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

by in Fractals, Geometry, Mathematics, Pentagons, Polygons, Squares, Triangles and tagged , , , , , , , , | Leave a comment

Suppose you trace all possible routes followed by a point inside a triangle jumping halfway towards one or another of the three vertices of the triangle. If you mark each jump, you get a famous geometrical shape called the Sierpiński triangle (or Sierpiński sieve).

Sierpiński triangle found by tracing all possible routes for a point jumping halfway towards the vertices of a triangle

The Sierpiński triangle is a fractal, because it contains copies of itself at smaller and smaller scales. Now try the same thing with a square. If you trace all possible routes followed by a point inside a square jumping halfway towards one or another of the four vertices of the square, you don’t get an obvious fractal. Instead, the interior of the square fills steadily (and will eventually be completely solid):

Routes of a point jumping halfway towards vertices of a square

Try a variant. If the point is banned from jumping towards the same vertex twice or more in a row, the routes trace out a fractal that looks like this:

Ban on choosing same vertex twice or more in a row

If the point is banned from jumping towards the vertex one place anti-clockwise of the vertex it’s just jumped towards, you get a fractal like this:

Ban on jumping towards vertex one place anti-clockwise of previously chosen vertex

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

Ban on jumping towards the vertex diagonally opposite of the previously chosen vertex

That ban is equivalent to banning the point from jumping from the vertex diagonally opposite to the vertex it’s just jumped towards. Finally, here’s the fractal created when you ban the point from jumping towards the vertex one place clockwise of the vertex it’s just jumped towards:

Ban on jumping towards vertex one place clockwise of previously chosen vertex

As you can see, the fractal is a mirror-image of the one-place-anti-clockwise-ban fractal.

I discovered the ban-construction of those fractals more than twenty years ago. Then I found that I was re-discovering the same fractals when I looked at what first seemed like completely different ways of constructing fractals. There are lots of different routes to the same result. I’ve recently discovered yet another route. Let’s try what seems like an entirely different way of constructing fractals. Take a square and erect four new half-sized squares, sq1, sq2, sq3, sq4, on each corner. Then erect three more quarter-sized squares on the outward facing corners of sq1, sq2, sq3 and sq4. Carry on doing that and see what happens at the end when you remove all the previous stages of construction:

Animation of the new construction

Animation in black-and-white

It’s the T-square fractal again. Now try rotating the squares you add at stage 3 and see what happens (the rotation means that two new squares are added on adjacent outward-facing corners and one new square on the inward-facing corner):

Animation of the construction

It’s the one-place-clockwise-ban fractal again. Now try rotating the squares two places, so that two new squares are added on diagonally opposite outward-facing corners and one new square on the inward-facing corner:

Animation of the construction

It’s the same-vertex-ban fractal again. Finally, rotate squares one place more:

Animation of the construction

It’s the one-place-clockwise-ban fractal again. And this method isn’t confined to squares. Here’s what happens when you add 5/8th-sized triangles to the corners of triangles:

Animation of the construction

And here’s what happens when you add 5/13th-sized pentagons to the corners of pentagons:

Animation of the construction

Finally, here’s a variant on that pentagonal fractal (adding two rather than four pentagons at stage 3 and higher):

Animation of the construction

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

Sóccrates Says…

by in Language, Quotations, Translation and tagged , , , , , , , , , , , , , | Leave a comment

“A beleza vem primeiro. A vitória é secundária. O que importa é a alegria.” — Sócrates, o futebolista brasileiro
• “Beauty comes first. Victory is secondary. What matters is joy.” — Brazilian footballer Sócrates

I’ve also found the quote as:

“A beleza está primeiro. A vitória é secundária. O que é interessa é o prazer.”
• “Beauty comes first. Victory is secondary. What matters is pleasure.”

Reflet de Robert

by in Literature, Psychology, Quotations, Robert de Montesquiou and tagged , , , , , , , , , , , | Leave a comment

[I]t was hard to pierce Robert de Montesquiou’s carapace — and he wouldn’t have wanted you to. He was perhaps at heart a melancholic: he liked to say that his mother had “given me the sad present of life”. His restlessness and furious inquisitiveness might have been a response to this. He was vain without being especially self-reflective, one of those who, rather than look inside to discover who they are, prefer to see themselves in the reflections that come back from others. — Julian Barnes, The Man in the Red Coat (2019), pp. 192-3

Elsewhere Other-Accessible…

• Portait of a Peacock — Cornelia Otis Skinner’s essay on Montesquiou
• Le Paon dans les Pyrénées — review of Barnes’ The Man in the Red Coat

Spiral Artefact #2

by in Arithmetic, Integer Sequences, Mathematics, Ulam Spirals and tagged , , , , , , , , , | Leave a comment

Why stop at primes? Those are the numbers the Ulam spiral is usually used for. You get a grid of square blocks, then move outward from the middle of the grid in a spiral, counting as you go. If the count matches a prime, you fill the block in. The first block is 1. Not filled. The second block is 2, which is prime. So the block is filled. The third block is 3, which is prime. Filled again. And so on. In the end, the Ulam spiral for primes looks like this:

The Ulam spiral of prime numbers

But why stop at primes? If you change the fill-test, you get different patterns. I’ve recently tried a test based on how many ways a number can be represented as the sum of consecutive integers. For example, 5, 208 and 536 can be represented in only one way:

5 = 2+3
208 = 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22
536 = sum(26..41) = 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41

Let’s use “runsum” to mean a sum of consecutive integers. If the function runsum(n) returns the count of runsums for n, then runsum(5) = runsum(208) = runsum(536) = 1. Here are spirals for runsum(n) = 1:

A spiral for runsum(n) = 1, i.e. numbers that are the sum of consecutive integers in only one way

runsum(n) = 1 (higher resolution)

runsum(n) = 1 (higher resolution still)

Now try runsum(n) = 2, i.e. numbers that are the sum of consecutive integers in exactly two ways:

A spiral for runsum(n) = 2

runsum(n) = 2 (hi-res #1)

runsum(n) = 2 (hi-res #2)

runsum(n) = 2 (hi-res #3)

Why do most of the numbers fall on a diagonal? I don’t know, but I know that the diagonal represents square numbers:

9 = sum(4..5) = sum(2..4)
25 = sum(12..13) = sum(3..7)
36 = sum(11..13) = sum(1..8)
49 = sum(24..25) = sum(4..10)

Now try runsum(n) = 3:

A spiral for runsum(n) = 3

runsum(n) = 3 (hi-res)

It’s a densely packed spiral, unlike the spiral for runsum(n) = 4:

A spiral for runsum(n) = 4

runsum(n) = 4 (hi-res)

Like the spiral for runsum(n) = 2, the numbers are disproportionately falling on the diagonal of square numbers:

81 = 9^2 = sum(40..41) = sum(26..28) = sum(11..16) = sum(5..13)
324 = 18^2 = sum(107..109) = sum(37..44) = sum(32..40) = sum(2..25)
2500 = 50^2 = sum(498..502) = sum(309..316) = sum(88..112) = sum(43..82)

Here are spirals for runsum(n) = 5:

A spiral for runsum(n) = 5 (note patterns in green)

runsum(n) = 5 (hi-res #1)

runsum(n) = 5 (hi-res #2)

There are two interesting patterns in the spiral, marked in green above and enlarged below:

Pattern #1 in spiral for runsum(n) = 5

Pattern #2 in spiral for runsum(n) = 5

Are the patterns merely artefacts or does one or both represent something mathematically significant? I don’t know.

More spirals:

A spiral for runsum(n) = 6

A spiral for runsum(n) = 7

runsum(n) = 7 (hi-res)

A spiral for runsum(n) = 8

runsum(n) = 8 (hi-res #1)

runsum(n) = 8 (hi-res #2)

Numbers in the spiral for runsum(n) = 8 are again falling disproportionately on the diagonal of square numbers. Here’s one of those squares:

441 = 21^2 = sum(220..221) = sum(146..148) = sum(71..76) = sum(60..66) = sum(45..53) = sum(25..38) = sum(16..33) = sum(11..31)

Previously Pre-Posted…

• Spiral Artefact #1 — a look at patterns in spirals with different tests

Performativizing Papyrocentricity #77

by in Book Reviews and tagged , , , , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents…

• Bestial at the Festival – The Festival, D.M. Mitchell (2021)

• Linkin’ Lawyer – Resurrection Walk, Michael Connelly (2023)

• Mini Miki – Maximal Mikita: The Mostly Morbid Memoirs of Mikita Brottman, Mikita Brottman (2024)

• Spider Guider – Britain’s Spiders: A Field Guide, Lawrence Bee, Geoff Oxford and Helen Smith (2020)

• Gnostalgie du Perdu – Gnosticism: An Anthology, ed. Robert M. Grant (1961)

Or Read a Review at Random: RaRaR

Chevaleurs Oniriques

by in French, Language, Literature, Quotations, Translation and tagged , , , , , , , , | Leave a comment

« Les valeurs oniriques l’ont définitivement emporté sur les autres et je demande à ce qu’on tienne pour un crétin celui qui se refuserait encore, par exemple, à voir un cheval galoper sur une tomate. » André Breton (1896-1966)
• “Oneiric values have definitely won out over the others, and I maintain that anyone who still refuses to see, for instance, a horse galloping on a tomato, must be an idiot.” — André Breton, viâ Soluble Fish by Incunabula Media