Information on Heesch tiling can be found here… https://en.wikipedia.org/wiki/Heesch%27s_problem
The examples I found, were drawn up in VectorEngineer Quick-Tools with the aid of a grid (regular hexagons connected point to point). I used Jasc PaintShop Pro V.7 to make them look pretty.
Here are the shapes that I decided upon. Numbers inside denote number of sides.
Heesch number 4
Heesch number 5
Heesch number 5
By dissecting the pentagon on the far right into three distinct shapes, three other pentagons can be produced, including the floret (second in from left). All but one can tile the plane – but which?
Below are a few examples of how the floret (coloured) and smaller dart, might fit together. All patterns should be able to extend further.
… and a couple more with small and large regular hexagons at the centre.
Can this chaotic pattern keep going?
Here’s a few extras that I dug up, with the smaller darts highlighted. The last pattern displays no symmetry.
A few patterns that were created from my doodlings, with the help of two bits of kit – Kaleidoscope 2.1 plugin from https://www.mehdiplugins.com/ and Selva 3D online (https://www.selva3d.com/).
… and a couple of life patterns (with user input).
Continuing with the prismatic chains theme…
Propellers – tri-fold rotational patterns with the latter exhibiting a dividing line. If you were to omit this line and rejoin the segments, would the pattern then become a mirror image of the first?
An interesting combination of links, chains and gears (aka Christmas trees).
At first glace, the inset image on the right, looks like a magnified version of the one on the left. In fact, it has grown due to more shapes being added to the motif.
Fort – a particular favourite of mine. By mirroring top, bottom, left and right, the pattern will produce a more detailed wood grain effect than the example above it.
Whirlpool – this pattern has no symmetry. If the seven spiral gears were to continue to form a closed loop, it would leave a regular hexagon void at its centre.