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JUNE Challenge - suggestion - draw a Regular dodecahedron

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draw a regular dodecahedron.

From Wikipedia, the free encyclopedia: https://en.wikipedia.org/wiki/Regular_dodecahedron

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular composed of twelve regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}.

Typ

Faces by sides

Conway notation

Schläfli symbols

Coxeter diagram

Properties

Dimensions

If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is

$r_u = a\frac{\sqrt{3}}{4} \left(1 + \sqrt{5}\right) \approx 1.401258538 \cdot a$

and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is

$r_i = a\frac{1}{2} \sqrt{\frac{5}{2} +\frac{11}{10}\sqrt{5}} \approx 1.113516364 \cdot a$

while the midradius, which touches the middle of each edge, is

$r_m = a\frac{1}{4} \left(3 +\sqrt{5}\right) \approx 1.309016994 \cdot a$

These quantities may also be expressed as

$r_u = a\, \frac{\sqrt{3}}{2} \phi$

$r_i = a\, \frac{\phi^2}{2 \sqrt{3-\phi}}$

$r_m = a\, \frac{\phi^2}{2}$

The regular dodecahedron can also be represented as a spherical tiling.

Uniform tiling 532-t0.png

Orthographic projection

Honeycomb of regular dodecahedra-cubes

Honeycomb of regular dodecahedra-cubes-J91.png

Edited by YLedEditor Tuesday, May 31, 2016 12:06 AM

Monday, May 30, 2016 11:48 PM

Answerer