Steps of this construction:
- Use the green circles and blue lines to construct the yellow pentagon, along with its green inscribed pentagram.
- Construct the equilateral triangle shown in gray. This is needed to obtain a twelve degree angle. The triangle is needed for its sixty degree angle, because 72 – 60 = 12. (The 72 degree angle is found inside the pentagon.)
- Identify the twelve degree angle shown in bold. A twelve degree angle is needed because 360 / 30 = 12.
- Use the red circles to complete the thirty sides of the regular triacontagon, which is shown with bold black segments, inscribed inside a large, bold, red circle.
This collection of curves was built around a tessellation of the plane using regular hexagons. To make the second version, I inverted the colors, except for the black circles and arcs.
I made these virtual models using Stella 4d: Polyhedron Navigator. If you’d like to try this program for yourself — free — the website to visit is http://www.software3d.com/Stella.php.
The snub dodecahedron may be thought of as a dozen regular pentagons, surrounded and separated by a sea of triangles. In this expansion of that Archimedean solid, thirty parallelograms are added to the mix, also surrounded by triangles. In the image above, coloring is by face type — for example, the yellow triangles are those triangles which share an edge with a pentagon. Other triangles have other colors.
The image shown below is of the same polyhedron, but with a different coloring-scheme. In it, all triangles are given the same color, even when their shapes are slightly different.
This polyhedron has an interesting all-pentagon dual, which is shown below. This dual has sixty each of both the small and large pentagons, for a total of 120 faces.
I used Stella 4d: Polyhedron Navigator to create these polyhedra, and to make these rotating images. This program may be tried for free at http://www.software3d.com/Stella.php.
Created using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.