Hour Power

Wednesday, 13th Nov, 2019 by Krilling for Company in Fractals, Geometry, Mathematics, Rep-Tiles, Squares, Triangles and tagged animated gifs, √5, fractal geometry, fractals, geometric dissection, geometry, hourglass fractal, L-tromino, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, rep-tile, Rep-Tiles, Sierpiński triangle, square, square root of 5, Triangles, triomino | Leave a comment

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)

Tri Again (Again)

Friday, 3rd Feb, 2017 by Krilling for Company in Fractals, Geometry, Mathematics, Rep-Tiles and tagged animated gif, animated gifs, crane fractal, fractal, fractal geometry, fractals, fractals from triangles, geometry, hourglass fractal, isosceles right triangles, isosceles triangles, math, mathematics, maths, mirror symmetry, mirror-symmetrical, recreational math, recreational mathematics, recreational maths, right triangles, simple fractals, Triangles | Leave a comment

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_chiral_1

Right triangle with labelled corner

right_triangle_chiral_2

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:

l-triomino_234

Crane fractal from L-triomino

Previously pre-posted:

• Square Routes
• Tri Again

Square Routes

Monday, 23rd Jan, 2017 by Krilling for Company in Fractals, Geometry, Mathematics, Rep-Tiles and tagged animated gif, animated gifs, fractal, fractal geometry, fractals, fractals from squares, geometry, hourglass fractal, L-tromino, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, Sierpiński triangle, simple fractals, Squares, tromino | Leave a comment

One of the pleasures of exploring an ancient city like York or Chester is that of learning new routes to the same destination. There are byways and alleys, short-cuts and diversions. You set off intending to go to one place and end up in another.

Maths is like that, even at its simplest. There are many routes to the same destination. I first found the fractal below by playing with the L-triomino, or the shape created by putting three squares in the shape of an L. You can divide it into four copies of the same shape and discard one copy, then do the same to each of the sub-copies, then repeat. I’ve decided to call it the hourglass fractal:

l-triomino_124

Hourglass fractal (animated)

l-triomino_124_upright_static1

Hourglass fractal (static)

Then I unexpectedly came across the fractal again when playing with what I call a proximity fractal:
v4_ban15_sw3_anim

Hourglass animated (proximity fractal)

v4_ban15_sw3_col

(Static image)

Now I’ve unexpectedly come across it for a third time, playing with a very simple fractal based on a 2×2 square. At first glance, the 2×2 square yields only one interesting fractal. If you divide the square into four smaller squares and discard one square, then do the same to each of the three sub-copies, then repeat, you get a form of the Sierpiński triangle, like this:

Sierpiński triangle stage 1

Sierpiński triangle #2

Sierpiński triangle #3

Sierpiński triangle #4

Sierpiński triangle animated

(Static image)

The 2×2 square seems too simple for anything more, but there’s a simple way to enrich it: label the corners of the sub-squares so that you can, as it were, individually rotate them 0°, 90°, 180°, or 270°. One set of rotations produces the hourglass fractal, like this:

Hourglass stage 1

Hourglass #2

Fractal #3

Hourglass #4

Hourglass #5

Hourglass #6

Hourglass animated

(Static image)

Here are some more fractals from the 2×2 square created using this technique (I’ve found some of them previously by other routes):

(Static image)

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Tag Archives: hourglass fractal

Hour Power

Tri Again (Again)

Square Routes