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.
There seems to be very few examples of these tilings around, so I decided to experiment a little and post my own. I’m not too sure on the symmetries, so any help with that would be helpful. Frank Morgan’s site is worth a read (I’ve been there a few hundred times to avoid blatant duplicates!)… https://sites.williams.edu/Morgan/2015/01/31/new-optimal-pentagonal-tilings/
I worked with the main clusters above. These quite often overlap. I have coloured the Cairo tiles in light grey throughout.
Prismatic tiles will always produce elongated hexagons. This example I have named the ‘Solar Power Generator’.
Both examples above, have four single prismatic hexagon chains emitting from the centre. The latter I have named ‘Union Jack’ (should have been red, white and blue – might change it).
Triple Pillbox is basically an extended version of Double Pillbox by Maggie Miller.
Chains can be of any length producing a stripey pattern. The second example above combines one, two and three length chains.
The ‘Kissing Gate’ – two fold rotational symmetry?
More examples to follow…
The numbers indicate how many birds are used to create the enclosures or voids. The resulting outlines create ovals, rounded rectangles, pentagons and a rounded hexagon. It looks like the patterns outside the voids will continue forever.
As with all the examples, more birds can sometimes be added within the void but will always leave gaps and may alter the symmetry. The bottom two below, have similar properties and will produce the same overall pattern for the first few iterations at least. One looks like the main core of ten birds has exploded and will soon produce a rounded decagon outline.
Michael Dowle uses the process of moving parts of a tessellation (that may be radial or planar) relative to each other along a plane (or line) to generate an alternative tessellation(s). That plane he calls a slip-plane.
Michael also drew all the illustrations for the book (see previous post) and many of the extensions of Robert’s original bird tessellation evolved during this activity. Many examples that Michael and Robert worked on, did not make it in print due to lack of space.
My examples below, combine mirrored and non-mirrored part cores, indicated by colour. I tend to work in a more haphazard way (not recommended).
With a slight modification, split core arrangements of 4/5, 4/6, 5/6 and 6/6 are also possible.
Two examples of alternative radial tiling seeds with a void at the centre.
I first came across this shape on a cover of a book, entitled ‘The Gentle Art of Filling Space (credited to Robert Reid, Michael Dowle and Anthony Steed).
I have since purchased the e-book of the same name (but different cover art) from Lulu Books, which was the less expensive option.
This intriguing shape is made up of six crowns (my drawings were based on observation only) and can be put together in many different ways.
The book only shows one radial tiling, so I spent some time exploring other possibilities. The examples below show a few ways the ‘birds’ can be tiled periodically (the last one is a hybrid).
Here, two tilings with radial symmetry using the ‘birds’ are illustrated (Robert Reid’s example is on the right). Components of these tilings may be mixed and matched to produce new radial tilings exhibiting no radial symmetry (not illustrated).
By splitting the core of 10 ‘birds’, more chaotic patterns can evolve. Some show the same type of core displacement.
I believe all the patterns above, can expand forever. Part two will show examples of mirrored/non-mirrored core combinations.
I first came across this 7-sided polygon here… http://isohedral.ca/heesch-numbers-part-3-bamboo-shoots-and-ice-cream-cones/
Eight of these fit into an 18-gon (octadecagon).
I replaced the two heptagons in the centre (and any space left over) with six ‘thorns’ (below).
There are now known to be 34 different arrangements of 6 ‘Bamboo Shoots’ and 6 ‘Thorns’ inside an 18-gon. There are an additional 26 if the mirror images of those arrangements without mirror symmetry are included, making a grand total of 60 (that’s ten more than I found)! The arrangements with grey colouring have mirror symmetry. Thank you to Michael Dowle for supplying this updated information and new diagram. The colours really help in understanding the method he used.
It seems that there are only three ways to tile the plane with the bamboo shoot and thorn, all creating ‘pulse’ tilings. The second example is a mirror of the first. Curiously though, mirroring the third is a duplicate of itself!?
You are welcome to correct me on any errors.
Below, a combination of dart, 15th pentagon and a regular hexagon.