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


Constructing a Sierpiński triangle (anim)


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


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


Constructing a T-square fractal (anim)


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

T-square fractal


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).

A Seed Indeed

Like plants, fractals grow from seeds. But plants start with a small seed that gets bigger. Fractals start with a big seed that gets smaller. For example, perhaps the most famous fractal of all is the Koch snowflake. The seed of the Koch snowflake is step #2 here:

Stages of the Koch snowflake (from Fractals and the coast of Great Britain)


To create the Koch snowflake, you replace each straight line in the initial triangle with the seed:

Creating the Koch snowflake (from Wikipedia)


Animated Koch snowflake (from Wikipedia)


Now here’s another seed for another fractal:

Fractal stage #1


The seed is like a capital “I”, consisting of a line of length l sitting between two lines of length l/2 at right angles. The rule this time is: Replace the center of the longer line and the two shorter lines with ½-sized versions of the seed:

Fractal stage #2


Try and guess what the final fractal looks like when this rule is applied again and again:

Fractal stage #3


Fractal stage #4


Fractal stage #5


Fractal stage #6


Fractal stage #7


Fractal stage #8


Fractal stage #9


Fractal stage #10


I call this fractal the hourglass. And there are a lot of ways to create it. Here’s an animated version of the way shown in this post:

Hourglass fractal (animated)


Hour Re-Re-Powered

In “Hour Power” I looked at my favorite fractal, the hourglass fractal:

The hourglass fractal


I showed three ways to create the fractal. Next, in “Hour Re-Powered”, I showed a fourth way. Now here’s a fifth (previously shown in “Tri Again”).

This is a rep-4 isosceles right triangle:

Rep-4 isosceles right triangle


If you divide and discard one of the four sub-triangles, then adjust one of the three remaining sub-triangles, then keep on dividing-and-discarding (and adjusting), you can create a certain fractal — the hourglass fractal:

Triangle to hourglass #1


Triangle to hourglass #2


Triangle to hourglass #3


Triangle to hourglass #4


Triangle to hourglass #5


Triangle to hourglass #6


Triangle to hourglass #7


Triangle to hourglass #8


Triangle to hourglass #9


Triangle to hourglass #10


Triangle to hourglass (anim) (open in new tab to see full-sized version)


And here is a zoomed version:

Triangle to hourglass (large)


Triangle to hourglass (large) (anim)


Hour Re-Powered

Pre-previously on Overlord in terms of the Über-Feral, I looked at my favorite member of the fractal community, the Hourglass Fractal:

The hourglass fractal


A real hourglass for comparison


As I described how I discovered the hourglass fractal indirectly and by accident, then showed how to create it directly, using two isosceles triangles set apex-to-apex in the form of an hourglass:

Triangles to hourglass #1


Triangles to hourglass #2


Triangles to hourglass #3


Triangles to hourglass #4


Triangles to hourglass #5


Triangles to hourglass #6

[…]

Triangles to hourglass #10


Triangles to hourglass #11


Triangles to hourglass #12


Triangles to hourglass (animated)


Now, here’s an even simpler way to create the hourglass fractal, starting with a single vertical line:

Line to hourglass #1


Line to hourglass #2


Line to hourglass #3


Line to hourglass #4


Line to hourglass #5


Line to hourglass #6


Line to hourglass #7


Line to hourglass #8


Line to hourglass #9


Line to hourglass #10


Line to hourglass #11


Line to hourglass (animated)


Hour Power

Would it be my favorite fractal if I hadn’t discovered it for myself? It might be, because I think it combines great simplicity with great beauty. I first came across it when I was looking at this rep-tile, that is, a shape that can be divided into smaller copies of itself:

Rep-4 L-Tromino


It’s called a L-tromino and is a rep-4 rep-tile, because it can be divided into four copies of itself. If you divide the L-tromino into four sub-copies and discard one particular sub-copy, then repeat again and again, you’ll get this fractal:

Tromino fractal #1


Tromino fractal #2


Tromino fractal #3


Tromino fractal #4


Tromino fractal #5


Tromino fractal #6


Tromino fractal #7


Tromino fractal #8


Tromino fractal #9


Tromino fractal #10


Tromino fractal #11


Hourglass fractal (animated)


I call it an hourglass fractal, because it reminds me of an hourglass:

A real hourglass


The hourglass fractal for comparison


I next came across the hourglass fractal when applying the same divide-and-discard process to a rep-4 square. The first fractal that appears is the Sierpiński triangle:

Square to Sierpiński triangle #1


Square to Sierpiński triangle #2


Square to Sierpiński triangle #3


[…]


Square to Sierpiński triangle #10


Square to Sierpiński triangle (animated)


However, you can rotate the sub-squares in various ways to create new fractals. Et voilà, the hourglass fractal appears again:

Square to hourglass #1


Square to hourglass #2


Square to hourglass #3


Square to hourglass #4


Square to hourglass #5


Square to hourglass #6


Square to hourglass #7


Square to hourglass #8


Square to hourglass #9


Square to hourglass #10


Square to hourglass #11


Square to hourglass (animated)


Finally, I was looking at variants of the so-called chaos game. In the standard chaos game, a point jumps half-way towards the randomly chosen vertices of a square or other polygon. In this variant of the game, I’ve added jump-towards-able mid-points to the sides of the square and restricted the point’s jumps: it can only jump towards the points that are first-nearest, seventh-nearest and eighth-nearest. And again the hourglass fractal appears:

Chaos game to hourglass #1


Chaos game to hourglass #2


Chaos game to hourglass #3


Chaos game to hourglass #4


Chaos game to hourglass #5


Chaos game to hourglass #6


Chaos game to hourglass (animated)


But what if you want to create the hourglass fractal directly? You can do it like this, using two isosceles triangles set apex-to-apex in the form of an hourglass:

Triangles to hourglass #1


Triangles to hourglass #2


Triangles to hourglass #3


Triangles to hourglass #4


Triangles to hourglass #5


Triangles to hourglass #6


Triangles to hourglass #7


Triangles to hourglass #8


Triangles to hourglass #9


Triangles to hourglass #10


Triangles to hourglass #11


Triangles to hourglass #12


Triangles to hourglass (animated)


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

For a good example of how more can be less, try the chaos game. You trace a point jumping repeatedly 1/n of the way towards a randomly chosen vertex of a regular polygon. When the polygon is a triangle and 1/n = 1/2, this is what happens:

Chaos triangle #1


Chaos triangle #2


Chaos triangle #3


Chaos triangle #4


Chaos triangle #5


Chaos triangle #6


Chaos triangle #7


As you can see, this simple chaos game creates a fractal known as the Sierpiński triangle (or Sierpiński sieve). Now try more and discover that it’s less. When you play the chaos game with a square, this is what happens:

Chaos square #1


Chaos square #2


Chaos square #3


Chaos square #4


Chaos square #5


Chaos square #6


Chaos square #7


As you can see, more is less: the interior of the square simply fills with points and no attractive fractal appears. And because that was more is less, let’s see how less is more. What happens if you restrict the way in which the point inside the square can jump? Suppose it can’t jump twice towards the same vertex (i.e., the vertex v+0 is banned). This fractal appears:

Ban on choosing vertex [v+0]


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

Ban on vertex [v+1] (or [v-1], depending on how you number the vertices)


And if the point can’t jump towards two places clockwise or anti-clockwise of the currently chosen vertex, this fractal appears:

Ban on vertex [v+2], i.e. the diagonally opposite vertex


At least, that is one possible route to those three particular fractals. You see another route, start with this simple fractal, where dividing and discarding parts of a square creates a Sierpiński triangle:

Square to Sierpiński triangle #1


Square to Sierpiński triangle #2


Square to Sierpiński triangle #3


Square to Sierpiński triangle #4


[…]


Square to Sierpiński triangle #10


Square to Sierpiński triangle (animated)


By taking four of these square-to-Sierpiński-triangle fractals and rotating them in the right way, you can re-create the three chaos-game fractals shown above. Here’s the [v+0]-ban fractal:

[v+0]-ban fractal #1


[v+0]-ban #2


[v+0]-ban #3


[v+0]-ban #4


[v+0]-ban #5


[v+0]-ban #6


[v+0]-ban #7


[v+0]-ban #8


[v+0]-ban #9


[v+0]-ban (animated)


And here’s the [v+1]-ban fractal:

[v+1]-ban fractal #1


[v+1]-ban #2


[v+1]-ban #3


[v+1]-ban #4


[v+1]-ban #5


[v+1]-ban #6


[v+1]-ban #7


[v+1]-ban #8


[v+1]-ban #9


[v+1]-ban (animated)


And here’s the [v+2]-ban fractal:

[v+2]-ban fractal #1


[v+2]-ban #2


[v+2]-ban #3


[v+2]-ban #4


[v+2]-ban #5


[v+2]-ban #6


[v+2]-ban #7


[v+2]-ban #8


[v+2]-ban #9


[v+2]-ban (animated)

And taking a different route means that you can find more fractals — as I will demonstrate.


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