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I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.

Update: you may wish to look also at this question, in particular at the paper mentioned by Thomas Riepe.

I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.

Update: you may wish to look also at this question, in particular at the paper mentioned by Thomas Riepe.

I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.

I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.

Update: you may wish to look also at this question, in particular at the paper mentioned by Thomas Riepe.

I'm not an expert and would gladly learn about any updates but, as far as I know, the problem of universality of the Dynkin digrams appearing in rather remote areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.

I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the A-D-E problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution.