# 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:

# Living Culler

When you replace a square with four smaller squares, each a quarter the size of the original, the smaller squares occupy the same area, because 4 * ¼ = 1. If you discard one sub-square, then divide each of the three remaining sub-squares into four sub-sub-square, discard one sub-sub-quare and repeat, you create fractals like those I looked at in Squaring and Paring. The fractals stay within a fixed boundary.

# ⇓

Square replaced with four smaller squares, each ¼th the size of the original

Animated fractal

Static fractal

This time I want to look at a slightly different process. Replace a square with nine smaller squares each a quarter the size of the original. Now the sub-squares occupy a larger area than the original, because 9 * ¼ = 2¼. If you discard — or cull — sub-squares and repeat, the resultant fractal grows beyond the original boundary. Indeed, sub-squares start to overlap, so you can use colours to represent how often a particular pixel has been covered with a square. Here is an example of this process in action:

# ⇓

Square replaced with nine smaller squares, each ¼th the size of the original

Animated fractal

Static fractal #1

Static fractal #2

Here are the individual stages of a more complex fractal that uses the second process:

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8

Stage 9 (compare Fingering the Frigit and Performativizing the Polygonic)

Stage 10

Animated version

Static version #1

Static version #2

And here are some more of the fractals you can create in a similar way:

Static version #1

Static version #2

Static version #2

Static version #2

Static version #3

Various fractals in an animated gif

# Squaring and Paring

Squares are often thought to be the most boring of all shapes. Yet every square holds a stunning secret – something that in legend prompted a mathematical cult to murder a traitor. If each side of a square is one unit long, how long is the square’s diagonal, that is, the line from one corner to the opposite corner?

By Pythagoras’ theorem, the answer is this:

• x^2 = 1^2 + 1^2
• x^2 = 2
• x = √2

But what is √2? Pythagoras and his followers thought that all numbers could be represented as either whole numbers or ratios of whole numbers. To their dismay, so it’s said, they discovered that they were wrong. √2 is an irrational number – it can’t be represented as a ratio. In modern notation, it’s an infinitely decimal that never repeats:

• √2 = 1·414213562373095048801688724209698…

A modern story, unattested in ancient records, says that the irrationality of √2 was a closely guarded secret in the Pythagorean cult. When Hippasus of Metapontum betrayed the secret, he was drowned at sea by enraged fellow cultists. Apocryphal or not, the story shows that squares aren’t so boring after all.

Nor are they boring when they’re caught in the fract. Divide one square into nine smaller copies of itself:

Discard three of the copies like this:

Stage 1
Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Then do the same to each of the sub-squares:

Stage 1

And repeat:

Stage 3

Stage 4

Stage 5

Stage 6

The result is a fractal of endlessly subdividing contingent hexagons:

Animated vesion

Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Here are a few more of the fractals you can create by squaring and paring:

Retain squares 1, 3, 5, 7, 9 (reading left-to-right, bottom-to-top)

Retain squares 2, 4, 5, 6, 8

Retain squares 1, 2, 4, 5, 6, 8, 9

Retain squares 1, 4, 6, 7, 10, 11, 13, 16

Retain squares 1, 3, 6, 7, 8, 9, 10, 11, 14, 16

Retain squares 2, 3, 5, 6, 8, 9, 11, 12, 14, 15

Retain squares 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25

Retain squares 1, 3, 7, 8, 11, 12, 14, 15, 18, 19, 23, 25

Retain squares 1, 5, 7, 8, 9, 12, 14, 17, 18, 19, 21, 25

Retain squares 2, 3, 4, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 22, 23, 24

Retain squares 1, 2, 5, 6, 7, 9, 13, 17, 19, 20, 21, 24, 25

# He Say, He Sigh, He Sow #44

H. Rider Haggard describes fractals:

Out of the vast main aisle there opened here and there smaller caves, exactly, Sir Henry said, as chapels open out of great cathedrals. Some were large, but one or two — and this is a wonderful instance of how nature carries out her handiwork by the same unvarying laws, utterly irrespective of size — were tiny. One little nook, for instance, was no larger than an unusually big doll’s house, and yet it might have been a model for the whole place, for the water dropped, tiny icicles hung, and spar columns were forming in just the same way. — King Solomon’s Mines, 1885, ch. XVI, “The Place of Death”

# 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