A Radial Tessellation of Regular Pentagons and Their Expanding Gap-Polygons

pentagonsI 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.

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