Would it be my favorite fractal if I hadn’t discovered it for myself? It might be, because I think it combines great simplicity with great beauty. I first came across it when I was looking at this rep-tile, that is, a shape that can be divided into smaller copies of itself:
Rep-4 L-Tromino
It’s called a L-tromino and is a rep-4 rep-tile, because it can be divided into four copies of itself. If you divide the L-tromino into four sub-copies and discard one particular sub-copy, then repeat again and again, you’ll get this fractal:
Tromino fractal #1
Tromino fractal #2
Tromino fractal #3
Tromino fractal #4
Tromino fractal #5
Tromino fractal #6
Tromino fractal #7
Tromino fractal #8
Tromino fractal #9
Tromino fractal #10
Tromino fractal #11
Hourglass fractal (animated)
I call it an hourglass fractal, because it reminds me of an hourglass:
A real hourglass
The hourglass fractal for comparison
I next came across the hourglass fractal when applying the same divide-and-discard process to a rep-4 square. The first fractal that appears is the Sierpiński triangle:
Square to Sierpiński triangle #1
Square to Sierpiński triangle #2
Square to Sierpiński triangle #3
[…]
Square to Sierpiński triangle #10
Square to Sierpiński triangle (animated)
However, you can rotate the sub-squares in various ways to create new fractals. Et voilà, the hourglass fractal appears again:
Square to hourglass #1
Square to hourglass #2
Square to hourglass #3
Square to hourglass #4
Square to hourglass #5
Square to hourglass #6
Square to hourglass #7
Square to hourglass #8
Square to hourglass #9
Square to hourglass #10
Square to hourglass #11
Square to hourglass (animated)
Finally, I was looking at variants of the so-called chaos game. In the standard chaos game, a point jumps half-way towards the randomly chosen vertices of a square or other polygon. In this variant of the game, I’ve added jump-towards-able mid-points to the sides of the square and restricted the point’s jumps: it can only jump towards the points that are first-nearest, seventh-nearest and eighth-nearest. And again the hourglass fractal appears:
Chaos game to hourglass #1
Chaos game to hourglass #2
Chaos game to hourglass #3
Chaos game to hourglass #4
Chaos game to hourglass #5
Chaos game to hourglass #6
Chaos game to hourglass (animated)
But what if you want to create the hourglass fractal directly? You can do it like this, using two isosceles triangles set apex-to-apex in the form of an hourglass:
Triangles to hourglass #1
Triangles to hourglass #2
Triangles to hourglass #3
Triangles to hourglass #4
Triangles to hourglass #5
Triangles to hourglass #6
Triangles to hourglass #7
Triangles to hourglass #8
Triangles to hourglass #9
Triangles to hourglass #10
Triangles to hourglass #11
Triangles to hourglass #12
Triangles to hourglass (animated)
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