Allus Pour, Horic

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

Performativizing Paronomasticity

The title of this incendiary intervention 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).

Bat out of L

Pre-previously on Overlord-in-terms-of-the-Über-Feral, I’ve looked at intensively interrogated issues around the L-triomino, a shape created from three squares that can be divided into four copies of itself:

An L-triomino divided into four copies of itself

I’ve also interrogated issues around a shape that yields a bat-like fractal:

A fractal full of bats

Bat-fractal (animated)

Now, to end the year in spectacular fashion, I want to combine the two concepts pre-previously interrogated on Overlord-in-terms-of-the-Über-Feral (i.e., L-triominoes and bats). The L-triomino can also be divided into nine copies of itself:

An L-triomino divided into nine copies of itself

If three of these copies are discarded and each of the remaining six sub-copies is sub-sub-divided again and again, this is what happens:

Fractal stage 1

Fractal stage 2

Fractal #3

Fractal #4

Fractal #5

Fractal #6

Et voilà, another bat-like fractal:

L-triomino bat-fractal (static)

L-triomino bat-fractal (animated)

Elsewhere other-posted:

Tridentine Math

The Tridentine Mass is the Roman Rite Mass that appears in typical editions of the Roman Missal published from 1570 to 1962. — Tridentine Mass, Wikipedia

A 30°-60°-90° right triangle, with sides 1 : √3 : 2, can be divided into three identical copies of itself:

30°-60°-90° Right Triangle — a rep-3 rep-tile…

And if it can be divided into three, it can be divided into nine:

…that is also a rep-9 rep-tile

Five of the sub-copies serve as the seed for an interesting fractal:

Fractal stage #1

Fractal stage #2

Fractal stage #3

Fractal #4

Fractal #5

Fractal #6

Fractal #6

Tridentine cross (animated)

Tridentine cross (static)

This is a different kind of rep-tile:

Noniamond trapezoid

But it yields the same fractal cross:

Fractal #1

Fractal #2

Fractal #3

Fractal #4

Fractal #5

Fractal #6

Tridentine cross (animated)

Tridentine cross (static)

Elsewhere other-available:

Holey Trimmetry — another fractal cross

Square Routes Re-Re-Revisited

This is an L-triomino, or shape created from three squares laid edge-to-edge:

When you divide each square like this…

You can create a fractal like this…

Stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Stage #9

Stage #10

Animated fractal

Here are more fractals created from the triomino:

Animated

Static

Animated

Static

Animated

Static

And here is a different shape created from three squares:

And some fractals created from it:

Animated

Static

Animated

Static

Animated

Static

And a third shape created from three squares:

And some fractals created from it:

Animated

Static

Animated

Static

Animated

Static

Animated

Static

Animated

Static

Animated

Static

Animated

Static

Animated

Static

Bats and Butterflies

I’ve used butterfly-images to create fractals. Now I’ve found a butterfly-image in a fractal. The exciting story begins with a triabolo, or shape created from three isoceles right triangles:

The triabolo is a rep-tile, or shape that can be divided into smaller copies of itself:

In this case, it’s a rep-9 rep-tile, divisible into nine smaller copies of itself. And each copy can be divided in turn:

But what happens when you sub-divide, then discard copies? A fractal happens:

Fractal crosses (animated)

Fractal crosses (static)

That’s a simple example; here is a more complex one:

Fractal butterflies #1

Fractal butterflies #2

Fractal butterflies #3

Fractal butterflies #4

Fractal butterflies #5

Fractal butterflies (animated)

Some of the gaps in the fractal look like butterflies (or maybe large moths). And each butterfly is escorted by four smaller butterflies. Another fractal has gaps that look like bats escorted by smaller bats:

Fractal bats (animated)

Fractal bats (static)

Elsewhere other-posted:

Gif Me Lepidoptera — fractals using butterflies
Holey Trimmetry — more fractal crosses

Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1

Fractal cross #2

Fractal cross #3

Fractal cross #4

Fractal cross #5

Fractal cross #6

Fractal cross (animated)

Fractal cross (static)

If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1

Fractal Hex-Ring #2

Fractal Hex-Ring #3

Fractal Hex-Ring #4

Fractal Hex-Ring #5

Fractal Hex-Ring #6

Fractal Hex-Ring (animated)

Fractal Hex-Ring (static)

Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:

Nonomino

It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)

If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1

Fractal cross #2

Fractal cross #3

Fractal cross #4

Fractal cross #5

Fractal cross #6

Fractal cross (animated)

Fractal cross (static)

If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1

Fractal Hex-Ring #2

Fractal Hex-Ring #3

Fractal Hex-Ring #4

Fractal Hex-Ring #5

Fractal Hex-Ring (animated)

Fractal Hex-Ring (static)

Elsewhere other-available:

Square Routes Re-Revisited

This is a very simple fractal:

It has four orientations:

Any orientation can be turned into any other by a rotation of 90°, 180° or 270°, either clockwise or anticlockwise. If you mix orientations and rotations, you can create much more complex fractals. Here’s a selection of them:

Animated fractal

Static fractal

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Animated fractal

Static

Elsewhere other-posted:

Phrallic Frolics

It’s a classic of low literature:

There was a young man of Devizes
Whose balls were of different sizes:
The one was so small
’Twas no use at all;
But t’other won several prizes.

But what if he had been a young man with balls of different colours? This is a core question I want to interrogate issues around in terms of the narrative trajectory of this blog-post. Siriusly. But it’s not the keyliest core question. More corely keyly still, I want to ask what a fractal phallus might look like. Or a phrallus, for short. The narrative trajectory initializes with this fractal, which is known as a pentaflake (so-named from its resemblance to a snowflake):

Pentaflake — a pentagon-based fractal

It’s created by repeatedly replacing pentagons with six smaller pentagons, like this:

Pentaflake stage 0

Pentaflake stage 1

Pentaflake stage 2

Pentaflake stage 3

Pentaflake stage 3

Pentaflake stage 4

Pentaflake (animated)

Pentaflake (static)

This is another version of the pentaflake, missing the central pentagon of the six used in the standard pentaflake:

No-Center Pentaflake stage 0

No-Center Pentaflake stage 1

Stage 2

Stage 3

Stage 4

No-Center Pentaflake (animated)

No-Center Pentaflake (static #1)

No-Center Pentaflake (static #2)

The phrallus, or fractal phallus, begins with an incomplete version of the first stage of the pentaflake (note balls of different colours):

Phrallus stage 1

Phrallus stage 1 (monochrome)

Phrallus stage 2

Phrallus stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8

And there you have it: a fractal phallus, or phrallus. Here is an animated version:

Phrallus (animated)

Phrallus (static)

But the narrative trajectory is not over. The center of the phrallus can be rotated to yield mutant phralloi. Stage #1 of the mutants looks like this:

Phrallus (mutation #1)

Phrallus (mutation #2)

Phrallus (mutation #3)

Phrallus (mutation #4)

Phrallus (mutation #5)

Mutant phralloi (rotating)

Here are some animations of the mutant phralloi:

Phrallus (mutation #3) (animated)

Phrallus (mutation #5) (animated)

This mutation doesn’t position the pentagons in the usual way:

Phrallus (another upright version) (animated)

The static mutant phralloi look like this:

Phrallus (mutation #2)

Phrallus (mutation #3)

Phrallus (upright #2)

And if the mutant phralloi are combined in a single image, they rotate like this:

Mutant phralloi (rotating)

Coloured mutant rotating phralloi #1

Coloured mutant rotating phralloi #2

Square Routes Revisited

Take a square, divide it into four smaller squares, and discard the smaller square on the top right. Do the same to each of the subsquares, dividing it into four sub-subsquares, then discarding the one on the top right. And repeat with the sub-subsquares. And the sub-sub-squares. And the sub-sub-sub-squares. And so on. The result is a fractal like this:

Stage 1

Stage 2

Stage 3

Stage 4

Animated fractal

Final fractal (static)

It looks as though this procedure isn’t very fertile. But you can enrich it by rotating each of the subsquares in a different way, so that the discarded sub-subsquare is different. Here’s an example:

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Animated fractal

Final fractal (static)

Here are more examples of how rotating the subsquares in different ways produces different fractals:

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Animated fractal

Static fractal

Previously pre-posted:

Square Routes — first look at this kind of fractal

Corralling Chaos

“Down through the aether I saw the accursed earth turning, ever turning, with angry and tempestuous seas gnawing at wild desolate shores and dashing foam against the tottering towers of deserted cities.” — “The Crawling Chaos” (1921), Winifred Jackson and H. P. Lovecraft.

All the best people brood incessantly on the fact that a point inside a square jumping half-way towards a randomly chosen vertex will not create a fractal. Inside a triangle, yes: a fractal appears. Inside a pentagon too. But not inside a square:

Point jumping half-way towards a randomly chosen vertex

Instead, the interior of the square fills with random points: it crawls with chaos, you might say. However, fractals appear inside a square if the point is restricted in some way: banned from jumping towards a vertex twice in a row; banned from jumping towards the second-nearest vertex; and so on. Those restrictions are what might be called soft, because they take place in software (or in the brain of someone following the rule as a game or piece of performance art). Here’s what might be called a hard restriction that creates a fractal: the point cannot jump towards a randomly vertex if its jump passes over any part of the red upright cross:

Point cannot pass over red lines

I call this a barrier fractal. It’s obvious that the point cannot jump from one corner of the square towards the opposite corner, which creates bare space stretching from each vertex towards the tips of the upright cross. Less obvious is the way in which this bare space “cascades” into other parts of the square, creating a repeatedly branching and shrinking pattern.

When the barrier is a circle, a similar fractal appears:

If the point can also jump towards the center of the circle, this is what happens:

Now here’s an upright cross with a gap in the middle:

Here’s an upright cross when the point can also jump towards the center of the cross:

A slanted cross with a central attractor:

And a single horizontal stroke:

A slanted stroke — note pentagons:

Even if the barrier is small and set on an edge of the square, it affects the rest of the square:

A more attractive example of edge-affects-whole:

Circles away from the edges

Detail of previous image

Here the point can also jump towards the center of the square’s edges:

A more subtle barrier fractal uses the previous jumps of the point to restrict its next jump. For example, if the point cannot jump across the line created by its previous-but-one jump, it moves like this:

Jump can’t cross track of last-but-one jump (animated gif)

The fractal itself looks like this:

Rule: on jump #3, cannot jump across the line created by jump #1; on jump #4, cannot cross the line created by jump #2; and so on.

And this is the fractal if the point cannot jump across the line created by its previous-but-two jump:

Rule: on jump #4, cannot jump across the line created by jump #2; on jump #5, cannot cross the line created by jump #3; and so on