Thirty Golden Rectangles, Rotating About a Common Axis

Posted on 7 December 2014 by RobertLovesPi

The third image in the last post is a faceting of the icosidodecahedron. In that faceting, the faces used are equilateral triangles, star pentagons, and golden rectangles. To make these two new images, starting with that particular faceting of the icosidodecahedron, I rendered its triangles and star pentagons invisible, leaving only the thirty golden rectangles. It’s shown twice below, simply because I wanted to show it using two different coloring-schemes.

I would not be able to create images like this without the use of my favorite computer program, Stella 4d, written by a friend of mine who lives in Australia. You can try this program yourself, as a free trial download, at http://www.software3d.com/Stella.php.

A Polyhedral Demonstration of the Fact That Nine Times Thirty Equals 270, Along with Its Interesting Dual

Posted on 16 November 2014 by RobertLovesPi

It would really be a pain to count the faces of this polyhedron, in order to verify that there are 270 of them. Fortunately, it isn’t necessary to do so. The polyhedron above is made of rhombus-shaped panels which correspond to the thirty faces of the rhombic triacontahedron. Each of these panels contains nine faces: one square, surrounded by eight triangles. Since (9)(30) = 270, it is therefore possible to see that this polyehdron has 270 faces, without actually going to the trouble to count them, one at a time.

The software I used to make this polyhedron may be found at http://www.software3d.com/Stella.php, and is called Stella 4d. With Stella 4d, a single mouse-click will let you see the dual of a polyhedron. Here’s the dual of the one above.

This polyhedron is unusual, in that it has faces with nine sides (enneagons, or nonagons), as well as fifteen sides (pentadecagons). However, these enneagons and pentadecagons aren’t regular — yet — but they will be in the next post.