A spinning Sierpiński tetrahedron or tetrix
Tetrahedrotor
Category Archives: Fractals
Trigging Triangles
A fractal is a shape in which a part looks like the whole. Trees are fractals. And lungs. And clouds. But there are man-made fractals too and probably the most famous of them all is the Sierpiński triangle, invented by the Polish mathematician Wacław Sierpiński (1882-1969):
Sierpiński triangle
There are many ways to create a Sierpiński triangle, but one of the simplest is to trace all possible routes followed by a point jumping halfway towards the vertices of an equilateral triangle. If you mark the endpoint of the jumps, the Sierpiński triangle appears as the routes get longer and longer, like this:
Point jumping 1/2 way towards vertices of an equilateral triangle (animated)
Once you’ve created a Sierpiński triangle like that, you can play with it. For example, you can use simple trigonometry to stretch the triangle into a circle:
Sierpiński triangle to circle stage #1
Sierpiński triangle to circle #2
Sierpiński triangle to circle #3
Sierpiński triangle to circle #4
Sierpiński triangle to circle #5
Sierpiński triangle to circle #6
Sierpiński triangle to circle #7
Sierpiński triangle to circle #8
Sierpiński triangle to circle #9
Sierpiński triangle to circle #10
Sierpiński triangle to Sierpiński circle (animated)
But the trigging of the triangle can go further. You can expand the Sierpiński circle further, like this:
Sierpiński circle expanded
Or shrink the Sierpiński triangle like this:
Shrinking Sierpiński triangle stage #1
Shrinking Sierpiński triangle #2
Shrinking Sierpiński triangle #3
Shrinking Sierpiński triangle #4
Shrinking Sierpiński triangle #5
Shrinking Sierpiński triangle #6
Shrinking Sierpiński triangle (animated)
You can also create new shapes using the jumping-point technique. Suppose that, as the point is jumping, you adjust its position outwards into the circumscribed circle whenever it lands within the boundaries of the governing triangle. But if the point lands outside those boundaries, you leave it alone. Using this adapted technique, you get a shape like this:
Adjusted Sierpiński circle
And if the point is swung by 60° after it’s adjusted into the circle, you get a shape like this:
Adjusted Sierpiński circle (60° swing)
Here are some animated gifs showing these shapes rotating in a full circle at various speeds:
Adjusted Sierpiński circle (swinging animation) (fast)
Adjusted Sierpiński circle (swinging animation) (medium)
Adjusted Sierpiński circle (swinging animation) (slow)
Arty Fish Haul
When is a fish a reptile? When it looks like this:
Fish from four isosceles right triangles
The fish-shape can be divided into eight identical sub-copies of itself. That is, it can be repeatedly tiled with copies of itself, so it’s an example of what geometry calls a rep-tile:
Fish divided into eight identical sub-copies
Fish divided again
Fish divided #4
Fish divided #5
Fish divided #6
Fish (animated rep-tiling)
Now suppose you divide the fish, then discard one of the sub-copies. And carry on dividing-and-discarding like that:
Fish discarding sub-copy 7 (#1)
Fish discarding sub-copy 7 (#2)
Fish discarding sub-copy 7 (#3)
Fish discarding sub-copy 7 (#4)
Fish discarding sub-copy 7 (#5)
Fish discarding sub-copy 7 (#6)
Fish discarding sub-copy 7 (#7)
Fish discarding sub-copy 7 (animated)
Here are more examples of the fish sub-dividing, then discarding sub-copies:
Fish discarding sub-copy #1
Fish discarding sub-copy #2
Fish discarding sub-copy #3
Fish discarding sub-copy #4
Fish discarding sub-copy #5
Fish discarding sub-copy #6
Fish discarding sub-copy #7
Fish discarding sub-copy #8
Fish discarding sub-copies (animated)
Now try a square divided into four copies of the fish, then sub-divided again and again:
Fish-square #1
Fish-square #2
Fish-square #3
Fish-square #4
Fish-square #5
Fish-square #6
Fish-square (animated)
The fish-square can be used to create more symmetrical patterns when the divide-and-discard rule is applied. Here’s the pattern created by dividing-and-discarded two of the sub-copies:
Fish-square divide-and-discard #1
Fish-square divide-and-discard #2
Fish-square divide-and-discard #3
Fish-square divide-and-discard #4
Fish-square divide-and-discard #5
Fish-square divide-and-discard #6
Fish-square divide-and-discard #7
Fish-square divide-and-discard #8 (delayed discard)
Fish-square divide-and-discard (animated)
Using simple trigonometry, you can convert the square pattern into a circular pattern:
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Circular version
Square to circle (animated)
Here are more examples of divide-and-discard fish-squares:
Fish-square divide-and-discard #1
Fish-square divide-and-discard #2
Fish-square divide-and-discard #3
Fish-square divide-and-discard #4
Fish-square divide-and-discard #5
Fish-square divide-and-discard #6
And more examples of fish-squares being converted into circles:
Fish-square to circle #1 (animated)
Fish-square to circle #2
Fish-square to circle #3
Fish-square to circle #4
Fish-square to circle #5
Fish-square to circle #6
Square Routes Re-Re-Re-Re-Re-Re-Revisited
Suppose you trace all possible routes followed by a point inside a triangle jumping halfway towards one or another of the three vertices of the triangle. If you mark each jump, you get a famous geometrical shape called the Sierpiński triangle (or Sierpiński sieve).
Sierpiński triangle found by tracing all possible routes for a point jumping halfway towards the vertices of a triangle
The Sierpiński triangle is a fractal, because it contains copies of itself at smaller and smaller scales. Now try the same thing with a square. If you trace all possible routes followed by a point inside a square jumping halfway towards one or another of the four vertices of the square, you don’t get an obvious fractal. Instead, the interior of the square fills steadily (and will eventually be completely solid):
Routes of a point jumping halfway towards vertices of a square
Try a variant. If the point is banned from jumping towards the same vertex twice or more in a row, the routes trace out a fractal that looks like this:
Ban on choosing same vertex twice or more in a row
If the point is banned from jumping towards the vertex one place anti-clockwise of the vertex it’s just jumped towards, you get a fractal like this:
Ban on jumping towards vertex one place anti-clockwise of previously chosen vertex
And if the point can’t jump towards two places clockwise or anti-clockwise of the currently chosen vertex, this fractal appears (called a T-square fractal):
Ban on jumping towards the vertex diagonally opposite of the previously chosen vertex
That ban is equivalent to banning the point from jumping from the vertex diagonally opposite to the vertex it’s just jumped towards. Finally, here’s the fractal created when you ban the point from jumping towards the vertex one place clockwise of the vertex it’s just jumped towards:
Ban on jumping towards vertex one place clockwise of previously chosen vertex
As you can see, the fractal is a mirror-image of the one-place-anti-clockwise-ban fractal.
I discovered the ban-construction of those fractals more than twenty years ago. Then I found that I was re-discovering the same fractals when I looked at what first seemed like completely different ways of constructing fractals. There are lots of different routes to the same result. I’ve recently discovered yet another route. Let’s try what seems like an entirely different way of constructing fractals. Take a square and erect four new half-sized squares, sq1, sq2, sq3, sq4, on each corner. Then erect three more quarter-sized squares on the outward facing corners of sq1, sq2, sq3 and sq4. Carry on doing that and see what happens at the end when you remove all the previous stages of construction:
Animation of the new construction
Animation in black-and-white
It’s the T-square fractal again. Now try rotating the squares you add at stage 3 and see what happens (the rotation means that two new squares are added on adjacent outward-facing corners and one new square on the inward-facing corner):
Animation of the construction
It’s the one-place-clockwise-ban fractal again. Now try rotating the squares two places, so that two new squares are added on diagonally opposite outward-facing corners and one new square on the inward-facing corner:
Animation of the construction
It’s the same-vertex-ban fractal again. Finally, rotate squares one place more:
Animation of the construction
It’s the one-place-clockwise-ban fractal again. And this method isn’t confined to squares. Here’s what happens when you add 5/8th-sized triangles to the corners of triangles:
Animation of the construction
And here’s what happens when you add 5/13th-sized pentagons to the corners of pentagons:
Animation of the construction
Finally, here’s a variant on that pentagonal fractal (adding two rather than four pentagons at stage 3 and higher):
Animation of the construction
Previously pre-posted (please peruse):
• Square Routes
• Square Routes Revisited
• Square Routes Re-Revisited
• Square Routes Re-Re-Revisited
• Square Routes Re-Re-Re-Revisited
• Square Routes Re-Re-Re-Re-Revisited
• Square Routes Re-Re-Re-Re-Re-Revisited
Fractal-Ize!
Vision Crystal by the American artist Alex Grey (born November 29, 1953)
The Hex Fractor #3
In “Diamonds to Dust”, I showed how the Mitsubishi logo could be turned into a fractal, like this:
The Mitsubishi diamonds (source)
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↓
Mitsubishi logo to fractal (animated)
Now I want to look at another famous symbol and its fractalization. Here’s the symbol, the hexagram:
Hexagram, a six-pointed star
The hexagram can be dissected into twelve equilateral triangles like this:
Hexagram dissected into 12 equilateral triangles
If each triangle in the dissection is replaced by a hexagram, then the hexagram is dissected again into twelve triangles, you get a famous fractal, the Koch snowflake:
The Koch snowflake
The Koch snowflake again
Hexagram to Koch snowflake (animated)
If you color the triangles, you get something like this:
Colored hexagram to fractal (animated)
Of course, this is a very inefficient way to create a Koch snowflake, because the interior hexagrams consume processing time while not contributing to the fractal boundary of the snowflake. But in a way you can fully fractalize the hexagram if you draw only the point at the center of each triangle and then color it according to how many times the pixel in question has been drawn on before. To see how this works, first look at what happens when the center-points are represented in white:
White center-points (animated)
And here’s the fully fractalized hexagram, with colored center-points:
Colored center-points (animated)
Previously Pre-Posted…
• The Hex Fractor #1 — hexagons and fractals
• The Hex Fractor #2 — hexagons and fractals again
• Diamonds to Dust — turning the Mitsubishi logo into a fractal
Trim Pickings
Here is an equilateral triangle divided into nine smaller equilateral triangles:
Rep-9 equilateral triangle
The triangle is a rep-tile — it’s tiled with repeating copies of itself. In this case, it’s a rep-9 triangle. Each of the nine smaller triangles can obviously be divided in their turn:
Rep-81 equilateral triangle
Rep-729 equilateral triangle
Rep-729 equilateral triangle again
Rep-6561 equilateral triangle
Rep-9 triangle repeatedly subdividing (animated)
How try trimming the original rep-9 triangle, picking one of the trimmings, and repeating in finer detail. If you choose six triangles in this pattern, you can create a symmetrical braided fractal:
Triangular fractal stage 1
Triangular fractal #2
Triangular fractal #3
Triangular fractal #3 (cleaning up)
Triangular fractal #3 (cleaning up more)
Triangular fractal #4
Triangular fractal #5
Triangular fractal #6
Triangular fractal (animated)
But this fractal using a three-triangle trim-picking isn’t symmetrical:
Trim-picking #1
Trim-picking #2
Trim-picking #3
Trim-picking #4
Trim-picking #5
To make it symmetric, you have to delay the trim, using the full rep-9 trim for the first stage:
Delayed trim-picking #1
Delayed trim-picking #2
Delayed trim-picking #3
Delayed trim-picking #4
Delayed trim-picking #5
Delayed trim-picking #6 (with first two stages as rep-9)
Delayed trim-picking (animated)
Here are some more delayed trim-pickings used to created symmetrical patterns:
Polykoch (Kontinued)
In “Polykoch!”, I looked at variants on the famous Koch snowflake, which is created by erecting new triangles on the sides of an equilaternal triangle, like this:
Koch snowflake #1
Koch snowflake #2
Koch snowflake #3
Koch snowflake #4
Koch snowflake #5
Koch snowflake #6
Koch snowflake #7
Koch snowflake (animated)
One variant is simple: the new triangles move inward rather than outward:
Inverted Koch snowflake #1
Inverted Koch snowflake #2
Inverted Koch snowflake #3
Inverted Koch snowflake #4
Inverted Koch snowflake #5
Inverted Koch snowflake #6
Inverted Koch snowflake #7
Inverted Koch snowflake (animated)
Or you can alternate between moving the new triangles inward and outward. When they always move outward and have sides 1/5 the length of the sides of the original triangle, the snowflake looks like this:
When they move inward, then always outward, the snowflake looks like this:
And so on:
Now here’s a Koch square with its new squares moving inward:
Inverted Koch square #1
Inverted Koch square #2
Inverted Koch square #3
Inverted Koch square #4
Inverted Koch square #5
Inverted Koch square #6
Inverted Koch square (animated)
And here’s a pentagon with squares moving inwards on its sides:
Pentagon with squares #1
Pentagon with squares #2
Pentagon with squares #3
Pentagon with squares #4
Pentagon with squares #5
Pentagon with squares #6
Pentagon with squares (animated)
And finally, an octagon with hexagons on its sides. First the hexagons move outward, then inward, then outward, then inward, then outward:
Octagon with hexagons #1
Octagon with hexagons #2
Octagon with hexagons #3
Octagon with hexagons #4
Octagon with hexagons #5
Octagon with hexagons (animated)
Polykoch!
This is how you form the famous Koch snowflake, in which at each stage you erect a new triangle on the middle of each line whose sides are 1/3 the length of the line:
Koch snowflake #1
Koch snowflake #2
Koch snowflake #3
Koch snowflake #4
Koch snowflake #5
Koch snowflake #6
Koch snowflake #7
Koch snowflake (animated)
Here’s a variant of the Koch snowflake, with new mid-triangles whose sides are 1/2 the length of the lines:
Koch snowflake (1/2 side) #1
Koch snowflake (1/2 side) #2
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Stage #8
Koch snowflake (1/2 side) (animated)
But why stop at triangles? This is a Koch square, in which at each stage you erect a new 1/3 square on the middle of each line:
Koch square #1
Koch square #2
Koch square #3
Koch square #4
Koch square #5
Koch square #6
Koch square (animated)
And a Koch pentagon, in which at each stage you erect a pentagon on the middle of each line whose sides are 1 – (1/φ^2 * 2) = 0·236067977… the length of the line (I used 55/144 as an approximation of 1/φ^2):
Koch pentagon (side 55/144) #1
Koch pentagon #2
Koch pentagon #3
Koch pentagon #4
Koch pentagon #5
Koch pentagon #6
Koch pentagon (animated)
In this close-up, you can see how precisely the sprouting pentagons kiss at each stage:
Koch pentagon (close-up) #1
Koch pentagon (close-up) #2
Koch pentagon (close-up) #3
Koch pentagon (close-up) #4
Koch pentagon (close-up) #5
Koch pentagon (close-up) #6
Koch pentagon (close-up) (animated)
Blancmange Butterfly
Blancmange butterfly. Is that a ’60s psychedelic band? No, it’s one of the shapes you can get by playing with blancmange curves. As I described in “White Rites”, a blancmange curve is a fractal created by summing the heights of successively smaller and more numerous zigzags, like this:
Zigzags 1 to 10
Zigzags 1 to 10 (animated)
Blancmange curve
In the blancmange curves below, the height (i.e., the y co-ordinate) has been normalized so that all the images are the same height:
Construction of a normalized blancmange curve (animated)
This is the solid version:
Solid normalized blancmange curve (animated)
I wondered what happens when you wrap a blancmange curve around a circle. Well, this happens:
Construction of a blancmange circle (animated)
You get what might be called a blancmange butterfly. The solid version looks like this (patterns in the circles are artefacts of the graphics program I used):
Solid blancmange circle (animated)
Next I tried using arcs rather zigzags to construct the blancmange curves and blancmange circles:
Arching blancmange curve (i.e., constructed with arcs) (animated)
And below is the circular version of a blancmange curve constructed with arcs. The arching circular blancmanges look even more like buttocks and then intestinal villi (the fingerlike projections lining our intestines):
Arching blancmange circle (animated)
The variations on blancmange curves don’t stop there — in fact, they’re infinite. Below is a negative arching blancmange curve, where the heights of the original arching blancmange curve are subtracted from the (normalized) maximum height:
Negative arching blancmange curve (animated)
And here’s an arching blancmange curve that’s alternately negative and positive:
Negative-positive arching blancmange curve (animated)
The circular version looks like this:
Negative-positive arching blancmange circle (animated)
Finally, here’s an arching blancmange curve that’s alternately positive and negative:
Positive-negative arching blancmange curve (animated)
And the circular version:
Positive-negative arching blancmange circle (animated)
Elsewhere Other-Accessible…
• White Rites — more variations on blancmange curves
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 