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:

Square Routes Re-Re-Visited

Rep-tilian Rites

A pentomino is one of the shapes created by laying five squares edge-to-edge. There are twelve of them (not counting reflections) and this is the P-pentomino:

p_pentomino

But it’s not just a pentomino, it’s also a rep-tile, or a shape that can divided into smaller copies of itself. There are two ways of doing this (I’ve rotated the pentomino 90° to make the images look better):

p_pentomino_a


p_pentomino_b


Once you’ve divided the shape into four copies, you can divide the copies, then the copies of the copies, and the copies of the copies of the copies, and so on for ever:

p_pentomino_a_anim


p_pentomino_a_anim


And if you’ve got a reptile, you can turn it into a fractal. Simply divide the shape, discard one or more copies, and continue:

p_pentomino_a_124_1

Pentomino-based fractal stage 1


p_pentomino_a_124_2

Pentomino-based fractal stage 2


p_pentomino_a_124_3

Pentomino-based fractal stage 3


p_pentomino_a_124_4

Stage 4


p_pentomino_a_124_5

Stage 5


p_pentomino_a_124_6

Stage 6


p_pentomino_a_124_7

Stage 7


p_pentomino_a_124_8

Stage 8


p_pentomino_a_124_9

Stage 9


p_pentomino_a_124_10

Stage 10


Here are some more fractals created using the same divide-and-discard process:

p_pentomino_b_234

p_pentomino_b_234anim

Animated version


p_pentomino_b_134

p_pentomino_b_134anim

Animated version


p_pentomino_b_124

p_pentomino_b_124anim


p_pentomino_b_123

p_pentomino_b_123anim


p_pentomino_a_134anim

p_pentomino_a_134


p_pentomino_a_234anim

p_pentomino_a_234


p_pentomino_a_124

p_pentomino_a_124anim


p_pentomino_a_123

p_pentomino_a_123anim


You can also use variants on a standard rep-tile dissection, like rotating the copies or trying different patterns of dissection at different levels to see what new shapes appear:

p_pentomino_adj_13

p_pentomino_adj_anim13


p_pentomino_adj_6

p_pentomino_adj_anim6


p_pentomino_adj_anim5

p_pentomino_adj_5


p_pentomino_adj_3

p_pentomino_adj_anim3


p_pentomino_adj_2

p_pentomino_adj_anim2


p_pentomino_adj_1

p_pentomino_adj_anim1


p_pentomino_adj_17

p_pentomino_adj_anim17


p_pentomino_adj_15

p_pentomino_adj_anim15


p_pentomino_adj_16

p_pentomino_adj_anim16


p_pentomino_adj_8

p_pentomino_adj_anim8


p_pentomino_adj_10

p_pentomino_adj_anim10


p_pentomino_adj_11

p_pentomino_adj_anim11


p_pentomino_adj_14

p_pentomino_adj_anim14


p_pentomino_adj_anim4


p_pentomino_adj_anim12


p_pentomino_adj_anim9


p_pentomino_adj_anim7

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:

sphinx_hexiamond

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):

sphinx4

sphinx9

So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:

double_triangle_rep-tile

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:

equilateral_triangle_fish_rep-tile

right_triangle_fish_rep-tile

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.

right triangle rep-tiles

right_triangle_fish

equilateral_triangle_reptiles

equilateral_triangle_rocket

(Please open the following images in a new window if they fail to animate)

duodeciamond

triangle mosaic


Previously pre-posted (please peruse):

Rep-Tile Reflections