6a. Truncated cuboctahedron 12/4 8/6 6/8 from truncation of cube 6/4

(Great rhombicuboctahedron)

Data

The distance from the centre of the cube to the center of the cube square is EP/2. The distance from the centre of the truncated cuboctahedron to the centre of the octagon (red) is EA×1.91

These two distances are equal, thus EA = EP× 0.26.

Divide the EP in five parts x + x + xÖ2 + x + x; x= EP / (4+Ö2) =EP×0.18; EA = EP×Ö2 /(4+Ö2) = EP×0.26

. The distance from the octagon edge to the cube square edge is x = EP× 0.18

Construction

Six octagons (red) are applied on the six cube squares.

The eight corners (yellow) are cut off, resulting in eight hexagons (green-blue). The twelve edges are cut off , giving

twelve squares (green-blue).

.

(If the cuboctahedron is truncated by dividing the edges in two parts, eight triangles on the cuboctahedron triangles

and six squares on the cuboctahedron squares and twelve rectangles (edges: EP/2 and EP Ö2/2) are formed. Kepler gave

this polyhedron the name truncated cuboctahedron , but he knew that it was not possible to perform truncation in this way.)

Line 1: Cube, net and solid. A drawing for calculation of EA

. and for the construction

Line 2: Truncated cuboctahedron,

net and solid

6b. Truncated cuboctahedron from truncation of octahedron 8/3

Data

The distance from the centre of the octahedron to the centre of the triangle is EP×0.41. The distance from the centre

of the truncated cuboctahedron to the centre of the hexagon (red) is EA×2.09.

These two distances are equal, thus EA = EP 0.19.

The distance between the mid-edge of the hexagon and the mid-edge of the octahedron triangle is in-circle radius

of the octagon triangle minus in-circle radius of the hexagon; 0.29 EP – 0.87 EA = 0.12 EP

The distance between the corner of the octahedron triangle and the mid-edge of the hexagon is circus-circle radius

of the octahedon triangle and the in-circle radius of the hexagon ; 0.58 EP – 0.87 EA = 0.41 EP.

Construction

Eight hexagons (red) are applied on the triangles of the octahedron. The corners (yellow) and the edges (yellow) are

cut off resulting in twelve squares and six octagons (blue-green).(See 6a)

Drawing demonstrating the site of the hexagon