I composed this shape (in 2019) by overlapping decagons, a technique that was employed by Robert Reid in the book ‘The Gentle Art of Filling Space’. However, my motif generating procedure differs, in that the decagon overlaps occur at midpoints between decagon vertices as opposed to Robert’s coincident vertices (thank you to Michael Dowle for pointing that out).
I later stumbled across a similar shape, discovered by Tim Lexen, that uses only three arcs, called a tricurve. Both the penguin and tricurve can tile in the same way but according to Tim, the penguin has more tiling possibilities.
I like the tricurve for its smoothness but I find the penguin better to work with, as it has points of reference, such as edge midpoints and vertices.
Penguins can be put together in a number of ways.
A typical radial tiling.
A random selection of central cores that can expand forever.
The beginnings of a single tailed zigzag spiral.
Modified central cores.
Tilings with holes. Small and large decagons and octagons.
An attractive decagram and five pointed star.
Just found this five tailed spiral. I’m quite sure there are others.