“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
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
This is a beautiful and interesting shape, reminiscent of a piece of jewellery:
Pentagons in a ring
I came across it in this tricky little word-puzzle:
Word puzzle using pentagon-ring
Here’s a printable version of the puzzle:
Printable puzzle
Let’s try placing some other regular polygons with s sides around regular polygons with s*2 sides:
Hexagonal ring of triangles
Octagonal ring of squares
Decagonal ring of pentagons
Dodecagonal ring of hexagons
Only regular pentagons fit perfectly, edge-to-edge, around a regular decagon. But all these polygonal-rings can be used to create interesting and beautiful fractals, as I hope to show in a future post.
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
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
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”.
Here’s an interesting shape that looks like a distorted and dissected capital S:
A distorted and dissected capital S
If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below):
The shape contains three smaller versions of itself
The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself:
Five more sub-fractals
But how do you create the shape? You start by selecting three lines from this divided equilateral triangle:
A divided equilateral triangle
These are the three lines you need to create the shape:
Fractangle seed (the three lines correspond to the three sub-fractals seen above)
Now replace each line with a half-sized set of the same three lines:
Fractangle stage #2
And do that again:
Fractangle stage #3
And again:
Fractangle stage #4
And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle:
Fractangle stage #5
Fractangle stage #6
Fractangle stage #7
Fractangle stage #8
Fractangle stage #9
Fractangle stage #10
Fractangle stage #11
Here’s an animation of the process:
Creating the fractangle (animated)
And here are more fractangles created in a similar way from three lines of the divided equilateral triangle:
Fractangle #2
Fractangle #2 (anim)
(open in new window if distorted)
Fractangle #2 (seed)
Fractangle #3
Fractangle #3 (anim)
Fractangle #3 (seed)
Fractangle #4
Fractangle #4 (anim)
Fractangle #4 (seed)
You can also use a right triangle to create fractangles:
Divided right triangle for fractangles
Here are some fractangles created from three lines chosen of the divided right triangle:
Fractangle #5
Fractangle #5 (anim)
Fractangle #5 (seed)
Fractangle #6
Fractangle #6 (anim)
Fractangle #6 (seed)
Fractangle #7
Fractangle #7 (anim)
Fractangle #7 (seed)
Fractangle #8
Fractangle #8 (anim)
Fractangle #8 (seed)
Pre-previously on Overlord in terms of the Über-Feral, I looked at my favorite member of the fractal community, the Hourglass Fractal:
The hourglass fractal
A real hourglass for comparison
As I described how I discovered the hourglass fractal indirectly and by accident, then showed how to create it directly, 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 #10
Triangles to hourglass #11
Triangles to hourglass #12
Triangles to hourglass (animated)
Now, here’s an even simpler way to create the hourglass fractal, starting with a single vertical line:
Line to hourglass #1
Line to hourglass #2
Line to hourglass #3
Line to hourglass #4
Line to hourglass #5
Line to hourglass #6
Line to hourglass #7
Line to hourglass #8
Line to hourglass #9
Line to hourglass #10
Line to hourglass #11
Line to hourglass (animated)
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)
Here’s a three-armed star made with three lines radiating at intervals of 120°:
Triangular fractal stage #1
At the end of each of the three lines, add three more lines at half the length:
Triangular fractal #2
And continue like this:
Triangular fractal #3
Triangular fractal #4
Triangular fractal #5
Triangular fractal #6
Triangular fractal #7
Triangular fractal #8
Triangular fractal #9
Triangular fractal #10
Triangular fractal (animated)
Because this fractal is created from a series of stars, you could call it a fractar. Here’s a black-and-white version:
Triangular fractar (black-and-white)
Triangular fractar (black-and-white) (animated)
(Open in a new window for larger version if the image seems distorted)
A four-armed star doesn’t yield an easily recognizable fractal in a similar way, so let’s try a five-armed star:
Pentagonal fractar stage #1
Pentagonal fractar #2
Pentagonal fractar #3
Pentagonal fractar #4
Pentagonal fractar #5
Pentagonal fractar #6
Pentagonal fractar #7
Pentagonal fractar (animated)
Pentagonal fractar (black-and-white)
Pentagonal fractar (bw) (animated)
And here’s a six-armed star:
Hexagonal fractar stage #1
Hexagonal fractar #2
Hexagonal fractar #3
Hexagonal fractar #4
Hexagonal fractar #5
Hexagonal fractar #6
Hexagonal fractar (animated)
Hexagonal fractar (black-and-white)
Hexagonal fractar (bw) (animated)
And here’s what happens to the triangular fractar when the new lines are rotated by 60°:
Triangular fractar (60° rotation) #1
Triangular fractar (60°) #2
Triangular fractar (60°) #3
Triangular fractar (60°) #4
Triangular fractar (60°) #5
Triangular fractar (60°) #6
Triangular fractar (60°) #7
Triangular fractar (60°) #8
Triangular fractar (60°) #9
Triangular fractar (60°) (animated)
Triangular fractar (60°) (black-and-white)
Triangular fractar (60°) (bw) (animated)
Triangular fractar (60°) (no lines) (black-and-white)
A four-armed star yields a recognizable fractal when the rotation is 45°:
Square fractar (45°) #1
Square fractar (45°) #2
Square fractar (45°) #3
Square fractar (45°) #4
Square fractar (45°) #5
Square fractar (45°) #6
Square fractar (45°) #7
Square fractar (45°) #8
Square fractar (45°) (animated)
Square fractar (45°) (black-and-white)
Square fractar (45°) (bw) (animated)
Without the lines, the final fractar looks like the plan of a castle:
Square fractar (45°) (bw) (no lines)
And here’s a five-armed star with new lines rotated at 36°:
Pentagonal fractar (36°) #1
Pentagonal fractar (36°) #2
Pentagonal fractar (36°) #3
Pentagonal fractar (36°) #4
Pentagonal fractar (36°) #5
Pentagonal fractar (36°) #6
Pentagonal fractar (36°) #7
Pentagonal fractar (36°) (animated)
Again, the final fractar without lines looks like the plan of a castle:
Pentagonal fractar (36°) (no lines) (black-and-white)
Finally, here’s a six-armed star with new lines rotated at 30°:
Hexagonal fractar (30°) #1
Hexagonal fractar (30°) #2
Hexagonal fractar (30°) #3
Hexagonal fractar (30°) #4
Hexagonal fractar (30°) #5
Hexagonal fractar (30°) #6
Hexagonal fractar (30°) (animated)
And the hexagonal castle plan:
Hexagonal fractar (30°) (black-and-white) (no lines)
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 