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.
I designed a new wavy style of kite and dart, without the need of any colour matching, as long as no tiles are flipped.
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…