Construction of a Sierpiński tetrahedron (from WikiMedia)
Post-Performative Post-Scriptum
The toxic title of this incendiary intervention radically references George Harrison’s album Extra Texture (1975).
Extra Tetra
Category Archives: Geometry
Extra Tetra
First Whirled Warp
Imagine two points moving clockwise around the circumference of a circle. Find the midpoint between the two points when one point is moving twice as fast as the other. The midpoint will trace this shape:
Midpoint of two points moving around circle at speeds s and s*2
(n.b. to make things easier to see, the red circle shown here and elsewhere is slightly larger than the virtual circle used to calculate the midpoints)
Now suppose that one point is moving anticlockwise. The midpoint will now trace this shape:
Midpoint for s, -s*2
Now try three points, two moving at the same speed and one moving twice as fast:
Midpoint for s, s, s*2
When the point moving twice as fast is moving anticlockwise, this shape appears:
Midpoint for s, s, -s*2
Here are more of these midpoint-shapes:
Midpoint for s, s*3
Midpoint for s, -s*3
Midpoint for s*2, s*3
Midpoint for s, -s, s*2
Midpoint for s, s*2, -s*2
Midpoint for s, s*2, s*2
Midpoint for s, -s*3, -s*5
Midpoint for s, s*2, s*3
Midpoint for s, s*2, -s*3
Midpoint for s, -s*3, s*5
Midpoint for s, s*3, s*5
Midpoint for s, s, s, s*3
Midpoint for s, s, s, -s*3
Midpoint for s, s, -s, s*3
Midpoint for s, s, -s, -s*3
But what about points moving around the perimeter of a polygon? Here are the midpoints of two points moving clockwise around the perimeter of a square, with one point moving twice as fast as the other:
Midpoint for square with s, s*2
And when one point moves anticlockwise:
Midpoint for square with s, -s*2
If you adjust the midpoints so that the square fills a circle, they look like this:
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Midpoint for square with s, s*2, with square adjusted to fill circle
When the red circle is removed, the midpoint-shape is easier to see:
Midpoint for square with s, s*2, circ-adjusted
Here are more midpoint-shapes from squares:
Midpoint for s, s*3
Midpoint for s, -s*3
Midpoint for s, s*4
And some more circularly adjusted midpoint-shapes from squares:
Midpoint for s, s*3, circ-adjusted
Midpoint for s*2, s*3, circ-adjusted
Midpoint for s, s*5, circ-adjusted
Midpoint for s, s*6, circ-adjusted
Midpoint for s, s*7, circ-adjusted
Finally (for now), let’s look at triangles. If three points are moving clockwise around the perimeter of a triangle, one moving four times as fast as the other two, the midpoint traces this shape:
Midpoint for triangle with s, s, s*4
Now try one of the points moving anticlockwise:
Midpoint for s, s, -s*4
Midpoint for s, -s, s*4
If you adjust the midpoints so that the triangular space fills a circle, they look like this:
Midpoint for s, s, s*4, with triangular space adjusted to fill circle
Midpoint for s, -s, s*4, circ-adjusted
Midpoint for s, s, -s*4, circ-adjusted
There are lots more (infinitely more!) midpoint-shapes to see, so watch this (circularly adjusted) space.
Previously pre-posted (please peruse)
• We Can Circ It Out — more on converting polygons into circles
Strange “S” in the Light
Unexpected discoveries are one of the joys of mathematics, even for amateurs. And computers help you make more of them, because computers make it easy to adjust variables or search faster and further through math-space than any unaided human ever could (on the downside, computers can make you lazy and blunt your intuition). Here’s an unexpected discovery I made using a computer in November 2020:
A distorted and dissected capital “S”
It’s a strange “S” that looks complex but is constructed very easily from three simple lines. And it’s a fractal, a shape that contains copies of itself at smaller and smaller scales:
Five sub-fractals of the Strange “S”
Elsewhere Other-Accessible…
• Fractangular Frolics — the Strange “S” in more light
We Can Circ It Out
It’s a pretty little problem to convert this triangular fractal…
Sierpiński triangle (Wikipedia)
…into its circular equivalent:
Sierpiński triangle as circle
Sierpiński triangle to circle (animated)
But once you’ve circ’d it out, as it were, you can easily adapt the technique to fractals based on other polygons:
T-square fractal (Wikipedia)
⇓
T-square fractal as circle
T-square fractal to circle (animated)
Elsewhere other-accessible…
• Dilating the Delta — more on converting polygonic fractals to circles…
Diamonds to Dust
It’s one of the most famous and easily recognizable logos in the world:
The Mitsubishi diamonds (source)
Those are the three diamonds of Mitsubishi, whose name itself means “three diamonds” or “three rhombi” in Japanese (see 三菱). But if you look at the Mitsubishi diamonds with a mathematical eye, you can see how to create them in two simple steps. First, you divide an equilateral triangle into nine smaller equilateral triangles. Then you discard three of the sub-triangles, like this:
Equilateral triangle divided into nine sub-triangles
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Six sub-triangles left after three are discarded
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But why stop there? Once you’ve discarded three triangles, six triangles are left. Now do the same to the remaining six: divide each into nine sub-triangles and discard three of the sub-triangles. Then do it again and again. When you’ve reduced the diamonds to dust, you’ve got a fractal, a shape that repeats itself at smaller and smaller scales:
Diamond fractal stage #1
Diamond fractal stage #2
Diamond fractal stage #3
Diamond fractal stage #4
Diamond fractal stage #5
Diamond fractal stage #6
Diamond fractal (animated)
After that, you can convert the fractal-within-a-triangle into a fractal-within-a-circle:
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Diamond fractal, triangular to circular (animated)
You can create other fractals by dividing-and-discarding sub-triangles from a rep-9 equilaterial triangle. Here’s what I call a rep9-tri grid fractal:
Grid fractal stage #1
Grid fractal stage #2
Grid fractal stage #3
Grid fractal stage #4
Grid fractal stage #5
Grid fractal stage #6
Grid fractal stage #7
Grid fractal (animated)
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Grid fractal, triangular to circular (animated)
And here’s a rep9-tri hexagon fractal:
Hexagon fractal (initial form)
Hexagon fractal stage #1
Hexagon fractal stage #2
Hexagon fractal stage #3
Hexagon fractal stage #4
Hexagon fractal stage #5
Hexagon fractal stage #6
Hexagon fractal (animated)
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Hexagon fractal, triangular to circular (animated)
The Belles of El
Title page of Sir Henry Billingsley’s first English version of Euclid’s Elements, 1570, with personifications of Geometria, Astronomia, Arithmetica and Musica as beautiful young women
The Elements of Geometrie of the Moſt Aucient Philoſopher Evclide of Megara.
Faithfully (now first) tranʃlated into the Engliʃhe toung, by H. Billingſley, Citizen of London.
Whereunto are annexed certaine Scolies, Annotations, and Inuentions, of the best Mathematiciens, both of times past, and in this our age.
With a very fruitfull Præface made by M.I. Dee, ʃpecifying the chiefe Mathematicall Sciences, what they are, and wherunto commodious: where, alʃo, are diʃcloʃed certaine new Secrets Mathematicall and Mechanicall, untill theʃe our daies, greatly miʃʃed.
Imprinted at London by Iohn Daye.
The title of this incendiary intervention is a paronomasia on “The Bells of Hell…”, a British airmen’s song in terms of core issues around World War I.
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.
Scaffscapes
A fractal is a shape that contains copies of itself on smaller and smaller scales. You can find fractals everywhere in nature. Part of a fern looks like the fern as a whole:
Fern as fractal (source)
Part of a tree looks like the tree as a whole:
Tree as fractal (source)
Part of a landscape looks like the landscape as a whole:
Landscape as fractal (source)
You can also create fractals for yourself. Here are three that I’ve constructed:
Fractal #1
Fractal #2
Fractal #3 — the T-square fractal
The three fractals look very different and, in one sense, that’s exactly what they are. But in another sense, they’re the same fractal. Each can morph into the other two:
Fractal #1 → fractal #2 → fractal #3 (animated)
Here are two more fractals taken en route from fractal #2 to fractal #3, as it were:
Fractal #4
Fractal #5
To understand how the fractals belong together, you have to see what might be called the scaffolding. The construction of fractal #3 is the easiest to understand. First you put up the scaffolding, then you take it away and leave the final fractal:
Fractal #3, scaffolding stage 1
Fractal #3, stage 2
Fractal #3, stage 3
Fractal #3, stage 4
Fractal #3, stage 5
Fractal #3, stage 6
Fractal #3, stage 7
Fractal #3, stage 8
Fractal #3, stage 9
Fractal #3, stage 10
Fractal #3 (scaffolding removed)
Construction of fractal #3 (animated)
Now here’s the construction of fractal #1:
Fractal #1, stage 1
Fractal #1, stage 2
Fractal #1, stage 3
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Construction of fractal #1 (animated)
Fractal #1 (static)
And the constructions of fractals #2, #4 and #5:
Fractal #2, stage 1
Fractal #2, stage 2
Fractal #2, stage 3
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Fractal #2 (animated)
Fractal #2 (static)
Fractal #4, stage 1
Fractal #4, stage 2
Fractal #4, stage 3
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Fractal #4 (animated)
Fractal #4 (static)
Fractal #5, stage 1
Fractal #5, stage 2
Fractal #5, stage 3
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Fractal #5 (animated)
Fractal #5
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
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 
