Circles, Triangles, and Hexagons

Nine Circles and Their Inscribed Octagrams

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Nine Circles and Their Inscribed Octagrams

52 Triangles Inscribed in 13 Circles

39 Squares Inscribed in 13 Circles

Star Polygon Enumeration

Posted on 17 February 2013 by RobertLovesPi

For any given regular convex polygon, how many star polygons exist?

That’s the beginning. Here are two more:

Since the pattern continues in this manner, it is easy to find the number of possible star polygons for a regular polygon with n sides. First, if (and only if) n is odd, add one. Next, divide by two. Lastly, subtract two, and you have your answer.

Starflower of Seven

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Starflower of Five

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Twenty Interpenetrating, Rotating Equiangular Golden Hexagons

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Alternating sides of these hexagons have lengths in the Golden Ratio. I created this as a faceting of the rhombicosidodecahedron. It also has faces which are star pentagons and triangles, but those faces are hidden in this view.

(Created using software from www.software3d.com/stella.php)

Zome Icosahedron Encasing a Rhombic Enneacontahedron

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Zome Icosahedron Encasing a Rhombic Enneacontahedron

Zome is a three-dimensional ball-and-stick geometrical modeling system based on the Golden Ratio. I have a large collection, and have used it for years, both as a teaching tool, and for my own investigations. Zome is available for sale at www.zometool.com.

Here is a close-up shot, so you can better see the interior of this figure: