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

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.