Category Archives: Fractals

Tright Sights

by in Circles, Fractals, Geometry, Mathematics, Square Roots, Triangles and tagged , , , , , , | Leave a comment

Here’s a right triangle, where a^2 + b^2 = c^2. But what are the exact values of a, b, and c?

You might be able to guess by eye, but could you prove your guess? Now try the same right triangle tiled with three identical copies of itself:

1-√3-2 triangle as rep3 rep-tile

Now you can prove the exact values of a, b, and c. If the vertical side, a, is 1, then the hypotenuse, c, is 2, because the length that fits once into a fits twice into c. Therefore 2^2 = 1^2 + b^2 → 4 = 1 + b^2 → 4-1 = b^2 → 3 = b^2 → √3 = b. The horizontal side, b, has a length of √3 = 1.73205080757… So the right triangle is 1-√3-2. And if it’s rep3, that is, can be divided into three identical copies of itself, then it’s also rep9, rep27, and so on:

1-√3-2 triangle as rep9 rep-tile

1-√3-2 triangle as rep27 rep-tile

1-√3-2 triangle as rep81 rep-tile

1-√3-2 triangle as rep243 rep-tile

1-√3-2 triangle as rep729 rep-tile

Once you’ve got a rep-tile, you can create fractals. But the 1-√3-2 triangle is cramped. You need more space to work with. And it’s easy to find that space when you realize that a standard equilateral triangle can be divided into six 1-√3-2 triangles:

Equilateral triangle divided into six 1-√3-2 triangles

Equilateral triangle tiled with 1-√3-2 triangles (stage 1)

(please open in new window if image is distorted)

Equilateral triangle tiled with 1-√3-2 triangles (stage 2)

Equilateral triangle tiled with 1-√3-2 triangles (stage 3)

Here are variant colorings of the stage-3 tiled triangle:

But where are the fractals? In one way, you’ve already seen them. But they get more obvious like this:

Fractal stage 1

Fractal #2

Fractal #3

Fractal #4

Fractal #5

Fractal #6

Fractal #7

Fractal #8

Fractal (animated)

Another fractal stage 1

[…]

↓

[…]

Another fractal #8

Another fractal (animated)

And when you have a fractal created using an equilateral triangle, it’s easy to expand the fractal into a circle, like this:

Original fractal

↓

Fractal expanded into circle

Triangular fractal to circular fractal (animated)

Extra Tetra

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Construction of a Sierpiński tetrahedron (from WikiMedia)

Post-Performative Post-Scriptum

The toxic title of this incendiary intervention radically references George Harrison’s album Extra Texture (1975).

Strange “S” in the Light

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Unexpected discoveries are one of the joys of mathematics, even for amateurs. And computers help you make more of them, because computers make it easy to adjust variables or search faster and further through math-space than any unaided human ever could (on the downside, computers can make you lazy and blunt your intuition). Here’s an unexpected discovery I made using a computer in November 2020:

A distorted and dissected capital “S”

It’s a strange “S” that looks complex but is constructed very easily from three simple lines. And it’s a fractal, a shape that contains copies of itself at smaller and smaller scales:

Five sub-fractals of the Strange “S”

Elsewhere Other-Accessible…

• Fractangular Frolics — the Strange “S” in more light

We Can Circ It Out

by in Circles, Fractals, Geometry, Mathematics, Polygons, Squares, Triangles and tagged , , , , , , , , , , , , , , | Leave a comment

It’s a pretty little problem to convert this triangular fractal…

Sierpiński triangle (Wikipedia)

…into its circular equivalent:

Sierpiński triangle as circle

Sierpiński triangle to circle (animated)

But once you’ve circ’d it out, as it were, you can easily adapt the technique to fractals based on other polygons:

T-square fractal (Wikipedia)
⇓

T-square fractal as circle

T-square fractal to circle (animated)

Elsewhere other-accessible…

• Dilating the Delta — more on converting polygonic fractals to circles…

Fylfy Fractals

by in Fractals, Geometry, Mathematics, Rep-Tiles, Squares, Triangles and tagged , , , , , , , , , , | Leave a comment

An equilateral triangle is a rep-tile, because it can be tiled completely with smaller copies of itself. Here it is as a rep-4 rep-tile, tiled with four smaller copies of itself:

Equilateral triangle as rep-4 rep-tile

If you divide and discard one of the sub-copies, then carry on dividing-and-discarding with the sub-copies and sub-sub-copies and sub-sub-sub-copies, you get the fractal seen below. Alas, it’s not a very attractive or interesting fractal:

Divide-and-discard fractal stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Stage #9

Divide-and-discard fractal (animated)

You can create more attractive and interesting fractals by rotating the sub-triangles clockwise or anticlockwise. Here are some examples:

Now try dividing a square into four right triangles, then turning each of the four triangles into a divide-and-discard fractal. The resulting four-fractal shape is variously called a swastika, a gammadion, a cross cramponnée, a Hakenkreuz and a fylfot. I’m calling it a fylfy fractal:

Divide-and-discard fractals in the four triangles of a divided square stage #1

Fylfy fractal #2

Fylfy fractal #3

Fylfy fractal #4

Fylfy fractal #5

Fylfy fractal #6

Fylfy fractal #7

Fylfy fractal #8

Fylfy fractal (animated)

Finally, you can adjust the fylfy fractals so that each point in the square becomes the equivalent point in a circle:

Absolutely Sabulous

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The Hourglass Fractal (animated gif optimized at ezGIF)

Performativizing Paronomasticity

The title of this incendiary intervention is a paronomasia on the title of the dire Absolutely Fabulous. The adjective sabulous means “sandy; consisting of or abounding in sand; arenaceous” (OED).

Elsewhere Other-Accessible

• Hour Re-Re-Re-Re-Powered — more on the hourglass fractal
• Allus Pour, Horic — an earlier paronomasia for the fractal

Game of Throwns

by in Chaos Game, Fractals, Geometry, Mathematics, Squares, Triangles and tagged , , , , , , , , , , , , | Leave a comment

In “Scaffscapes”, I looked at these three fractals and described how they were in a sense the same fractal, even though they looked very different:

Fractal #1

Fractal #2

Fractal #3

But even if they are all the same in some mathematical sense, their different appearances matter in an aesthetic sense. Fractal #1 is unattractive and seems uninteresting:

Fractal #1, unattractive, uninteresting and unnamed

Fractal #3 is attractive and interesting. That’s part of why mathematicians have given it a name, the T-square fractal:

Fractal #3 — the T-square fractal

But fractal #2, although it’s attractive and interesting, doesn’t have a name. It reminds me of a ninja throwing-star or shuriken, so I’ve decided to call it the throwing-star fractal or ninja-star fractal:

Fractal #2, the throwing-star fractal

A ninja throwing-star or shuriken

This is one way to construct a throwing-star fractal:

Throwing-star fractal, stage 1

Throwing-star fractal, #2

Throwing-star fractal, #3

Throwing-star fractal, #4

Throwing-star fractal, #5

Throwing-star fractal, #6

Throwing-star fractal, #7

Throwing-star fractal, #8

Throwing-star fractal, #9

Throwing-star fractal, #10

Throwing-star fractal, #11

Throwing-star fractal (animated)

But there’s another way to construct a throwing-star fractal. You use what’s called the chaos game. To understand the commonest form of the chaos game, imagine a ninja inside an equilateral triangle throwing a shuriken again and again halfway towards a randomly chosen vertex of the triangle. If you mark each point where the shuriken lands, you eventually get a fractal called the Sierpiński triangle:

Chaos game with triangle stage 1

Chaos triangle #2

Chaos triangle #3

Chaos triangle #4

Chaos triangle #5

Chaos triangle #6

Chaos triangle #7

Chaos triangle (animated)

When you try the chaos game with a square, with the ninja throwing the shuriken again and again halfway towards a randomly chosen vertex, you don’t get a fractal. The interior of the square just fills more or less evenly with points:

Chaos game with square, stage 1

Chaos square #2

Chaos square #3

Chaos square #4

Chaos square #5

Chaos square #6

Chaos square (anim)

But suppose you restrict the ninja’s throws in some way. If he can’t throw twice or more in a row towards the same vertex, you get a familiar fractal:

Chaos game with square, ban on throwing towards same vertex, stage 1

Chaos square, ban = v+0, #2

Chaos square, ban = v+0, #3

Chaos square, ban = v+0, #4

Chaos square, ban = v+0, #5

Chaos square, ban = v+0, #6

Chaos square, ban = v+0 (anim)

But what if the ninja can’t throw the shuriken towards the vertex one place anti-clockwise of the vertex he’s just thrown it towards? Then you get another familiar fractal — the throwing-star fractal:

Chaos square, ban = v+1, stage 1

Chaos square, ban = v+1, #2

Chaos square, ban = v+1, #3

Chaos square, ban = v+1, #4

Chaos square, ban = v+1, #5

Game of Throwns — throwing-star fractal from chaos game (static)

Game of Throwns — throwing-star fractal from chaos game (anim)

And what if the ninja can’t throw towards the vertex two places anti-clockwise (or two places clockwise) of the vertex he’s just thrown the shuriken towards? Then you get a third familiar fractal — the T-square fractal:

Chaos square, ban = v+2, stage 1

Chaos square, ban = v+2, #2

Chaos square, ban = v+2, #3

Chaos square, ban = v+2, #4

Chaos square, ban = v+2, #5

T-square fractal from chaos game (static)

T-square fractal from chaos game (anim)

Finally, what if the ninja can’t throw towards the vertex three places anti-clockwise, or one place clockwise, of the vertex he’s just thrown the shuriken towards? If you can guess what happens, your mathematical intuition is much better than mine.

Post-Performative Post-Scriptum

I am not now and never have been a fan of George R.R. Martin. He may be a good author but I’ve always suspected otherwise, so I’ve never read any of his books or seen any of the TV adaptations.

Scaffscapes

by in Fractals, Geometry, Mathematics, Squares and tagged , , , , , , , , , | Leave a comment

A fractal is a shape that contains copies of itself on smaller and smaller scales. You can find fractals everywhere in nature. Part of a fern looks like the fern as a whole:

Fern as fractal (source)

Part of a tree looks like the tree as a whole:

Tree as fractal (source)

Part of a landscape looks like the landscape as a whole:

Landscape as fractal (source)

You can also create fractals for yourself. Here are three that I’ve constructed:

Fractal #1

Fractal #2

Fractal #3 — the T-square fractal

The three fractals look very different and, in one sense, that’s exactly what they are. But in another sense, they’re the same fractal. Each can morph into the other two:

Fractal #1 → fractal #2 → fractal #3 (animated)

Here are two more fractals taken en route from fractal #2 to fractal #3, as it were:

Fractal #4

Fractal #5

To understand how the fractals belong together, you have to see what might be called the scaffolding. The construction of fractal #3 is the easiest to understand. First you put up the scaffolding, then you take it away and leave the final fractal:

Fractal #3, scaffolding stage 1

Fractal #3, stage 2

Fractal #3, stage 3

Fractal #3, stage 4

Fractal #3, stage 5

Fractal #3, stage 6

Fractal #3, stage 7

Fractal #3, stage 8

Fractal #3, stage 9

Fractal #3, stage 10

Fractal #3 (scaffolding removed)

Construction of fractal #3 (animated)

Now here’s the construction of fractal #1:

Fractal #1, stage 1

Fractal #1, stage 2

Fractal #1, stage 3

↓

Construction of fractal #1 (animated)

Fractal #1 (static)

And the constructions of fractals #2, #4 and #5:

Fractal #2, stage 1

Fractal #2, stage 2

Fractal #2, stage 3

↓

Fractal #2 (animated)

Fractal #2 (static)

Fractal #4, stage 1

Fractal #4, stage 2

Fractal #4, stage 3

↓

Fractal #4 (animated)

Fractal #4 (static)

Fractal #5, stage 1

Fractal #5, stage 2

Fractal #5, stage 3

↓

Fractal #5 (animated)

Fractal #5