Category Archives: Fractals

Get Your Prox Off #3

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I’ve looked at lot at the fractals created when you randomly (or quasi-randomly) choose a vertex of a square, then jump half of the distance towards it. You can ban jumps towards the same vertex twice in a row, or jumps towards the vertex clockwise or anticlockwise from the vertex you’ve just chosen, and so on.

But you don’t have to choose vertices directly: you can also choose them by distance or proximity (see “Get Your Prox Off” for an earlier look at fractals-by-distance). For example, this fractal appears when you can jump half-way towards the nearest vertex, the second-nearest vertex, and the third-nearest vertex (i.e., you can’t jump towards the four-nearest or most distant vertex):

vertices = 4, distance = (1,2,3), jump = 1/2

It’s actually the same fractal as you get when you choose vertices directly and ban jumps towards the vertex diagonally opposite from the one you’ve just chosen. But this fractal-by-distance isn’t easy to match with a fractal-by-vertex:

v = 4, d = (1,2,4), j = 1/2

Nor is this one:

v = 4, d = (1,3,4)

This one, however, is the same as the fractal-by-vertex created by banning a jump towards the same vertex twice in a row:

v = 4, d = (2,3,4)

The point can jump towards second-nearest, third-nearest and fourth-nearest vertices, but not towards the nearest. And the nearest vertex will be the one chosen previously.

Now let’s try squares with an additional point-for-jumping-towards on each side (the points are numbered 1 to 8, with points 1, 3, 5, 7 being the true vertices):

v = 4 + s1 point on each side, d = (1,2,3)

v = 4 + s1, d = (1,2,5)

v = 4 + s1, d = (1,2,7)

v = 4 + s1, d = (1,3,8)

v = 4 + s1, d = (1,4,6)

v = 4 + s1, d = (1,7,8)

v = 4 + s1, d = (2,3,8)

v = 4 + s1, d = (2,4,8)

And here are squares where the jump is 2/3, not 1/2, and you can choose only the nearest or third-nearest jump-point:

v = 4, d = (1,3), j = 2/3

v = 4 + s1, d = (1,3), j = 2/3

Now here are some pentagonal fractals-by-distance:

v = 5, d = (1,2,5), j = 1/2

v = 5 + s1, d = (1,2,7)

v = 5 + s1, d = (1,2,8)

v = 5 + s1, d = (1,2,9)

v = 5 + s1, d = (1,9,10)

v = 5 + s1, d = (1,10), j = 2/3

v = 5 + s1, d = (various), j = 2/3 (animated)

And now some hexagonal fractals-by-distance:

v = 6, d = (1,2,4), j = 1/2

v = 6, d = (1,3,5)

v = 6, d = (1,3,6)

v = 6, d = (1,2,3,4)

v = 6 + central point, d = (1,2,3,4)

v = 6, d = (1,2,3,6)

v = 6, d = (1,2,4,6)

v = 6, d = (1,3,4,5)

v = 6, d = (1,3,4,6)

v = 6, d = (1,4,5,6)

Elsewhere other-accessible:

• Get Your Prox Off — an earlier look at fractals-by-distance
• Get Your Prox Off # 2 — and another

Fractal + Star = Fractar

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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 star, 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)

Performativizing the Polygonic #3

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

Pre-previously in my passionate portrayal of polygonic performativity, I showed how a single point jumping randomly (or quasi-randomly) towards the vertices of a polygon can create elaborate fractals. For example, if the point jumps 1/φth (= 0.6180339887…) of the way towards the vertices of a pentagon, it creates this fractal:

Point jumping 1/φth of the way to a randomly (or quasi-randomly) chosen vertex of a pentagon

But as you might expect, there are different routes to the same fractal. Suppose you take a pentagon and select a single vertex. Now, measure the distance to each vertex, v(1,i=1..5), of the original pentagon (including the selected vertex) and reduce it by 1/φ to find the position of a new vertex, v(2,i=1..5). If you do this for each vertex of the original pentagon, then to each vertex of the new pentagons, and so on, in the end you create the same fractal as the jumping point does:

Shrink pentagons by 1/φ, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Shrink by 1/φ (animated) (click for larger if blurred)

And here is the route to a centre-filled variant of the fractal:

Central pentagon, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Central pentagon (animated) (click for larger if blurred)

Using this shrink-the-polygon method, you can reach the same fractals by a third route. This time, use vertex v(1,i) of the original polygon as the centre of the new polygon with its vertices v(2,i=1..5). Creation of the fractal looks like this:

Pentagons over vertices, shrink by 1/φ, stage #1 (no pentagons over vertices)

Stage #2

Stage #3

Stage #4

Stage #4

Stage #5

Stage #7

Pentagons over vertices (animated) (click for larger if blurred)

And here is a third way of creating the centre-filled pentagonal fractal:

Pentagons over vertices and central pentagon, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Pentagons over vertices with central pentagon (animated) (click for larger if blurred)

And here is a fractal created when there are three pentagons to a side and the pentagons are shrunk by 1/φ^2 = 0.3819660112…:

Pentagon at vertex + pentagon at mid-point of side, shrink by 1/φ^2

Final stage

Pentagon at vertex + pentagon at mid-point of side (animated) (click for larger if blurred)

Pentagon at vertex + pentagon at mid-point of side + central pentagon, shrink by 1/φ^2 and c. 0.5, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Pentagon at vertex + mid-point + center (animated) (click for larger if blurred)

Previously pre-posted:

• Performativizing the Polygonic #2
• Performativizing the Polygonic #1

Performativizing the Polygonic #2

by in √2, Base Behaviour, Fractals, Geometry, φ, Mathematics, Phi, Square root of 2 and tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Suppose a café offers you free drinks for three days. You can have tea or coffee in any order and any number of times. If you want tea every day of the three, you can have it. So here’s a question: how many ways can you choose from two kinds of drink in three days? One simple way is to number each drink, tea = 1, coffee = 2, then count off the choices like this:


1: 111
2: 112
3: 121
4: 122
5: 211
6: 212
7: 221
8: 222

Choice #1 is 111, which means tea every day. Choice #6 is 212, which means coffee on day 1, tea on day 2 and coffee on day 3. Now look at the counting again and the way the numbers change: 111, 112, 121, 122, 211… It’s really base 2 using 1 and 2 rather than 0 and 1. That’s why there are 8 ways to choose two drinks over three days: 8 = 2^3. Next, note that you use the same number of 1s to count the choices as the number of 2s. There are twelve 1s and twelve 2s, because each number has a mirror: 111 has 222, 112 has 221, 121 has 212, and so on.

Now try the number of ways to choose from three kinds of drink (tea, coffee, orange juice) over two days:


11, 12, 13, 21, 22, 23, 31, 32, 33 (c=9)

There are 9 ways to choose, because 9 = 3^2. And each digit, 1, 2, 3, is used exactly six times when you write the choices. Now try the number of ways to choose from three kinds of drink over three days:


111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333 (c=27)

There are 27 ways and (by coincidence) each digit is used 27 times to write the choices. Now try three drinks over four days:


1111, 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1311, 1312, 1313, 1321, 1322, 1323, 1331, 1332, 1333, 2111, 2112, 2113, 2121, 2122, 2123, 2131, 2132, 2133, 2211, 2212, 2213, 2221, 2222, 2223, 2231, 2232, 2233, 2311, 2312, 2313, 2321, 2322, 2323, 2331, 2332, 2333, 3111, 3112, 3113, 3121, 3122, 3123, 3131, 3132, 3133, 3211, 3212, 3213, 3221, 3222, 3223, 3231, 3232, 3233, 3311, 3312, 3313, 3321, 3322, 3323, 3331, 3332, 3333 (c=81)

There are 81 ways to choose and each digit is used 108 times. But the numbers don’t have represent choices of drink in a café. How many ways can a point inside an equilateral triangle jump four times half-way towards the vertices of the triangle? It’s the same as the way to choose from three drinks over four days. And because the point jumps toward each vertex in a symmetrical way the same number of times, you get a nice even pattern, like this:

vertices = 3, jump = 1/2

Every time the point jumps half-way towards a particular vertex, its position is marked in a unique colour. The fractal, also known as a Sierpiński triangle, actually represents all possible choices for an indefinite number of jumps. Here’s the same rule applied to a square. There are four vertices, so the point is tracing all possible ways to choose four vertices for an indefinite number of jumps:

v = 4, jump = 1/2

As you can see, it’s not an obvious fractal. But what if the point jumps two-thirds of the way to its target vertex and an extra target is added at the centre of the square? This attractive fractal appears:

v = 4 + central target, jump = 2/3

If the central target is removed and an extra target is added on each side, this fractal appears:

v = 4 + 4 midpoints, jump = 2/3

That fractal is known as a Sierpiński carpet. Now up to the pentagon. This fractal of endlessly nested contingent pentagons is created by a point jumping 1/φ = 0·6180339887… of the distance towards the five vertices:

v = 5, jump = 1/φ

With a central target in the pentagon, this fractal appears:

v = 5 + central, jump = 1/φ

The central red pattern fits exactly inside the five that surround it:

v = 5 + central, jump = 1/φ (closeup)

v = 5 + c, jump = 1/φ (animated)

For a fractal of endlessly nested contingent hexagons, the jump is 2/3:

v = 6, jump = 2/3

With a central target, you get a filled variation of the hexagonal fractal:

v = 6 + c, jump = 2/3

And for a fractal of endlessly nested contingent octagons, the jump is 1/√2 = 0·7071067811… = √½:

v = 8, jump = 1/√2

Previously pre-posted:

• Performativizing the Polygonic

Back to Drac’

by in √2, Fractals, Geometry, Mathematics, Square root of 2, Squares and tagged , , , , , , , , , , , , , , , , , | Leave a comment

draconic, adj. /drəˈkɒnɪk/ pertaining to, or of the nature of, a dragon. [Latin draco, -ōnem, < Greek δράκων dragon] — The Oxford English Dictionary

In Curvous Energy, I looked at the strange, beautiful and complex fractal known as the dragon curve and showed how it can be created from a staid and sedentary square:

A dragon curve

Here are the stages whereby the dragon curve is created from a square. Note how each square at one stage generates a pair of further squares at the next stage:

Dragon curve from squares #1

Dragon curve from squares #2

Dragon curve from squares #3

Dragon curve from squares #4

Dragon curve from squares #5

Dragon curve from squares #6

Dragon curve from squares #7

Dragon curve from squares #8

Dragon curve from squares #9

Dragon curve from squares #10

Dragon curve from squares #11

Dragon curve from squares #12

Dragon curve from squares #13

Dragon curve from squares #14

Dragon curve from squares (animated)

The construction is very easy and there’s no tricky trigonometry, because you can use the vertices and sides of each old square to generate the vertices of the two new squares. But what happens if you use lines rather than squares to generate the dragon curve? You’ll discover that less is more:

Dragon curve from lines #1

Dragon curve from lines #2

Dragon curve from lines #3

Dragon curve from lines #4

Dragon curve from lines #5

Each line at one stage generates a pair of further lines at the next stage, but there’s no simple way to use the original line to generate the new ones. You have to use trigonometry and set the new lines at 45° to the old one. You also have to shrink the new lines by a fixed amount, 1/√2 = 0·70710678118654752… Here are further stages:

Dragon curve from lines #6

Dragon curve from lines #7

Dragon curve from lines #8

Dragon curve from lines #9

Dragon curve from lines #10

Dragon curve from lines #11

Dragon curve from lines #12

Dragon curve from lines #13

Dragon curve from lines #14

Dragon curve from lines (animated)

But once you have a program that can adjust the new lines, you can experiment with new angles. Here’s a dragon curve in which one new line is at an angle of 10°, while the other remains at 45° (after which the full shape is rotated by 180° because it looks better that way):

Dragon curve 10° and 45°

Dragon curve 10° and 45° (animated)

Dragon curve 10° and 45° (coloured)

Here are more examples of dragon curves generated with one line at 45° and the other line at a different angle:

Dragon curve 65°

Dragon curve 65° (anim)

Dragon curve 65° (col)

Dragon curve 80°

Dragon curve 80° (anim)

Dragon curve 80° (col)

Dragon curve 135°

Dragon curve 135° (anim)

Dragon curve 250°

Dragon curve 250° (anim)

Dragon curve 250° (col)

Dragon curve 260°

Dragon curve 260° (anim)

Dragon curve 260° (col)

Dragon curve 340°

Dragon curve 340° (anim)

Dragon curve 340° (col)

Dragon curve 240° and 20°

Dragon curve 240° and 20° (anim)

Dragon curve 240° and 20° (col)

Dragon curve various angles (anim)

Previously pre-posted:

• Curvous Energy — a first look at dragon curves

Rigging in the Trigging

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

Here’s a simple pattern of three triangles:

Three-Triangle Pattern

Now replace each triangle in the pattern with the same pattern at a smaller scale:

Replacing triangles

If you keep on doing this, you create what I’ll call a trigonal fractal (trigon is Greek for “triangle”):

Trigonal Fractal stage #3 (click for larger)

Trigonal Fractal stage #4

Trigonal Fractal stage #5

Trigonal Fractal #6

Trigonal Fractal #7

Trigonal Fractal #8

Trigonal Fractal (animated) (click for larger)

You can use the same pattern to create different fractals by rotating the replacement patterns in different ways. I call this “rigging the trigging” and here are some of the results:


You can also use a different seed-pattern to create the fractals:

Trigonal fractal (animated)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)


Trigonal fractal (anim)

Note: The title of this incendiary intervention is of course a paronomasia on the song “Frigging in the Rigging”, also known as “Good Ship Venus” and performed by the Sex Pistols on The Great Rock ’n’ Roll Swindle (1979).

Curvous Energy

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

Here is a strange and beautiful fractal known as a dragon curve:

A dragon curve (note: this is a twin-dragon curve or Davis-Knuth dragon)

And here is the shape generally regarded as the dullest and most everyday of all:

A square

But squares are square, so let’s go back to dragon-curves. This particular kind of dragon-curve looks a lot like a Chinese dragon. You can see the same writhing energy and scaliness:

Chinese dragon

Dragon-curve for comparison

Dragon-curves also look like some species of soft coral:

Red soft-coral

In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself

Dragon-curve rep-4

Dragon-curve rep-8

Dragon-curve rep-16

Dragon-curve rep-32

Dragon-curve self-tiling (animated)

From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)

And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1

Dragon-curve tiling #2

But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square

Then you see how the shape can be replaced by five smaller copies of itself:

Square overlaid by five smaller squares

Square replaced by five smaller squares

Then you set about replacing it with two of those smaller copies:

Replacing squares Stage #0

Replacing squares Stage #1

Then you do it again to each of the copies:

Replacing squares Stage #2

And again:

Replacing squares #3

And again:

Replacing squares #4

And keep on doing it:

Replacing squares #5

Replacing squares #6

Replacing squares #7

Replacing squares #8

Replacing squares #9

Replacing squares #10

Replacing squares #11

Replacing squares #12

Replacing squares #13

Replacing squares #14

Replacing squares #15

And in the end you’ve got a dragon-curve:

Dragon-curve built from squares

Dragon-curve built from squares (animated)

Phrock and Roll

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

What does a fractal phallus look like?

Millions of people have axed this corely key question.

The Overlord of the Über-Feral can answer it — keyly, corely and comprehensively dot dot dot

And here is the answer: Phrallic Frolics…

Tright Treeing

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

Here is a very simple tree with two branches:

Two-branch tree

These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

Two-branch tree stage 1

2-Tree stage 2

2-Tree stage 3

2-Tree stage 4

2-Tree (animated)

If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1, draw a left branch to node 2 and colour the node green.
2. Return to node 1.
3. Draw a right branch to node 3 and colour the node green.
4. Finish.

Now try a slightly less simple tree with branches that fork twice:

Four-branch tree (static)

These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

4-Tree #1

4-Tree #2

4-Tree #3

4-Tree #4

4-Tree #5

4-Tree #6

4-Tree #7

4-Tree #8

4-Tree #9

4-Tree #10

4-Tree #11

4-Tree (animated)

If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1 and draw a left branch to node 2.
2. Draw a left branch to node 3 and colour it green.
3. Return to node 2.
4. Draw a right branch to node 4 and colour it green.
5. Return to node 2.
6. Return to node 1.
7. Draw a right branch to node 5.
8. Draw a left branch to node 6.
9. Draw a left branch to node 7 and colour it green.
10. Return to node 6.
11. Draw a left branch to node 8 and colour it green.
12. Finish.

It’s easy to see that the list of instructions would be much bigger for a tree with branches that fork three times, let alone four times or you. But you don’t need to give a full set of explicit instructions: you can use a program, or a list of instructions using variables. Suppose the tree has branches that fork f times. If f = 4, you will need an array variable level() with four values, level(1), level(2), level(3) and level(4). Now follow these instructions:

1. li = 1, level(1) = 0, level(2) = 0, ... level(f+1) = 0
2. level(li) = level(li) + 1
3. If level(li) = 1, draw a branch to the left and jump to step 7
4. If level(li) = 2, draw a branch to the right and jump to step 7
5. li = li - 1 (note that this line is reached if the tests fail in lines 3 and 4)
6. If li > 0, jump to step 2, otherwise jump to step 11
7. If li = f, draw a green node and jump to step 5
9. li = li + 1
10. Jump to step 2
11. Finish.

By changing the value of f, a computer can use those eleven basic instructions to draw any size of tree (I’ve left out details like changes in the length of branches and so on). When f = 4, the tree will look like this:

16-Tree (static)

16-Tree (animated)

With simple adjustments, the program can be used for other shapes whose underlying structure can be represented symbolically as a tree. The program is in fact a fractalizer, that is, it draws a fractal. So if you use a version of the program to draw fractals based on right-triangles, you can say you are “tright treeing” (tright = triangle-that-is-right).

Here is some tright treeing. Start with a simple isoceles right-triangle. It can be divided into smaller isoceles right-triangles by finding the midpoint of the hypotenuse, then repeating:

Right-triangle rep-2 stage 1

Right-triangle #2

Tright #3

Tright #4

Tright #5

Tright #6

Tright #7

Tright #7 (no internal lines)

You can distort the isoceles right-triangle in interesting ways by finding the midpoint of a side other than the hypotenuse, like this:

Right-triangle (distorted) #1

Distorted tright #2

Distorted tright #3

Distorted tright #4

Distorted tright #5

Distorted tright #6

Distorted tright #7

Distorted tright #8

Distorted tright #9

Distorted tright #10

Distorted tright #11

Distorted tright #12

Distorted tright #13

Distorted tright (animated)

Here’s a different right-triangle. When you divide it regularly, it looks like this:

Right-triangle rep-3 stage 1

Rep-3 Tright #2

3-Tright #3

3-Tright #4

3-Tright #5

3-Tright #6

3-Tright #7

3-Tright #8

3-Tright #9

3-Tright (one colour)

When you distort the divisions, you can create interesting fractals (click on images for larger versions):

Distorted 3-Tright

Distorted 3-Tright

Distorted 3-Tright

Distorted 3-Tright

Distorted 3-Tright

Distorted 3-Tright

Distorted 3-Tright (animated)

And when four of the distorted right-triangles (rep-2 or rep-3) are joined in a diamond, you can create shapes like these:

Creating a diamond #1

Creating a diamond #2

Creating a diamond #3

Creating a diamond #4

Creating a diamond (animated)

Rep-3 right-triangle diamond (divided)

Rep-3 right-triangle diamond (single colour)

Distorted rep-3 right-triangle diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond

Distorted 3-tright diamond (animated)

Distorted rep-2 right-triangle

Distorted 2-tright diamond

Distorted 2-tright diamond

Distorted 2-tright diamond

Distorted 2-tright diamond

Distorted 2-tright diamond (animated)

Square Routes Re-Re-Re-Revisited

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

Discovering something that’s new to you in recreational maths is good. But so is re-discovering it by a different route. I’ve long been passionate about what happens when a point is allowed to jump repeatedly halfway towards the randomly chosen vertices of a square. If the point can choose any vertex any number of times, the interior of the square fills slowly and completely with points, like this:

Point jumping at random halfway towards vertices of a square

However, if the point is banned from jumping towards the same vertex twice or more in a row, an interesting fractal appears:

Fractal #1 — ban on jumping towards vertex vi twice or more

If the point can’t jump towards the vertex one place clockwise of the vertex it’s just jumped towards, this fractal appears:

Fractal #2 — ban on jumping towards vertex vi+1

If the point can’t jump towards the vertex two places clockwise of the vertex it’s just jumped towards, this fractal appears (two places clockwise is also two places anticlockwise, i.e. the banned vertex is diagonally opposite):

Fractal #3 — ban on jumping towards vertex vi+2

Now I’ve discovered a new way to create these fractals. You take a filled square, divide it into smaller squares, then remove some of them in a systematic way. Then you do the same to the smaller squares that remain. For fractal #1, you do this:

Fractal #1, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Fractal #1 (animated)

For fractal #2, you do this:

Fractal #2, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Fractal #2 (animated)

For fractal #3, you do this:

Fractal #3, stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Fractal #3 (animated)

If the sub-squares are coloured, it’s easier to understand how, say, fractal #1 is created:

Fractal #1 (coloured), stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Fractal #1 (coloured and animated)

The fractal is actually being created in quarters, with one quarter rotated to form the second, third and fourth quarters:

Fractal #1, quarter

Here’s an animation of the same process for fractal #3:

Fractal #3 (coloured and animated)

So you can create these fractals either with a jumping point or by subdividing a square. But in fact I discovered the subdivided-square route by looking at a variant of the jumping-point route. I wondered what would happen if you took a point inside a square, allowed it to trace all possible routes towards the vertices without marking its position, then imposed the restriction for Fractal #1 on its final jump, namely, that it couldn’t jump towards the vertex it jumped towards on its previous jump. If the point is marked after its final jump, this is what appears (if the routes chosen had been truly random, the image would be similar but messier):

Fractal #1, restriction on final jump

Then I imposed the same restriction on the point’s final two jumps:

Fractal #1, restriction on final 2 jumps

And final three jumps:

Fractal #1, restriction on final 3 jumps

And so on:

Fractal #1, restriction on final 4 jumps

Fractal #1, restriction on final 5 jumps

Fractal #1, restriction on final 6 jumps

Fractal #1, restriction on final 7 jumps

Here are animations of the same process applied to fractals #2 and #3:

Fractal #2, restrictions on final 1, 2, 3… jumps

Fractal #3, restrictions on final 1, 2, 3… jumps

The longer the points are allowed to jump before the final restriction is imposed on their n final jumps, the more densely packed the marked points will be:

Fractal #1, packed points #1

Packed points #2

Packed points #3

Eventually, the individual points will form a solid mass, like this:

Fractal #1, solid mass of points

Fractal #1, packed points (animated)

Previously pre-posted (please peruse):

• Square Routes
• Square Routes Revisited
• Square Routes Re-Revisited
• Square Routes Re-Re-Revisited