Robert Reid created many beautiful tilings by overlapping n-gons. I tried this using three 16-gons which produced a 12-gon crescent. I believe the resulting shape can tile forever in various guises.
The radial tilings (and spiral) below, are all different. Three with imbalanced cores.
The Witch’s hat (far left) cannot tile the plane. But, by splitting it into two shapes (a house and one other), various new tilings can evolve.
All the drawings below were kindly sent to me by Michael Dowle. They are all based on the house and cookie fortune example (second in from the left). I edited the garish colours a little!
Michael stated that the radial tilings, proved to be quite challenging. The first and second examples below, start off with the same core (the first was abandoned).
I didn’t realise at the time but a shoveler can be cut from a enneamond. The waste produces another shape which I have called a coot. Further cuts produce a cygnet (more on this later perhaps).
The shoveler and coot in combination can produce a regular hexagon. Other permutations are possible.
It is interesting to note that the shoveler has 8 sides, the coot 7 and the cygnet 9.
Michael Dowle has redrawn his original findings of the new shape. Most of the angles are rudimentary but the others are quite odd.
I have since found a couple of motifs that can be tiled periodically. The coloured double hexes help to show how it was constructed.
A combination of the two (with one modification) produces something like this…
The new shape has been dubbed the ‘Shoveler’ as it resembles a duck on water. I missed off a couple of internal angles (D) last time round. The two examples on the right show that B+C+D=360 and A+D=360 (creating a double hex).
Here are a few more ‘seeds’…
Next up is a few small periodic/radial tilings. I coloured in the double hexes in the bottom right image, to show the chaotic nature of the pattern.
Following on from the previous post, the new shape (below) can be thought of as ‘parts’ (A, B and C) to aid pattern making.
Below are some of the combinations that could be used as starting points for possibly much larger tilings.
Here’s a few more…
I still have others to show but have run out of time.
This particular heptiamond can produce an unlimited array of interesting patterns.
Furthermore, by placing three in a particular arrangement, a new shape emerges (currently unnamed).
However, there were a couple of angles that I could not determine, so I asked the help of Michael Dowle who very kindly came up with the answers.
Drawing by Michael Dowle
Disappointingly, initial experimentation with the new shape, didn’t seem to work for radial/chaotic tiling. As you can see below, the shape (highlighted in green), doesn’t quite fit in two of the examples and the other is miles out!
Curiously though and to my surprise, I have since discovered many ways that the new shape can tile, seemingly very chaotically. Results of this coming soon…