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).
A radial tiling can display two types of mirror symmetry (Robert Reid’s example is on the right) or can be mixed and matched to produce no symmetry (not shown).
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.
I divided a square into two parts producing a kite and dart and includes angles of 30, 60, 90, 150 and 210. The small equilateral triangle (below right) is used later on as a centre piece.
The kite and dart can be arranged in a certain way, to create an infinite tiling, radiating from a single point. Zig-zags are also possible.
Six fold rotational symmetry works using six ‘V’ tilings.
By combining groups of 60 and 90 degree angled sub tilings, larger patterns evolve. Bright yellow areas are basically just squares (a kite and dart for each).
‘V’ tilings can also overlap (white dart in centre below).
Other permutations used solely or in combination, are also possible. The fourth example below, is a mix from two out of the three patterns above it.
I tried creating a triangle but it left a hole. However, by filling this with a small equilateral triangle (in blue), a pattern can still emanate from it, without any triangle repetition.
Continuing on from the previous post, Michael Dowle made up some drawings of his own and has provided explanations to help illustrate the chirality property of the wavy kite and darts.
On left, my tessellation and far right, the same turned through 180° and mirrored.
The example in the middle, is drawn using a mixture of the two enantiomers of the blue wavy kites and only one of the two enantiomeric darts. But you cannot mix these enantiomers, since their placement/relative positioning, now breaks Penrose’s rules (not easy to spot – ed).
The illustration above (5-fold rotational symmetry), uses 5 copies of my tessellation with some extra kites and darts to fill spaces.
Credit – Michael Dowle.