Twi-Phi

Here’s a pentagon:

Stage #1

And here’s the pentagon with smaller pentagons on its vertices:

Stage #2

And here’s more of the same:

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Animated fractal

At infinity, the smaller pentagons have reached out like arms to exactly fill the gaps between themselves without overlapping. But how much smaller is each set of smaller pentagons than its mother-pentagon when the gaps are exactly filled? Well, if the radius of the mother-pentagon is r, then the radius of each daughter-pentagon is r * 1/(φ^2) = r * 0·38196601125…

But what happens if the radius relationship of mother to daughter is r * 1/φ = r * 0·61803398874 = r * (φ-1)? Then you get this fractal:

Stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Stage #9

Animated fractal

Middlemath

Suppose you start at the middle of a triangle, then map all possible ways you can jump eight times half-way towards one or another of the vertices of the triangle. At the end of the eight jumps, you mark your final position with a dot. You could jump eight times towards the same vertex, or once towards vertex 1, once towards vertex 2, and once again towards vertex 1. And so on. If you do this, the record of your jumps looks something like this:

The shape is a fractal called the Sierpiński triangle. But if you try the same thing with a square — map all possible jumping-routes you can follow towards one or another of the four vertices — you simply fill the interior of the square. There’s no interesting fractal:

So you need a plan with a ban. Try mapping all possible routes where you can’t jump towards the same vertex twice in a row. And you get this:

Ban on jumping towards same vertex twice in a row, v(t) ≠ v(t-1)

If you call the current vertex v(t) and the previous vertex v(t-1), the ban says that v(t) ≠ v(t-1). Now suppose you can’t jump towards the vertex one place clockwise of the previous vertex. Now the ban is v(t)-1 ≠ v(t-1) or v(t) ≠ v(t-1)+1 and this fractal appears:

v(t) ≠ v(t-1)+1

And here’s a ban on jumping towards the vertex two places clockwise (or counterclockwise) of the vertex you’ve just jumped towards:

v(t) ≠ v(t-1)+2

And finally the ban on jumping towards the vertex three places clockwise (or one place counterclockwise) of the vertex you’ve just jumped towards:

v(t) ≠ v(t-1)+3 (a mirror-image of v(t) ≠ v(t-1)+1, as above)

Now suppose you introduce a new point to jump towards at the middle of the square. You can create more fractals, but you have to adjust the kind of ban you use. The central point can’t be included in the ban or the fractal will be asymmetrical. So you continue taking account of the vertices, but if the previous jump was towards the middle, you ignore that jump. At least, that’s what I intended, but I wonder whether my program works right. Anyway, here are some of the fractals that it produces:

v(t) ≠ v(t-1) with central point (wcp)

v(t) ≠ v(t-1)+1, wcp

v(t) ≠ v(t-1)+2, wcp

And here are some bans taking account of both the previous vertex and the pre-previous vertex:

v(t) ≠ v(t-1) & v(t) ≠ v(t-2), wcp

v(t) ≠ v(t-1) & v(t-2)+1, wcp

v(t) ≠ v(t-1)+2 & v(t-2), wcp

v(t) ≠ v(t-1) & v(t-2)+1, wcp

v(t) ≠ v(t-1)+1 & v(t-2)+1, wcp

v(t) ≠ v(t-1)+2 & v(t-2)+1, wcp

v(t) ≠ v(t-1)+3 & v(t-2)+1, wcp

v(t) ≠ v(t-1) & v(t-2)+2, wcp

v(t) ≠ v(t-1)+1 & v(t-2)+2, wcp

v(t) ≠ v(t-1)+2 & v(t-2)+2, wcp

Now look at pentagons. They behave more like triangles than squares when you map all possible jumping-routes towards one or another of the five vertices. That is, a fractal appears:

All possible jumping-routes towards the vertices of a pentagon

But the pentagonal-jump fractals get more interesting when you introduce jump-bans:

v(t) ≠ v(t-1)

v(t) ≠ v(t-1)+1

v(t) ≠ v(t-1)+2

v(t) ≠ v(t-1) & v(t-2)

v(t) ≠ v(t-1)+2 & v(t-2)

v(t) ≠ v(t-1)+1 & v(t-2)+1

v(t) ≠ v(t-1)+3 & v(t-2)+1

v(t) ≠ v(t-1)+1 & v(t-2)+2

v(t) ≠ v(t-1)+2 & v(t-2)+2

v(t) ≠ v(t-1)+3 & v(t-2)+2

Finally, here are some pentagonal-jump fractals using a central point:

Post-Performative Post-Scriptum

I’m not sure if I’ve got the order of some bans right above. For example, should v(t) ≠ v(t-1)+1 & v(t-2)+2 really be v(t) ≠ v(t-1)+2 & v(t-2)+1? I don’t know and I’m not going to check. But the idea of jumping-point bans is there and that’s all you need if you want to experiment with these fractal methods for yourself.

Fractal + Star = Fractar

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)

Create a triangle. Find a point somewhere inside it. Choose a corner at random and move halfway towards it. Mark the new point. Repeat the procedure: choose, move, mark. Repeat again and again. In time, a fractal will appear:

However, if you try the same thing with a square – choose a corner at random, move halfway towards it, mark the new point, repeat – no fractal appears. Instead, the points fill the interior of the square:

But what happens if you impose restrictions on the randomly chosen corner (or chorner)? Suppose you can’t choose the same corner twice in a row. If this rule is applied to the square, this fractal appears:

Now apply the no-corner-twice-in-a-row rule to a square that contains a central chorner. This fractal appears:

And if the rule is that you can choose a corner twice in a row but not thrice? This fractal appears:

Here is the rule is that a corner can’t be chosen if it was chosen two moves ago:

But what if the restriction is based not on how often or when a corner is chosen, but on its proximity, i.e. how near it is to the marked point? If the nearest corner can’t be chosen, the result is the same as the no-corner-twice-in-a-row rule:

But if the second-nearest corner can’t be chosen, this fractal appears:

This is the fractal when the third-nearest corner can’t be chosen:

And this is the fractal when the fourth-nearest, or most distant, corner can’t be chosen:

Here are the same restrictions applied to a pentagon:

Nearest corner forbidden

Second-nearest corner forbidden

Third corner forbidden

Fourth corner forbidden

Fifth corner forbidden

Fifth corner forbidden (animated)

And a pentagon with a central chorner:

Now try excluding more than one corner. Here are pentagons excluding the n-nearest and n+1-nearest corners (for example, the nearest and second-nearest corners; the second-nearest and third-nearest; and so on):

But what if the moving point is set equal to the n-nearest corner before it moves again? If the corner is the second-nearest and the shape is a triangle with a central chorner, this is the fractal that appears:

Animated version

And here is the same rule applied to various n-nearest corners in a pentagon: