‘Sphere’ is an applet I found on Jaap’s Puzzle Page some years ago… https://www.jaapsch.net/puzzles/sphere.htm
I used it to create fictitious petanque boules (although one or two I recognise) with mostly ‘over the top’ designs. Striations help players tell their boules apart. I played the game for about 15 years.
I used a couple of filters in Paintshop Pro 7 to vary the colour and to create some depth.
One of a few polykites that I found around the time of the hat. It reminds me of a woodlouse or perhaps some sort of dinosaur.
Woodlice can form attractive radial patterns with 2, 3 and 4-fold symmetries. Each use all six rotations and their mirrors.
Two types of 2-fold symmetry
Two types of 3-fold symmetry
4-fold symmetry (left) and next to it, a star shaped hole at the centre.
Small and large hexagonal holes.
A decorated radial to finish.
Thanks to the super human effort of Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss, the hat and turtle polykites have finally arrived on the scene… https://arxiv.org/abs/2303.10798
This article is intended to be a temporary ‘scrapbook’ of how the polykites were conceived. It will contain many raw images and is likely to look messy until I get round to producing something more presentable.
The hat and turtle can only tile the infinite plane aperiodically. They do not permit periodic tilings. They should be treated with respect!
The shaded areas below indicate the similarities between the two polykites. The hat is composed of 8 kites and the turtle, 10.
Jaap’s polyform software came in very handy (I have been using it for years)… https://www.jaapsch.net/puzzles/polysolver.htm
As you can see in one solution, there’s not much to go on. There’s a few clumps so worth investigating further.
Smaller grid but basically same end product.
I left Jaap’s solver program to run a bit longer, time to place them manually.
There’s bound to be a periodic pattern in there somewhere, right? Or, maybe a non tiler perhaps with a large Heesch number?
It was around this time when I asked for Craig Kaplan’s help, as I knew of his Sat Solver program that deals with Heesch numbers. Next it was time to get some shapes cut on the Silhouette card cutter.
I then tried a few radial patterns with three-fold symmetry. The example far right is a section taken out from the one adjacent to it, as I noticed signs of periodicity but the tessellation could not continue.
In the meantime, Craig’s Sat Solver was crunching away at the numbers (it would have usually spat out any periodic tilings by this time). He threw some of the computational drawings my way (I remember it to be an exciting time). We both started to observe some interesting formations and at the same time, looking out for any repetitions.
Below is another large patch of hat tiles generated and drawn by Craig Kaplan’s software. I added the rectangular white markers to indicate the “odd” side of the small and large triangular arrangements. As you can see, they are never opposite one another.
At a glance the patches below may look the same but the directional flow on the outer rings are different.
The poor quality print out on the right was from a section from another of Craig’s drawings that does not quite display three-fold symmetry. I had an idea to first modify it (below left) and then extend the pattern outwards over to one side. Then copy, paste, rotate and move this new batch of tiles to join to the other two sides but it wasn’t an exact match. I don’t think it is possible for any radial symmetries to tile the plane but will check that with Joseph.
Chains produce a mesh that have triangle and parallelogram like features. I noticed a deep periodic band from left to right, so I played around with 180 degree rotations of it to see if they would fit together but the orientations of chains top and bottom stayed the same.
A computerised Craig Kaplan special on the left. Adjacent to that is a clever substitution rule by Craig which didn’t make it to the paper. I joined up the small triangles (that were there to fill the gaps) which produced “2” like constellations.
It was around this time that I tried other polykites. This one below really stood out in Jaap’s solver program, but putting it together was quite difficult at first.
So, I got some cut and sent a picture of it to Craig with the message “… look familiar?!”
More of Craig’s sensational computational drawings followed. I re-coloured the one on the right to show off the clusters. Notice how they never overlap but sometimes leave gaps. These can be filled with lone tiles or combinations of tiles
Below is an artwork that I made up for “A Hat for Einstien” MoMath talk just recently. The clusters of one reflected tile surrounded by six unreflected tiles is a single piece, really to speed up the work. I have played with both the hat and turtle for a long while now but I have still yet to master them. Even though I know what sorts of configurations I expect to see, I still get stuck and have to back track quite a bit.
More to follow soon…
Drafters are 30-60-90 degree triangles and can produce some neat patterns. The examples that follow are all grid based. From left to right, duck, meerkat and electric eel.
Three ducks can be assembled into boomerang arrangements. These can be placed adjacent to one another within a concertina strip. Ducks in light grey and gold on the outer edges are reflected tiles.
The concertina strips can be shuffled around to create subtle variations.
Meercats produce a ‘ring’ around an offset central six-pointed star resulting in a regular hexagon – very nice. Colours denote left and right versions.
A paved structure without the colours.
The electric eel tessellation is comprised of jagged orbs of both non-reflected and reflected tiles intermixed with six-pointed star-like motifs (only one completed).
Within the same pattern, there are also large regular hexagon clusters that overlap. The one in dark grey is constructed differently to the other two on show (which are 180 degree rotations of each other).
I used the Polyform Puzzle Solver by Jaap Scherphuis for all of these. An application I have been using for many a year, thanks Jaap.
I’ve spent many hours playing with this text prompt to image AI generator – it’s amazing… https://www.artbreeder.com/
Here’s just a few of the images I have created (in no particular order). Click on one to enlarge, then use arrow keys.
I may change the gallery from time to time.
I was going through some stuff around Christmas time and found some really old dot matrix prints. I think I had a Atari 520 STFM and a Lexmark printer at the time. The first is a Heighway dragon curve discovered by John Heighway in 1966.
I’m not familiar with the other two but they look like they are based on six sided stars. They are interesting to study, as they are both made up of a single line from start to finish. Very decorative.
I put together these two polydrafters using the Polyform Puzzle Solver by Jaap Scherphuis… https://www.jaapsch.net/puzzles/polysolver.htm
The red squirrel can produce an attractive periodic tessellation. It is made up of a large motif containing all six rotations and their mirrors.
Six grey squirrels that rotate around a common point, produce a polyhex (a regular heptahex). These motifs, whether mirrored or not, can abut one another in endless arrangements. The greyed out areas denote reflections.
Also, polyhex motifs can be aligned differently around a central polyhex, which will alter the overall appearance.
Here’s another pattern that can only be put together one way (two, if you include its mirrored counterpart).
I created a new shape from Paul Bourke’s concave decagon (third in from left). You can view it here… http://paulbourke.net/geometry/tilingplane/
A selection of periodic tilings below.
There are various ways to string tiles together to form three types of ‘slip planes’. These can be used as building blocks in some situations. The first example (zigzag) includes reflections.
Wavy (low profile). I have found six unique configurations (reflections not shown).
Wavy (high profile). Seven unique configurations (reflections not shown).
Central cores change when combining different configurations of slip-planes.
Grouping of decagons can produce ever expanding ‘motifs’, which can produce some interesting formations.
A single decagon can completely surround itself in various different ways.
The group of decagons on the left produce a ‘sunflower’. A maximum of five adjacent ‘petals’ can be used in a pattern. Two groups of four are also possible.
The thick black line or ‘handlebar’ indicates where a choice can be made. For example, by positioning a decagon (third from the left) another ‘handlebar’ forms. Placing of single decagons thereafter, are more likely to become forced.
The decagons at the heart of this pattern display mirror symmetry. All other decagons placed outside of this are forced.
This impressive double pentagon star tessellation utilises all five decagon rotations and their mirrors.
The excellent ‘Tyler’ app by Melinda Green and Don hatch, lets you explore planar tilings using regular polygons (from 3 to 12 sides). The zoom facility is fantastic. Get it here… https://superliminal.com/geometry/
There are some really good examples, as shown in the gallery… https://superliminal.com/geometry/tyler/gallery/ but in this instance, I am using only hexagons and squares.
To make it more interesting, I put a constraint on the placement of both hexagons and squares, i.e., no hexagon may touch another (except at vertices) and likewise for squares. As well as this, the gaps must be consistent; in this case, thin rhombi.
Below left shows the only possible dodecahedron configuration of hexagons and squares excluding rotations. Also, the ‘propeller’ to the right of it, will always appear in any pattern.
If we consider only squares when continuing a pattern from a dodecahedron at its centre, there arises many combinations.
In the example below, the ‘disk’, far right, is not permissible, as a hexagon within the dodecahedron will touch another on the outer rim.
When extending each of the twenty five arrangements a little further, ‘bites’ sometimes occur. These can be filled in three different ways.
Below – six examples of central cores other than the dodecagon arrangement.
I finish with an attractive aperiodic tessellation.
There is also another article here on these type of formations but without the constraints that I set. http://followinglearning.blogspot.com/2015/07/tiling-again.html
Joshua Socolar discovered a substitution tiling rule for this set of shapes… https://tilings.math.uni-bielefeld.de/substitution/socolar/
It turns out that the ‘penguin’ nonagon can be constructed using any regular polygons, except pentagons and hexagons. I have ignored triangles and squares. You can view the original post here… https://hedraweb.wordpress.com/2021/04/02/penguin-nonagon/
So to recap, overlaps of polygons occur at midpoints between vertices. Only three polygons are needed but the visuals look quite attractive so I left them in. It is also worth noting that all other shapes produced within the regular polygon surrounds (except centres with odd numbers), can also tile but in less interesting ways.
The penguin will always be a nonagon but internal angles will change. The original penguin from overlapping decagons (below right) has identical internal angles for both feet (the only occurrence).
Because the pentadecagon is a factor of five, pentagon and decagon radials can still be achieved if you include central voids.
Here is one example with a pentagonal shaped void inner.
In the the examples below, I have used penguin from heptagons. I believe that concentric circles/loops, spirals and central voids will work in the same way with all regular polygons. However penguins from odd numbered polygons do not allow mirrored patterns.
Here I have produced two and three small spiral designs from the same penguin. Alas, a ‘galaxy’ spiral eluded me.
hedraweb - David Smith 