., EP×0.19.

Th distance between the decagon mid-point and the corner of the pentagon is the

circum-circle radius of the pentagon (EP×0.85) minus the in-circle radius of the decagon

(EA×1.54), i.e., EP×0.36.

Construction

Twelve truncated icosidodecahedron decagons (red) are applied on the twelve

dodecahedron pentagons.

The twenty corners and the thirty edges between them are cut off, giving twenty

hexagons and thirty squares (green-blue)

Drawing demonstrating the site

of the truncated icosidodecahedron decagon(red)

on the dodecahedron pentagon