r/mathpics

Inspired by Numberphile’s “Red & Black Knights” problem: players take turns placing pieces along a square spiral, and each new piece must respect the threat relations already created on the board.

I expanded the original knight-only system into a more general field of chess and fairy-chess leapers: knight, fers, vazir, camel, zebra, antelope, eland, satrap, aspbad, spehbed, marzban (following Jonas Karlsson’s generalized Stendhal variants). Each color becomes a player, each player has its own movement logic, and the image grows as a record of where conflict allows matter to exist.

For this series I used two modes: "symmetric" and "random rivalry".
Random rivalry creates a seeded matrix of antagonisms between colors: some players threaten each other, some ignore each other, and self-antagonism can also appear. Symmetric makes the rule stricter: a piece is rejected both if it would be threatened by an existing rival and if placing it would threaten an existing rival.

What moves me most is how much pattern emerges from such simple rules. A spiral, a few leaper moves, and a rivalry table start producing territories, borders, blooms, scars, almost like little frozen wars. I’m honestly ecstatic watching these structures appear from logic that still feels small enough to hold in my head.

Sources / inspiration:

• Red & Black Knights, Numberphile
• Jonas Karlsson’s Stendhal gallery
• OEIS A392177

Made with p5.js and GPT-5.5

IG: u/outertales ♥^(!)

Remove the center. Repeat forever.

The Sierpiński carpet starts with a single square and, with one recursive rule, punches an infinite number of holes into it.

No cut. No trick.

The loop is perfect because the mathematics demand it.

What you're looking at is real, just not from this dimension.

First experiments using FFT techniques to generate audio based on generative geometric structure. This example is a rendering as a polyphonic heavy metal nocturne, played pizzicato and tuned to an equal temperament Lochrian scale using a root frequency of 73.4Hz (D2). Each pixel is an oscillator and the surrounding pixels define its harmonic content. Main image is a section of the generative function and brighter centre section shows part of the the sonification data. Multi channel capability is obtained by a slight offset of the data for each channel. For example left data is plus two pixels offset on the y and right data is minus two pixels. Centre channel has no y offset and the final audio is a mix, right = 20% centre plus 80% right. This facilitates easy construction of 64 even 128 channel sound spaces. A 5.1 192 kHz audio version of this example can be located on this link at FreeSound.

The second figure originated with the goodly Stefan Felsner, & is actually the point–line dual of the figure @

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my previous post

https://www.reddit.com/r/mathpics/s/wwQ3Rxen5H

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. The rest are of a more technical nature – ancillary to the various reasonings adduced in the treatise the figures are from ...

... which is

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A Pseudoline Counterexample to the Strong Dirac Conjecture

by

Ben D Lund & George B Purdy & Justin W Smith

https://arxiv.org/pdf/1802.08015

¡¡ may download without prompting – PDF document – 139‧37㎅ !!

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. ＡＮＮＯＴＡＴＩＯＮＳ ＲＥＳＰＥＣＴＩＶＥＬＹ

•••••••••••••••••••••••••••••••••••••••••••••••••

Figure 4: The arrangement for j = 1, containing 3(6j + 2) + 1 = 25 pseudolines. Each pseudoline is incident to at most 10 vertices.

Figure 1: The dual of Felsner’s arrangement with 6k + 7 = 31 lines (including the line at infinity) and no line incident to more than 3k + 2 = 14 points of intersection.

Figure 2: A single wedge from Felsner’s arrangement.

Figure 3: The wedge for j = 1, the base case for our induction.

Figure 5: The wedge for j = 2.

In the No-3-in-line problem, no three points are in a line, in any direction or any slope.

"On 25th June 2026 Marijn Heule found a new solution with record grid size n=72 in the rot4 symmetry class."

MathWorld. Uni-bielefeld. Wikipedia.

... ie an arrangement, for each n , of the smallest possible № of points on an n×n grid such that adding a further point will necessarily induce some three in a line.

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By the goodly Robert Israel , from a reference found @

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Online Encyclopedia of Integer Sequences (OEIS) — A277433 Martin Gardner's minimal no-3-in-a-line problem, all slopes version.

https://oeis.org/search?q=A277433&language=english&go=Search

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Robert Israel, Examples for n <= 12 (provably optimal for n <= 10).

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I posted this earlier, & missed-off the last (n=12) one! 🙄

😆🤣

Poorly drawn by me