Fylfy Fractals

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:



















Game of Throwns

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.

Delta Skelta

“When I get to the bottom I go back to the top of the slide,
Where I stop and I turn and I go for a ride
Till I get to the bottom and I see you again.” — The Beatles, “Helter Skelter” (1968)


First stage of fractal #1











Animated fractal #1


First stage of fractal #2













Animated fractal #2

Six Mix Trix

Here’s an equilateral triangle divided into six smaller triangles:

Equilateral triangle divided into six irregular triangles (Stage #1)


Now keep on dividing:

Stage #2


Stage #3


Stage #4


Stage #5


Equilateral triangle dividing into six irregular triangles (animated)


But what happens if you divide the triangle, then discard some of the sub-triangles, then repeat? You get a self-similar shape called a fractal:

Divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Triangle fractal (animated)


Here’s another example:

Divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Triangle fractal (animated)


You can also delay the divide-and-discard to create a more symmetrical fractal, like this:

Delayed divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Triangle fractal (animated)


What next? You can use trigonometry to turn the cramped triangle into a circle:

Triangular fractal

Circular fractal
(Open in new window for full image)


Triangle-to-circle (animated)


Here’s another example:

Triangular fractal

Circular fractal


Triangle-to-circle (animated)


And below are some more circular fractals converted from triangular fractals. Some of them look like distorted skulls or transdimensional Lovecraftian monsters:

(Open in new window for full image)


















Previous Pre-Posted

Circus Trix — an earlier look at sextally-divided-equilateral-triangle fractals

Root Routes

Suppose a point traces all possible routes jumping half-way towards the three vertices of an equilateral triangle. A special kind of shape appears — a fractal called the Sierpiński triangle that contains copies of itself at smaller and smaller scales:

Sierpiński triangle, jump = 1/2


And what if the point jumps 2/3rds of the way towards the vertices as it traces all possible routes? You get this dull fractal:

Triangle, jump = 2/3


But if you add targets midway along each side of the triangle, you get this fractal with the 2/3rds jump:

Triangle, jump = 2/3, side-targets


Now try the 1/2-jump triangle with a target at the center of the triangle:

Triangle, jump = 1/2, central target


And the 2/3-jump triangle with side-targets and a central target:

Triangle, jump = 2/3, side-targets, central target


But why stop at simple jumps like 1/2 and 2/3? Let’s take the distance to the target, td, and use the function 1-(sqrt(td/7r)), where sqrt() is the square-root and 7r is 7 times the radius of the circumscribing circle:

Triangle, jump = 1-(sqrt(td/7r))


Here’s the same jump with a central target:

Triangle, jump = 1-(sqrt(td/7r)), central target


Now let’s try squares with various kinds of jump. A square with a 1/2-jump fills evenly with points:

Square, jump = 1/2 (animated)


The 2/3-jump does better with a central target:

Square, jump = 2/3, central target


Or with side-targets:

Square, jump = 2/3, side-targets


Now try some more complicated jumps:

Square, jump = 1-sqrt(td/7r)


Square, jump = 1-sqrt(td/15r), side-targets


And what if you ban the point from jumping twice or more towards the same target? You get this fractal:

Square, jump = 1-sqrt(td/6r), ban = prev+0


Now try a ban on jumping towards the target two places clockwise of the previous target:

Square, jump = 1-sqrt(td/6r), ban = prev+2


And the two-place ban with a central target:

Square, jump = 1-sqrt(td/6r), ban = prev+2, central target


And so on:

Square, jump = 1-sqrt(td/6.93r), ban = prev+2, central target


Square, jump = 1-sqrt(td/7r), ban = prev+2, central target


These fractals take account of the previous jump and the pre-previous jump:

Square, jump = 1-sqrt(td/4r), ban = prev+2,2, central target


Square, jump = 1-sqrt(td/5r), ban = prev+2,2, central target


Square, jump = 1-sqrt(td/6r), ban = prev+2,2, central target


Elsewhere other-accessible

Boole(b)an #2 — fractals created in similar ways

This Charming Dis-Arming

One of the charms of living in an old town or city is finding new routes to familiar places. It’s also one of the charms of maths. Suppose a three-armed star sprouts three half-sized arms from the end of each of its three arms. And then sprouts three quarter-sized arms from the end of each of its nine new arms. And so on. This is what happens:

Three-armed star


3-Star sprouts more arms


Sprouting 3-Star #3


Sprouting 3-Star #4


Sprouting 3-Star #5


Sprouting 3-Star #6


Sprouting 3-Star #7


Sprouting 3-Star #8


Sprouting 3-Star #9


Sprouting 3-Star #10


Sprouting 3-Star #11 — the Sierpiński triangle


Sprouting 3-star (animated)


The final stage is a famous fractal called the Sierpiński triangle — the sprouting 3-star is a new route to a familiar place. But what happens when you trying sprouting a four-armed star in the same way? This does:

Four-armed star #1


Sprouting 4-Star #2


Sprouting 4-Star #3


Sprouting 4-Star #4


Sprouting 4-Star #5


Sprouting 4-Star #6


Sprouting 4-Star #7


Sprouting 4-Star #8


Sprouting 4-Star #9


Sprouting 4-Star #10


Sprouting 4-star (animated)


There’s no obvious fractal with a sprouting 4-star. Not unless you dis-arm the 4-star in some way. For example, you can ban any new arm sprouting in the same direction as the previous arm:

Dis-armed 4-star (+0) #1


Dis-armed 4-Star (+0) #2


Dis-armed 4-Star (+0) #3


Dis-armed 4-Star (+0) #4


Dis-armed 4-Star (+0) #5


Dis-armed 4-Star (+0) #6


Dis-armed 4-Star (+0) #7


Dis-armed 4-Star (+0) #8


Dis-armed 4-Star (+0) #9


Dis-armed 4-Star (+0) #10


Dis-armed 4-star (+0) (animated)


Once again, it’s a new route to a familiar place (for keyly committed core components of the Overlord-of-the-Über-Feral community, anyway). Now try banning an arm sprouting one place clockwise of the previous arm:

Dis-armed 4-Star (+1) #1


Dis-armed 4-Star (+1) #2


Dis-armed 4-Star (+1) #3


Dis-armed 4-Star (+1) #4


Dis-armed 4-Star (+1) #5


Dis-armed 4-Star (+1) #6


Dis-armed 4-Star (+1) #7


Dis-armed 4-Star (+1) #8


Dis-armed 4-Star (+1) #9


Dis-armed 4-Star (+1) #10


Dis-armed 4-Star (+1) (animated)


Again it’s a new route to a familiar place. Now trying banning an arm sprouting two places clockwise (or anti-clockwise) of the previous arm:

Dis-armed 4-Star (+2) #1


Dis-armed 4-Star (+2) #2


Dis-armed 4-Star (+2) #3


Dis-armed 4-Star (+2) #4


Dis-armed 4-Star (+2) #5


Dis-armed 4-Star (+2) #6


Dis-armed 4-Star (+2) #7


Dis-armed 4-Star (+2) #8


Dis-armed 4-Star (+2) #9


Dis-armed 4-Star (+2) #10


Dis-armed 4-Star (+2) (animated)


Once again it’s a new route to a familiar place. And what happens if you ban an arm sprouting three places clockwise (or one place anti-clockwise) of the previous arm? You get a mirror image of the Dis-armed 4-Star (+1):

Dis-armed 4-Star (+3)


Here’s the Dis-armed 4-Star (+1) for comparison:

Dis-armed 4-Star (+1)


Elsewhere other-accessible

Boole(b)an #2 — other routes to the fractals seen above

Trifylfots

Here’s a simple fractal created by dividing an equilateral triangle into smaller equilateral triangles, then discarding (and rotating) some of those sub-triangles, then doing the same to the sub-triangles:

Fractangle (triangle-fractal) (stage 1)


Fractangle #2


Fractangle #3


Fractangle #4


Fractangle #5


Fractangle #6


Fractangle #7


Fractangle #8


Fractangle #9


Fractangle (animated)


I’ve used the same fractangle to create this shape, which is variously known as a swastika (from Sanskrit svasti, “good luck, well-being”), a gammadion (four Greek Γs arranged in a circle) or a fylfot (from the shape being used to “fill the foot” of a stained glass window in Christian churches):

Trifylfot


Because it’s a fylfot created ultimately from a triangle, I’m calling it a trifylfot (TRIFF-ill-fot). Here’s how you make it:

Trifylfot (stage 1)


Trifylfot #2


Trifylfot #3


Trifylfot #4


Trifylfot #5


Trifylfot #6


Trifylfot #7


Trifylfot #8


Trifylfot #9


Trifylfot (animated)


And here are more trifylfots created from various forms of fractangle:













































Elsewhere other-accessible

Fractangular Frolics — more on fractals from triangles

Dissing the Diamond

In “Fractangular Frolics” I looked at how you could create fractals by choosing lines from a dissected equilateral or isosceles right triangle. Now I want to look at fractals created from the lines of a dissected diamond, as here:

Lines in a dissected diamond


Let’s start by creating one of the most famous fractals of all, the Sierpiński triangle:

Sierpiński triangle stage 1


Sierpiński triangle #2


Sierpiński triangle #3


Sierpiński triangle #4


Sierpiński triangle #5


Sierpiński triangle #6


Sierpiński triangle #7


Sierpiński triangle #8


Sierpiński triangle #9


Sierpiński triangle #10


Sierpiński triangle (animated)


However, you can get an infinite number of Sierpiński triangles with three lines from the diamond:

Sierpińfinity #1


Sierpińfinity #2


Sierpińfinity #3


Sierpińfinity #4


Sierpińfinity #5


Sierpińfinity #6


Sierpińfinity #7


Sierpińfinity #8


Sierpińfinity #9


Sierpińfinity #10


Sierpińfinity (animated)


Here are some more fractals created from three lines of the dissected diamond (sometimes the fractals are rotated to looked better):



















And in these fractals one or more of the lines are flipped to create the next stage of the fractal:




Previously pre-posted:

Fractangular Frolics — fractals created in a similar way

Dissecting the Diamond — fractals from another kind of diamond

Circus Trix

Here’s a trix, or triangle divided into six smaller triangles:

Trix, or triangle divided into six smaller triangles


Now each sub-triangle becomes a trix in its turn:

Trix stage #2


And again:

Trix #3


Trix #4


Trix #5


Trix divisions (animated)


Now try dividing the trix and discarding sub-triangles, then repeating the process. A fractal appears:

Trix fractal #1


Trix fractal #2


Trix fractal #3


Trix fractal #4


Trix fractal #5


Trix fractal #6


Trix fractal #7


Trix fractal (animated)


But what happens if you delay the discarding, first dividing the trix completely into sub-triangles, then dividing completely again? You get a more attractive and symmetrical fractal, like this:

Trix fractal (delayed discard)


And it’s easy to convert the triangle into a circle, creating a fractal like this:

Delayed-discard trix fractal converted into circle


Delayed-discard trix fractal to circular fractal (animated)


Now a trix fractal that looks like a hawk-god:

Trix hawk-god #1


Trix hawk-god #2


Trix hawk-god #3


Trix hawk-god #4


Trix hawk-god #5


Trix hawk-god #6


Trix hawk-god #7


Trix hawk-god (animated)


Trix hawk-god converted to circle


Trix hawk-god to circle (animated)


If you delay the discard, you get this:

Trix hawk-god circle (delayed discard)


And here are more delayed-discard trix fractals:







Various circular trix-fractals (animated)


Post-Performative Post-Scriptum

In Latin, circus means “ring, circle” — the English word “circle” is actually from the Latin diminutive circulus, meaning “little circle”.