The only reason I am calling this simply a geometrical tiling, rather than a tessellation, is that I want to recognize the regular icosagons (twenty-sided polygons) as part of the pattern — and the icosagons here overlap, violating the established rules for tessellations.
Radial tessellations are described in the previous post. This is a continuation of the idea, but with both regular pentagons and hexagons included. As before, the largest resulting gap-polygons expand as one recedes from the center.
I call this sort of thing a “radial tessellation” — it follows definite rules that resemble those for regular or semi-regular tessellations, but possesses, primarily, radial symmetry. It also has lines of reflective symmetry, but these lines all meet at the radial-symmetry central point, which, in this case, is inside the central pentagon.
Moving out from the central point, the first gap-polygons encountered are black rhombi. The gaps exist because the 360 degrees necessary to surround a point cannot be divided by a whole number of 108 degree angles, from the regular pentagons, without leaving a remainder. This remainder, from arithmetic, is manifested geometrically as a gap between pentagons.
After the rhombi, moving further from the center, appear purple, non-convex equilateral hexadecagons, then, after that, larger, red polygons with more sides and indentations, and then the next, even-more-complex polygons after that, in yellow. Off the edges of the screen, this increase in gap-polygon size and complexity continues without limit, provided the pattern shown is followed. Here is the “recipe” for producing it:
1. Begin with a regular pentagon. Locate its center, and use it as the center point for all rotations.
2. Designate the line containing an outer edge of your figure as a line of reflection.
3. Reflect your entire figure over the designation line of reflection.
4. Take the newly-reflected figure, and rotate it around the central point by 72 degrees. Next, perform this same rotation, using the newest figure produced each time, three more times.
5. Return to step 2.
Without even checking, I know that my automatic tweet about this post (as @RobertLovesPi) will be retweeted by the @HexagonBot on Twitter. Why? Because @HexagonBot retweets any tweet containing the word “hexagon,” or “hexagons.” I have absolutely no idea why other polygons lack their own Twitterbots, though.
This is based on the well-known tessellation of the plane with hexagons. Each side of each hexagon has been replaced by a set of three semicircles.
To see a single four-part compound eye, look at the previous post.
The images on the faces of this icosahedron appeared on this blog three posts ago, and were created with Geometer’s Sketchpad and MS-Paint. To project this image onto this polyhedron, and then generate this rotating .gif, I used a third program, Stella 4d. You may try this last program yourself, as a free trial download, at www.software3d.com/Stella.php.
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