“The fact is, that, of all God’s gifts to the sight of man, color is the holiest, the most divine, the most solemn.” — John Ruskin, The Stones of Venice, Vol. 2 (1853)
Trim Pickings
Here is an equilateral triangle divided into nine smaller equilateral triangles:
Rep-9 equilateral triangle
The triangle is a rep-tile — it’s tiled with repeating copies of itself. In this case, it’s a rep-9 triangle. Each of the nine smaller triangles can obviously be divided in their turn:
Rep-81 equilateral triangle
Rep-729 equilateral triangle
Rep-729 equilateral triangle again
Rep-6561 equilateral triangle
Rep-9 triangle repeatedly subdividing (animated)
How try trimming the original rep-9 triangle, picking one of the trimmings, and repeating in finer detail. If you choose six triangles in this pattern, you can create a symmetrical braided fractal:
Triangular fractal stage 1
Triangular fractal #2
Triangular fractal #3
Triangular fractal #3 (cleaning up)
Triangular fractal #3 (cleaning up more)
Triangular fractal #4
Triangular fractal #5
Triangular fractal #6
Triangular fractal (animated)
But this fractal using a three-triangle trim-picking isn’t symmetrical:
Trim-picking #1
Trim-picking #2
Trim-picking #3
Trim-picking #4
Trim-picking #5
To make it symmetric, you have to delay the trim, using the full rep-9 trim for the first stage:
Delayed trim-picking #1
Delayed trim-picking #2
Delayed trim-picking #3
Delayed trim-picking #4
Delayed trim-picking #5
Delayed trim-picking #6 (with first two stages as rep-9)
Delayed trim-picking (animated)
Here are some more delayed trim-pickings used to created symmetrical patterns:
Bored Bard
Pol. How say you by that? Still harping on my daughter: yet he knew me not at first; he said I was a Fishmonger: he is farre gone, farre gone: and truly in my youth, I suffred much extreamity for loue: very neere this. Ile speake to him againe. What do you read my Lord?
Ham. Words, words, words. — Hamlet (c. 1600), Act 2, Scene 2
Polykoch (Kontinued)
In “Polykoch!”, I looked at variants on the famous Koch snowflake, which is created by erecting new triangles on the sides of an equilaternal triangle, like this:
Koch snowflake #1
Koch snowflake #2
Koch snowflake #3
Koch snowflake #4
Koch snowflake #5
Koch snowflake #6
Koch snowflake #7
Koch snowflake (animated)
One variant is simple: the new triangles move inward rather than outward:
Inverted Koch snowflake #1
Inverted Koch snowflake #2
Inverted Koch snowflake #3
Inverted Koch snowflake #4
Inverted Koch snowflake #5
Inverted Koch snowflake #6
Inverted Koch snowflake #7
Inverted Koch snowflake (animated)
Or you can alternate between moving the new triangles inward and outward. When they always move outward and have sides 1/5 the length of the sides of the original triangle, the snowflake looks like this:
When they move inward, then always outward, the snowflake looks like this:
And so on:
Now here’s a Koch square with its new squares moving inward:
Inverted Koch square #1
Inverted Koch square #2
Inverted Koch square #3
Inverted Koch square #4
Inverted Koch square #5
Inverted Koch square #6
Inverted Koch square (animated)
And here’s a pentagon with squares moving inwards on its sides:
Pentagon with squares #1
Pentagon with squares #2
Pentagon with squares #3
Pentagon with squares #4
Pentagon with squares #5
Pentagon with squares #6
Pentagon with squares (animated)
And finally, an octagon with hexagons on its sides. First the hexagons move outward, then inward, then outward, then inward, then outward:
Octagon with hexagons #1
Octagon with hexagons #2
Octagon with hexagons #3
Octagon with hexagons #4
Octagon with hexagons #5
Octagon with hexagons (animated)
Polykoch!
This is how you form the famous Koch snowflake, in which at each stage you erect a new triangle on the middle of each line whose sides are 1/3 the length of the line:
Koch snowflake #1
Koch snowflake #2
Koch snowflake #3
Koch snowflake #4
Koch snowflake #5
Koch snowflake #6
Koch snowflake #7
Koch snowflake (animated)
Here’s a variant of the Koch snowflake, with new mid-triangles whose sides are 1/2 the length of the lines:
Koch snowflake (1/2 side) #1
Koch snowflake (1/2 side) #2
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Stage #8
Koch snowflake (1/2 side) (animated)
But why stop at triangles? This is a Koch square, in which at each stage you erect a new 1/3 square on the middle of each line:
Koch square #1
Koch square #2
Koch square #3
Koch square #4
Koch square #5
Koch square #6
Koch square (animated)
And a Koch pentagon, in which at each stage you erect a pentagon on the middle of each line whose sides are 1 – (1/φ^2 * 2) = 0·236067977… the length of the line (I used 55/144 as an approximation of 1/φ^2):
Koch pentagon (side 55/144) #1
Koch pentagon #2
Koch pentagon #3
Koch pentagon #4
Koch pentagon #5
Koch pentagon #6
Koch pentagon (animated)
In this close-up, you can see how precisely the sprouting pentagons kiss at each stage:
Koch pentagon (close-up) #1
Koch pentagon (close-up) #2
Koch pentagon (close-up) #3
Koch pentagon (close-up) #4
Koch pentagon (close-up) #5
Koch pentagon (close-up) #6
Koch pentagon (close-up) (animated)
Hamble On
HAMBLEDON (n.)
The sound of a single-engined aircraft flying by, heard whilst lying in a summer field in England, which somehow concentrates the silence and sense of space and timelessness and leaves one with a profound feeling of something or other. — The Meaning of Liff, Douglas Adams and John Lloyd (1983)
Elsewhere Other-Accessible
• The Meaning of Liff — full text
• The Meaning of Liff — at Wikipedia
Unda the Spell
A big-wave surfer braves a big wave (image from The Daily Mail, 17xi17)
(click for larger)
Alt-Writer
“I’ve found a place halfway up the churchyard, near enough to the church to be aware of, in a spiritual sense, matins on Sunday morning, but also to be within reach of, in a temporal way, orgies on Saturday nights in The Woolpack. And alternating between the temporal and the spiritual is the way I wish to spend what eternity is left to me.” — Laurie Lee, Down in the Valley: A Writer’s Landscape (2019)
The Chimera Eye
Chimerical colors (from Wikipedia)
(click for larger)
Glowing Troppo
pombero, M. Á guar. En la tradición popular, duende imaginario de quien se dice que protege a los pájaros y a los cocuyos y rapta a niños que persiguen.
cocuyo, M. Insecto coleóptero de la América tropical, de unos tres centimétros de longitud, oblongo, pardo y con dos manchas amarillentas a los lados de tórax, por las cuales despide de noche una luz azulada bastante viva. — Diccionario esencial de la lengua española (2006)
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 