[WhitneyLand]

Yes, it's kind of hard to get more fundamental than waves and circles.

Although, as I was just about to say what math could you do without them on second thought, the answer seems to be quite a lot.

For example I couldn't say without researching it but how much did Gödel or Turing rely on them, at least for their most influential work?

That question will probably come back to bite hard given their footprints.

Another conjecture, most software developers will never have to use them extensively. Yes gaming, computer graphics, and I guess all signal processing would become suddenly more challenging but corporate IT and devops ought to be pretty safe.

Maybe it depends on a personal definition of fundamental. I think you could make an argument that Newton's results were more fundamental than Einstein's, however staggeringly less complete they might seem hundreds of years in the future.

[ISL]

Agreed -- I used the word "fundamental" to suggest that it emerges from pure mathematics. In that sense, Feigenbaum's constants are in the same class as \pi and e, which is distinct from the traditional "fundamental constants" of physics like c, \hbar, fundamental charge, etcetera.

The fundamental constants of physics have, as yet, no known mathematical origin. Feigenbaum's constants, on the other hand, emerge from pure mathematics and also appear in nature. That they are expected to be irrational, like \pi and e, only adds to the charm.

[jordigh]

Do we know the constant is irrational? I'd be surprised if we did.