Tri Again (Again)

I didn’t expect to find the hourglass fractal playing with squares. I even less expected it playing with triangles. Isosceles right triangles, to be precise. Then again, I found it first playing with the L-triomino, which is composed of three squares. And an isosceles triangle is half of a square. So it all fits. This is an isosceles right triangle:

Isosceles right triangle

It’s mirror-symmetrical, so it looks the same in a mirror unless you label one of the acute-angled corners in some way, like this:


Right triangle with labelled corner


Right triangle reflected

Reflection is how you find the hourglass fractal. First, divide a right triangle into four smaller right triangles.


Right triangle rep-tiled

Then discard one of the smaller triangles and repeat. If the acute corners of the smaller triangles have different orientations, one of the permutations creates the hourglass fractal, like this:


Hourglass #1


Hourglass #2


Hourglass #3


Hourglass #4


Hourglass #5


Hourglass #6


Hourglass #7


Hourglass #8


Hourglass #9


Hourglass animated

Another permutation of corners creates what I’ve decided to call the crane fractal, like this:

Crane fractal animated


Crane fractal (static)

The crane fractal is something else that I first found playing with the L-triomino:


Crane fractal from L-triomino

Previously pre-posted:

Square Routes
Tri Again

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.

right triangle rep-tiles




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


triangle mosaic

Previously pre-posted (please peruse):

Rep-Tile Reflections