Upright regular dodecahedron

There is a wealth of resources on the internet about the regular dodecahedron, most notably on Wikipedia. Wikipedia also provides extensive information about the construction of a regular dodecahedron with an inscribed cube of edge length 2 standing on one of its edges. However, what I needed, and perhaps someone else will need, is the construction of a regular dodecahedron standing on one of its vertices, an upright regular dodecahedron.

As I could not find a source for this, I had to do all the calculations myself, which took me several hours in total. I realised that the upright regular dodecahedron is easiest to describe in polar coordinates. All vertices lie on circles that have a common axis, at a total of six different heights. In the figure below, the north pole vertex is purple and the south pole vertex is red, while the blue, green, yellow and orange coloured vertices all lie on the same circle at the same height. The heights with respect to the centre for a regular dodecahedron with an inscribed cube of edge length 2 are given in the following table.

Number	Count	Colour	Height	Radius
i	n		h_i	r_i
1	1	violet	√ 9/3	0
2	3	blue	√ 5/3	√ 4/3
3	6	green	√ 1/3	√ 8/3
4	6	yellow	-√ 1/3	√ 8/3
5	3	orange	-√ 5/3	√ 4/3
6	1	red	-√ 9/3	0

While the vertices on planes 2 and 5 are equidistant with a separation angle 120°, they are rotated by an angle 60° between the two planes. On the other hand, the vertices of planes 3 and 4 are not equidistant, but they are rotated by the angle α = arctan(√ 3/5 ) with respect to the vertices of planes 2 and 5.

All other proportions are the same as on the Wikipedia link.

Created by Marko Pinteric. Please contact me for suggestions, new tricks, etc.

Updated . Web page has been read by visitors since February 2025.