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)