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Usually, in mathematics, diagrams appear to be pretty simple, even if they do look cool. Occasionally though, there are the instances where the diagrams are counterintuitive to what one should think they look like. Escape time fractals are good examples of diagrams which are counterintuitive because they have so much fascinating and unreasonable detail; they look nothing like one would expect.

When one looks to a mathematical field to produce images, they should expect mathematical images. However, even though mathematics is (arguably philosophically) the study of order, these images may sometimes bring chaos. Some examples of this are unreasonably detailed diagrams.

Examples:

  1. Escape time fractals (mandelbrot set, julia set, tetration fractal, tetration argument fractal...)
  2. The computational complexity hierarchy
  3. A lot of the commutative diagrams in abstract algebra
  4. The LCA diagrams in set theory

QUESTION: What are some other unreasonably chaotic yet intriguing diagrams in mathematics?

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  • $\begingroup$ Not quite a diagram but related to complexity and unexpectedness: the abelian sandpile model (see: en.wikipedia.org/wiki/Abelian_sandpile_model) and more generally, cellular automata). $\endgroup$ – orion2112 Sep 2 '18 at 5:00
  • $\begingroup$ That is quite interesting, and I would consider it a diagram. Just a very large scale version. In fact, I'd consider most chaotic dynamical systems to be diagrams in themselves. $\endgroup$ – Keith Millar Sep 2 '18 at 5:04
  • $\begingroup$ Just for clarification, commutative diagrams appear in many places beyond abstract algebra (e.g. differential geometry, (algebraic) toopology, ...) $\endgroup$ – Max Sep 2 '18 at 7:48

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