Pedal to the Medal

“Once, in a contest with a rival, he painted a blue curve on a huge sheet of paper. Then he dipped the feet of a chicken in red paint and persuaded the bird to walk all over the paper. The resulting image, he said, represented the Tatsuta river with red maple leaves floating in it. The judge gave him the prize.” — The Japanese artist Katsushika Hokusai (c. 1760-1849) described in Thomas W. Hodgkinson’s and Hubert van den Bergh’s How to Sound Cultured (2015).

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”.

Gold ’Lusk

Flat periwinkle, Littorina obtusa (Linnaeus, 1758)*


Previously pre-posted:

Walking Winkle

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