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Magic comes from shapes. For example, when you inscribe a regular octagram in a circle you get a heat effect, and the right straight line atop that summons a creature from beyond the mortal plane. Human interaction is not even required: the shapes themselves resolve into various magical effects. The laws of magi-physics just work that way.

That has some consequences. Patterns come about from mere chance, and in my world they produce magic. Piles of branches are found in the forest floor, and every now and then a subset of them might be in the right shape to produce a spell. There are many trees and spells may live for a long time, so due to the trees serving as infinite monkeys on typewriters, the forest is a place where you stand a decent chance of finding "wild magic".

Anthills and mole tunnels are other examples. There's less of them than forest branches but still you might find an unlucky subterranean mammal that dug a magic shape and had their home filled with rice pudding.

My question: where else may you find wild magic? Where might you find complex geometric shapes in nature, taking infinite monkeys into account?

Specifications:

  • A geometric shape is composed of (rough) lines made of one material, in a medium of one or more different materials. A bag of sand doesn't have any shapes because one particle isn't distinct from the rest; waterflows with varying salinity on the ocean floor also won't do. Marbling is barely enough to produce something measurable, and it won't summon creatures. The more distinct the materials the more potent the effect.
  • Stuff like patterns in sunflower seeds doesn't count, because the seeds define the vertices of a shape, and magic geometry is defined with the edges.
  • Disregard human activity. People are careful when bundling their firewood and have even begun laying out their village roads to form protection spells - but that is outside the question scope. This is only about "wild magic".
  • Assume regular earthly flora, fauna, and physics/geology. This effect has not been around for long enough to affect evolutionary development. The exception is that web-making spiders have gone extinct, for obvious reasons (many inscribed circles make an electricity spell). Bees and wasps lucked out on this; tessellated hexagons are too simple to make magic happen.
  • For the kinds of shapes that are magical, consider geometric mandala shapes (1, 2, 3, 4) - and certain subsets of those (like a pacman or a bowtie shape) and/or extensions (some extra lines atop). A bit complex for a random pile of branches, but again, infinite monkeys.
  • When a shape is more than half-way reached, the laws of probability bend and any remaining random contributions to the shape are much more likely to end up in the right locations. Nature wants magic shapes to happen.
  • Minimum size is 1 cm in diameter. Otherwise snowflakes would be a major issue. Furthermore, magic potency scales with physical size too (though less so than it does with medium contrast).
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    $\begingroup$ Would snowflakes or other naturally forming crystal structures count? What about "square waves" en.wikipedia.org/wiki/Cross_sea or cauliflower or the vascular systems and lungs of living creatures. What about river deltas? Seems like cool patterns might be everywhere. $\endgroup$
    – JonSG
    Sep 4 at 22:27
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    $\begingroup$ @JonSG I've just discounted snowflakes on account of their tiny size, but crystal structures, sure (as long as it is a shape in contrast with another medium). Never heard of square waves before, cool! But I'm going to say that they fall under the beehive rule, that they are just tessellations of a simple shape rather than one complex shape, and therefore not magical. River deltas and cauliflower aren't really geometrical; you can't readily construct them with compass and straight edge. $\endgroup$
    – KeizerHarm
    Sep 4 at 22:30
  • $\begingroup$ Lava can create octagonal basalt formations, sea shells can be huge and geometric, some turtles and tortoises have geometric patterns in their segmented shells $\endgroup$ Sep 4 at 23:27
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    $\begingroup$ Oh and in fact, a lot of flowers and plants are in fact natural mandalas $\endgroup$ Sep 4 at 23:32
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    $\begingroup$ I think the idea is cool, but you seem to have a moving target. I think it rather unlikely that a random pile of branches would make shapes that would be described as mandalas that are typically very intricate and precise (compass and ruler as you suggest) yet exclude (what I think are more impressive) natural pseudo-fractals like cauliflower, deltas, seashells or natural patterns like clouds or leopards and zebras. $\endgroup$
    – JonSG
    Sep 4 at 23:34
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Crystals

Crystals naturally form many geometrical shapes, often very smoothly. That's due to the way their molecules arrange in a lattice.

This is a picture from the world's largest crystals, in the Naica Cave in Mexico:

A man sitting among the world's largest crystals

Starfish

Some of them are pretty geometrical.

A simple starfish

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    $\begingroup$ And that's why caves have magical creatures in them :) Regarding eggs: single circles are not complex enough. $\endgroup$
    – KeizerHarm
    Sep 4 at 22:55
  • $\begingroup$ @KeizerHarm about circles, got it, I removed them from the answer. $\endgroup$ Sep 4 at 23:04
  • $\begingroup$ It applies to the feces too, to be honest. I'm less looking for simple shapes that are perfectly regular in nature, and more looking for complex shapes that might spontaneously arise if one gets really lucky (like a pile of tree branches landing in a pentagram). Starfish could be interesting - they might blow up if one of them goes to sit in the dead centre of a round anemone. $\endgroup$
    – KeizerHarm
    Sep 4 at 23:10
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    $\begingroup$ That's just many repetitions of the same simple shape. I'm not looking for patterns, I'm looking for complex geometric shapes - think mandalas. $\endgroup$
    – KeizerHarm
    Sep 4 at 23:18
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    $\begingroup$ @KeizerHarm What about basket stars? $\endgroup$ Sep 5 at 1:35
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Radial fauna is most commonly found in sea life. I encourage you to check out the gorgeous Kunstformen der Natur for gloriously etched examples. Many of these are microscopic in scale (i.e. radiolarians, diatoms) but larger radially symmetrical organisms or multi-organism structures exist, like jellyfish and corals.

https://upload.wikimedia.org/wikipedia/commons/b/bf/Haeckel_Diatomea_4.jpg

The smaller the structures, the closer they tend to resemble geometric shapes exactly, so you may want to think about scaling from a three-way perspective of size, precision and total numbers (e.g. a sea surface of millions of microscopic but geometrically perfect plankton may generate a sort of fuzzy, undirected but perhaps powerful magic field).

To get powerful, directed and purposeful geometric magic you want radial symmetry, pattern-generating abilities and intelligent will, which brings us to…

Magic-wielding octopodes

The octopus body plan is already eight-fold symmetric. Many octopuses (or your favourite plural) have the ability to change the pattern of their skin with exceptional control and precision, which they use for feats of camouflage that, as far as I’m concerned, already count as magic. They are also very, very clever. By shaping linear designs onto their arms, they would learn to generate magic patterns for hunting, protection etc. (Or potentially just to amuse themselves! Why not?) This doesn’t need anything evolutionarily beyond what they already have, and their learning abilities are already perfect to get them to experiment by trial and error (I seem to understand that your magic system is weaker for incomplete or imperfect forms, which would be great for experimenting as you could gradually sharpen your design and stop if you feel you’re catching fire or something)

Bonus: magic toadstools

Another group of radially symmetrical organisms, land-based this time, are the fruiting bodies of fungi. I’m not an expert in the least but I think, generally, extant mushrooms would be somewhat too simple for your requirements (mostly they have a pattern of straight radial gills underneath, and/or hexagonal textures on top like the bridal veil stinkhorn) but with a bit of selective pressure they may easily develop patterns for thaumaluminescence or spore-ejecting magic.

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Melting ice on rivers or drying mud mostly form irregular shapes, but could by chance form complex patterns.

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