An equilateral concave pentagon.
This particular shape was marketed as a tiling puzzle Pentalbi (by Naef Spielzeug -- a Swiss play-toy company) around 1980, as I recall. I had to buy a set (who could resist?). The tiles did surprising things together, and I'll be talking about some of those in the next few days, I hope.
In the meantime, here's what one does:
A. start with two regular pentagons. Align them so exactly two vertices are coincident.
B. Subtract the footprint of one from the other.
C. The result is equilateral, but concave, with interior angles: 36 (x2), 108 (x2) and 252 degrees. Note that the concave angle leaves 108 degrees on the exterior, which is enough for exactly one broad angle or three pointy ones to nestle inside.
D. If you look at the SVG code, you'll see that the polygons generated by Adobe Illustrator come with too many coordinates and the the coordinates have too much precision (and too much imprecision) so reworking those by hand will be useful if one wants a) precision and b) proper tessellation as one starts actually using them to tile in a space (like a computer screen) not known for a plenitude of infinitesimals. (The intuitionists among us might find this slightly humorous.) The superimposition of the red and translucent one reminded me a wee bit of a random superhero's face.
E. Form all rotations of the basic tile around 36 degrees from 36 up to (360-36)=324. There are ten of them. They nest nicely into a regular decagon. (but that's not all they do, as I'll show later).
F. Color each rotation with a separate but distinguishable color I've been interested in the psychophysics of how to choose N maximally distinguishable stimuli within a medium (like color) for some time. It is a tricky problem and would welcome pointers to extant research (@theanimatewoman @ninazumel @praystation @andreali @booksnips @shepazu) . I spoke to someone at the Color Institute in Rochester a few years ago about color and they knew of nothing objective at the time. For art, games, tiling, interface design, it's a practical question! I tried to alternate the colors (the planar dual of the tessellation is two-colorable, so I sort of alternated lighter and darker colors and tried to let pairs of tiles opposite each other (by 180) have similar hue. Aesthetics is a slightly separate issue: one wants pleasant color arrangements, but for sake of tessellation study, distinguishability is even more important.
G. One of these (F) was made in Illustrator, with rotations of tiles being output as actual x,y coordinates. Those then had to be hand edited to eliminate cruft and rounding errors (and to put into integer coordinates, so that tiles will actually fit when twirled and translated). The other (G) was done in SVG simply reusing and rotating a single base tile. I was pleased to see that the 9 virtual tiles played nicely with the one real one, having been concerned that the rounding error in the browsers might muck things up.
H. At one point, I goofed and used the wrong center of rotation. The result was sort of pleasant, so I kept it here and added some transparency so that the stacking order would not be so apparent.