# Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself: Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal: Fractal cross stage #1 Fractal cross #2 Fractal cross #3 Fractal cross #4 Fractal cross #5 Fractal cross #6 Fractal cross (animated) Fractal cross (static)

If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal: Fractal Hex-Ring #1 Fractal Hex-Ring #2 Fractal Hex-Ring #3 Fractal Hex-Ring #4 Fractal Hex-Ring #5 Fractal Hex-Ring #6 Fractal Hex-Ring (animated) Fractal Hex-Ring (static)

Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge: Nonomino

It can be divided into twelve smaller copies of itself, like this: Rep-12 Nonomino (discovered by Erich Friedman)

If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal: Fractal cross stage #1 Fractal cross #2 Fractal cross #3 Fractal cross #4 Fractal cross #5 Fractal cross #6 Fractal cross (animated) Fractal cross (static)

If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal: Fractal Hex-Ring #1 Fractal Hex-Ring #2 Fractal Hex-Ring #3 Fractal Hex-Ring #4 Fractal Hex-Ring #5 Fractal Hex-Ring (animated) Fractal Hex-Ring (static)

Elsewhere other-available:

# Hex Appeal

A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza: It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):  So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon: It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.

But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:  # Rep It Up

When I started to look at rep-tiles, or shapes that can be divided completely into smaller copies of themselves, I wanted to find some of my own. It turns out that it’s easy to automate a search for the simpler kinds, like those based on equilateral triangles and right triangles.    (Please open the following images in a new window if they fail to animate)  Previously pre-posted (please peruse):