https://www.youtube.com/watch?v=EK32jo7i5LQ
This cool video explains how coordinate artifacts in the polar coordinate system, just happened to single out residual classes that demonstrate Dirichlet theorem.
Having experience of both recreational mathematics and formal mathematics, seeing things like how one can construct certain n roots if you have another tool besides compass and straightedge, as well how visualisations sometimes make the problem easier to approach, I am wondering about a more general question that is implicitly raised in this video:
It is clear that the mere act of drawing a picture of a given mathematical problem $q$ actually imposes the constraint of the paper (basically $\Bbb{R}^2$ with continuous curves given different interpretations) onto $q$. This groups mathematical concidences such as $\frac{22}{7} \approx \pi$ together in some way to produce artefacts in the diagram, which sometimes illuminates the outcome of the underlying theory.
My question is,
- Are there any studies done on the nature of mathematical coincidences in various contexts and presentations, such that there are some known fact on how each class of mathematical concidences are related to the geometric artefacts they introduced?
- Are there field of mathematics or parts in the usual domains of e.g. real analysis, that actually exploit these artefacts in order to prove theorems. How should we engineer a suitable presentation method or algorithm so that mathematical concidence will be grouped in such a way to maximally illuminate the structure of the underlying thereom to the problem?