Endless Tiles of the Equilateral Triangle and Square
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    Infinite fractal tiles of the regular reptiles in all dimensions above 1. in 2d, tiles of the Equilateral Triangle (3 colours) and Square (2 and … Read More
    Infinite fractal tiles of the regular reptiles in all dimensions above 1. in 2d, tiles of the Equilateral Triangle (3 colours) and Square (2 and 4 colours). Read Less
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Infinite Tiles...
... of the regular reptiles in all dimensions above 1
(The regular reptiles in 2d are the Equliateral Triangle and Square:  Shapes with identical side lengths and angles, which can replicate themselves as smaller copies or tiles, hence the name reptiles.)
These are a type of fractal called L-systems, after Aristid Lindenmayer, also able to be rendered as Iterated Function Systems (IFS).  They tile the Triangle in 3 identical pieces, each with its own colour, and the Square in 2 or 4.  There are literally an infinite number of them... including an infinite number of animations from one tile to another, with every frame also a tile...
This tile of the equilateral triangle features, at the limit, an infinite number of pieces connected by points.
This tile of the square is "simply connected" - all in one obvious piece, albeit with fractal edges.
This tile of the equilateral triangle features, at the limit, an infinite number of pieces connected by points.  It's also a variation on the famous von Koch Snowflake curve, an early fractal.
This tile of the square features, at the limit, an infinite number of pieces connected by points.  Also, it tiles in 2 pieces rather than 4.
This tile of the equilateral triangle features, at the limit, an infinite number of unconnected pieces.  This is called a "dust", though not a "pure dust", in which each piece is exactly the same size - 1 holon.
This tile of the square is "simply connected" - all in one obvious piece, albeit with fractal edges.  It's also one of my personal favourites.
This tile of the equilateral triangle features, at the limit, an infinite number of pieces connected by points.
This tile of the square features, at the limit, an infinite number of pieces connected by points.  Also, it tiles in 2 pieces rather than 4.
This tile of the square features, at the limit, an infinite number of pieces connected by points.
This tile of the square is "simply connected" - all one obvious piece, albeit with fractal edges.
This tile of the square features, at the limit, an infinite number of pieces connected by points.