Fractangular Frolics

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)


Who Made Heu?

Fractal leaves of Heuchera “Red Lightning


Fractal river network in Shaanxi province, China


Post-Performative Post-Scriptum

Because “Heuchera” comes from the name of the German botanist J.H. Heucher (1677–1747), it should strictly speaking be pronounced something like “HOI-keh-ruh”. But people often say “HYOO-keh-ruh” or variations thereon.

Hour Re-Re-Re-Powered

Here’s a set of three lines:

Three lines


Now try replacing each line with a half-sized copy of the original three lines:

Three half-sized copies of the original three lines


What shape results if you keep on doing that — replacing each line with three half-sized new lines — over and over again? I’m not sure that any human is yet capable of visualizing it, but you can see the shape being created below:

Morphogenesis #3


Morphogenesis #4


Morphogenesis #5


Morphogenesis #6


Morphogenesis #7


Morphogenesis #8


Morphogenesis #9


Morphogenesis #10


Morphogenesis #11 — the Hourglass Fractal


Morphogenesis of the Hourglass Fractal (animated)


The shape that results is what I call the hourglass fractal. Here’s a second and similar method of creating it:

Hourglass fractal, method #2 stage #1


Hourglass fractal #2


Hourglass fractal #3


Hourglass fractal #4


Hourglass fractal #5


Hourglass fractal #6


Hourglass fractal #7


Hourglass fractal #8


Hourglass fractal #9


Hourglass fractal #10


Hourglass fractal #11


Hourglass fractal (animated)


And below are both methods in one animated gif, where you can see how method #1 produces an hourglass fractal twice as large as the hourglass fractal produced by method #2:

Two routes to the hourglass fractal (animated)


Elsewhere other-engageable:

Hour Power
Hour Re-Powered
Hour Re-Re-Powered

Tri Again (Again (Again))

Like the moon, mathematics is a harsh mistress. In mathematics, as on the moon, the slightest misstep can lead to disaster — as I’ve discovered again and again. My latest discovery came when I was looking at a shape called the L-tromino, created from three squares set in an L-shape. It’s a rep-tile, because it can be tiled with four smaller copies of itself, like this:

Rep-4 L-tromino


And if it can be tiled with four copies of itself, it can also be tiled with sixteen copies of itself, like this:

Rep-16 L-tromino


My misstep came when I was trying to do to a rep-16 L-tromino what I’d already done to a rep-4 L-tromino. And what had I already done? I’d created a beautiful shape called the hourglass fractal by dividing-and-discarding sub-copies of a rep-4 L-tromino. That is, I divided the L-tromino into four sub-copies, discarded one of the sub-copies, then repeated the process with the sub-sub-copies of the sub-copies, then the sub-sub-sub-copies of the sub-sub-copies, and so on:

Creating an hourglass fractal #1


Creating an hourglass fractal #2


Creating an hourglass fractal #3


Creating an hourglass fractal #4


Creating an hourglass fractal #5


Creating an hourglass fractal #6


Creating an hourglass fractal #7


Creating an hourglass fractal #8


Creating an hourglass fractal #9


Creating an hourglass fractal #10


Creating an hourglass fractal (animated)


The hourglass fractal


Next I wanted to create an hourglass fractal from a rep-16 L-tromino, so I reasoned like this:

• If one sub-copy of four is discarded from a rep-4 L-tromino to create the hourglass fractal, that means you need 3/4 of the rep-4 L-tromino. Therefore you’ll need 3/4 * 16 = 12/16 of a rep-16 L-tromino to create an hourglass fractal.

So I set up the rep-16 L-tromino with twelve sub-copies in the right pattern and began dividing-and-discarding:

A failed attempt at an hourglass fractal #1


A failed attempt at an hourglass fractal #2


A failed attempt at an hourglass fractal #3


A failed attempt at an hourglass fractal #4


A failed attempt at an hourglass fractal #5


A failed attempt at an hourglass fractal (animated)


Whoops! What I’d failed to take into account is that the rep-16 L-tromino is actually the second stage of the rep-4 triomino, i.e. that 4 * 4 = 16. It follows, therefore, that 3/4 of the rep-4 L-tromino will actually be 9/16 = 3/4 * 3/4 of the rep-16 L-tromino. So I tried again, setting up a rep-16 L-tromino with nine sub-copies, then dividing-and-discarding:

A third attempt at an hourglass fractal #1


A third attempt at an hourglass fractal #2


A third attempt at an hourglass fractal #3


A third attempt at an hourglass fractal #4


A third attempt at an hourglass fractal #5


A third attempt at an hourglass fractal #6


A third attempt at an hourglass fractal (animated)



Previously (and passionately) pre-posted:

Tri Again
Tri Again (Again)

At the Mountings of Mathness

Mounting n. a backing or setting on which a photograph, work of art, gem, etc. is set for display. — Oxford English Dictionary

Viewer’s advisory: If you are sensitive to flashing or flickering images, you should be careful when you look at the last couple of animated gifs below.


H.P. Lovecraft in some Mountings of Mathness






Agnathous Analysis

In Mandibular Metamorphosis, I looked at two distinct fractals and how you could turn one into the other in one smooth sweep. The Sierpiński triangle was one of the fractals:

Sierpiński triangle


The T-square fractal was the other:

T-square fractal (or part thereof)


And here they are turning into each other:

Sierpiński ↔ T-square (anim)
(Open in new window if distorted)


But what exactly is going on? To answer that, you need to see how the two fractals are created. Here are the stages for one way of constructing the Sierpiński triangle:

Sierpiński triangle #1


Sierpiński triangle #2


Sierpiński triangle #3


Sierpiński triangle #4


Sierpiński triangle #5


Sierpiński triangle #6


Sierpiński triangle #7


Sierpiński triangle #8


Sierpiński triangle #9


When you take away all the construction lines, you’re left with a simple Sierpiński triangle:


Constructing a Sierpiński triangle (anim)


Now here’s the construction of a T-square fractal:

T-square fractal #1


T-square fractal #2


T-square fractal #3


T-square fractal #4


T-square fractal #5


T-square fractal #6


T-square fractal #7


T-square fractal #8


T-square fractal #9


Take away the construction lines and you’re left with a simple T-square fractal:

T-square fractal


Constructing a T-square fractal (anim)


And now it’s easy to see how one turns into the other:

Sierpiński → T-square #1


Sierpiński → T-square #2


Sierpiński → T-square #3


Sierpiński → T-square #4


Sierpiński → T-square #5


Sierpiński → T-square #6


Sierpiński → T-square #7


Sierpiński → T-square #8


Sierpiński → T-square #9


Sierpiński → T-square #10


Sierpiński → T-square #11


Sierpiński → T-square #12


Sierpiński → T-square #13


Sierpiński ↔ T-square (anim)
(Open in new window if distorted)


Post-Performative Post-Scriptum

Mandibular Metamorphosis also looked at a third fractal, the mandibles or jaws fractal. Because I haven’t included the jaws fractal in this analysis, the analysis is therefore agnathous, from Ancient Greek ἀ-, a-, “without”, + γνάθ-, gnath-, “jaw”.

Mandibular Metamorphosis

Here’s the famous Sierpiński triangle:

Sierpiński triangle


And here’s the less famous T-square fractal:

T-square fractal (or part of it, at least)


How do you get from one to the other? Very easily, as it happens:

From Sierpiński triangle to T-square (and back again) (animated)
(Open in new window if distorted)


Now, here are the Sierpiński triangle, the T-square fractal and what I call the mandibles or jaws fractal:

Sierpiński triangle


T-square fractal


Mandibles / Jaws fractal


How do you cycle between them? Again, very easily:

From Sierpiński triangle to T-square to Mandibles (and back again) (animated)
(Open in new window if distorted)


FractAlphic Frolix

A fractal is a shape that contains smaller (and smaller) versions of itself, like this:

The hourglass fractal


Fractals also occur in nature. For example, part of a tree looks like the tree as whole. Part of a cloud or a lung looks like the cloud or lung as a whole. So trees, clouds and lungs are fractals. The letters of an alphabet don’t usually look like that, but I decided to create a fractal alphabet — or fractalphabet — that does.

The fractalphabet starts with this minimal standard Roman alphabet in upper case, where each letter is created by filling selected squares in a 3×3 grid:


The above is stage 1 of the fractalphabet, when it isn’t actually a fractal alphabet at all. But if each filled square of the letter “A”, say, is replaced by the letter itself, the “A” turns into a fractal, like this:








Fractal A (animated)


Here’s the whole alphabet being turned into fractals:

Full fractalphabet (black-and-white)


Full fractalphabet (color)


Full fractalphabet (b&w animated)


Full fractalphabet (color animated)


Now take a full word like “THE”:



You can turn each letter into a fractal using smaller copies of itself:







Fractal THE (b&w animated)


Fractal THE (color animated)


But you can also create a fractal from “THE” by compressing the “H” into the “T”, then the “E” into the “H”, like this:




Compressed THE (animated)



The compressed “THE” has a unique appearance and is both a letter and a word. Now try a complete sentence, “THE CAT BIT THE RAT”. This is the sentence in stage 1 of the fractalphabet:



And stage 2:



And further stages:





Fractal CAT (b&w animated)


Fractal CAT (color animated)


But, as we saw with “THE” above, that’s not the only fractal you can create from “THE CAT BIT THE RAT”. Here’s what I call a 2-compression of the sentence, where every second letter has been compressed into the letter that precedes it:


THE CAT BIT THE RAT (2-comp color)


THE CAT BIT THE RAT (2-comp b&w)


And here’s a 3-compression of the sentence, where every third letter has been compressed into every second letter, and every second-and-third letter has been compressed into the preceding letter:

THE CAT BIT THE RAT (3-comp color)


THE CAT BIT THE RAT (3-comp b&w)


As you can see above, each word of the original sentence is now a unique single letter of the fractalphabet. Theoretically, there’s no limit to the compression: you could fit every word of a book in the standard Roman alphabet into a single letter of the fractalphabet. Or you could fit an entire book into a single letter of the fractalphabet (with additional symbols for punctuation, which I haven’t bothered with here).

To see what the fractalphabeting of a longer text in the standard Roman alphabet might look like, take the first verse of a poem by A.E. Housman:

On Wenlock Edge the wood’s in trouble;
His forest fleece the Wrekin heaves;
The gale it plies the saplings double,
And thick on Severn snow the leaves. (“Poem XXXI” of A Shropshire Lad, 1896)

The first line looks like this in stage 1 of the fractalphabet:


Here’s stage 2 of the standard fractalphabet, where each letter is divided into smaller copies of itself:


And here’s stage 3 of the standard fractalphabet:


Now examine a colour version of the first line in stage 1 of the fractalphabet:


As with “THE” above, let’s try compressing each second letter into the letter that precedes it:


And here’s a 3-comp of the first line:


Finally, here’s the full first verse of Housman’s poem in 2-comp and 3-comp forms:

On Wenlock Edge the wood’s in trouble;
His forest fleece the Wrekin heaves;
The gale it plies the saplings double,
And thick on Severn snow the leaves. (“Poem XXXI of A Shropshire Lad, 1896)

“On Wenlock Edge” (2-comp)


“On Wenlock Edge” (3-comp)


Appendix

This is a possible lower-case version of the fractalphabet:

Controlled Chaos

The chaos game is a simple mathematical technique for creating fractals. Suppose a point jumps over and over again 1/2 of the distance towards a randomly chosen vertex of a triangle. This shape appears, the so-called Sierpiński triangle:

Sierpiński triangle created by the chaos game


But the jumps don’t have to be random: you can use an array to find every possible combination of jumps and so create a more even image. I call this controlled chaos. However, if you try the chaos game (controlled or otherwise) with a square, no fractal appears unless you restrict the vertex chosen in some way. For example, if the point can’t jump towards the same vertex twice or more in a row, this fractal appears:

Ban on jumping towards previously chosen vertex, i.e. v + 0


And if the point can’t jump towards the vertex one place clockwise of the previously chosen vertex, this fractal appears:

Ban on v + 1


If the point can’t jump towards the vertex two places clockwise of the previously chosen vertex, this fractal appears:

Ban on v + 2


If the point can’t jump towards the vertex three places clockwise, or one place anticlockwise, of the previously chosen vertex, this fractal appears (compare v + 1 above):

Ban on v + 3


You can also ban vertices based on how close the point is to them at any given moment. Suppose that the point can’t jump towards the nearest vertex, which means that it must choose to jump towards either the 2nd-nearest, 3rd-nearest or 4th-nearest vertex. A fractal we’ve already seen appears:

Must jump towards vertex at distance 2, 3 or 4


In effect, not jumping towards the nearest vertex means not jumping towards a vertex twice or more in a row. Another familiar fractal appears if the point can’t jump towards the most distant vertex:

d = 1,2,3


But new fractals also appear when the jumps are determined by distance:

d = 1,2,4


d = 1,3,4


And you can add more targets for the jumping point midway between the vertices of the square:

d = 1,2,8


d = 1,4,6


d = 1,6,8


d = 1,7,8


d = 2,3,6


d = 2,3,8


d = 2,4,8


d = 2,5,6


And what if you choose the next vertex by incrementing the previously chosen vertex? Suppose the initial vertex is 1 and the possible increments are 1, 2 and 2. This new fractal appears:

increment = 1,2,2 (for example, 1 + 1 = 2, 2 + 2 = 4, 4 + 2 = 6, and 6 is adjusted thus: 6 – 4 = 2)


And with this set of increments, it’s déjà vu all over again:

i = 2,2,3


And again:

i = 2,3,2


With more possible increments, familiar fractals appear in unfamiliar ways:

i = 1,3,2,3


i = 1,3,3,2


i = 1,4,3,3


i = 2,1,2,2


i = 2,1,3,4


i = 2,2,3,4


i = 3,1,1,2


Now try increments with midpoints on the sides:

v = 4 + midpoints, i = 1,2,4


As we saw above, this incremental fractal can also be created from a square with four vertices and no midpoints:

i = 1,3,3; initial vertex = 1


But the fractal changes when the initial vertex is set to 2, i.e. to one of the midpoints:

i = 1,3,3; initial vertex = 2


And here are more inc-fractals with midpoints:

i = 1,4,2 (cf. inc-fractal 1,2,4 above)


i = 1,4,8


i = 2,6,3


i = 3,2,6

<hr

i = 4,7,8


i = 1,2,3,5


i = 1,4,5,4


i = 6,2,4,1


i = 7,6,2,2


i = 7,8,2,4


i = 7,8,4,2


Allus Pour, Horic

*As a rotating animated gif (optimized at ezGIF).


Performativizing Paronomasticity

The title of this incendiary intervention, which is perhaps my most contrived title yet, is a paronomasia on Shakespeare’s “Alas, poor Yorick!” (Hamlet, Act 5, scene 1). “Allus” is a northern form of “always”, “pour” has its standard meaning, and “Horic” is from the Greek ὡρῐκός, hōrikos, which strictly speaking means “in one’s prime, blooming”. However, it could also be interpreted as meaning “hourly”. So the paronomasia means “Always pour, O Hourly One!” (i.e. hourglass).