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Has the aforementioned constant ever been used in any major proofs? Can it be expressed in terms of $e$, $\pi$, or both? Does it appear in any sort of geometric sense like $\pi$ does? Or is it just a kind of pretty number somebody thought would be nice to experiment with?

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2 Answers

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Or is it just a kind of pretty number somebody thought would be nice to experiment with?

This.

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More specifically, it is constructed in such a way that its (decimal) digits are easy to investigate. This allows to establish fairly easily that it is normal in its base. One might go as far saying it is a number constructed so that this works.

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For what it's worth I'm a software engineer and have ended up using Champernowne's constant (expressed in base 2) as a cheap stand-in for a random binary sequence.

For instance, to simulate grass growing on the floor of a dungeon, we can initialize the floor cells with Champernowne and then apply game of life. This has the advantage of being entirely deterministic. In functional programming contexts, you have to be really explicit when you want randomness since it consumes entropy and this counts as a side-effect.

Alternatively you end up having to weave a seed through everything, and when most of the methods don't need it it's a lot of extra bookkeeping. So computationally-simple methods to construct random "looking" binary sequences can prove quite helpful especially in a context where access to entropy is limited. It "looks" quite random but isn't nearly as expensive as a PRNG. Recall that the golden law of computer graphics states:

If it looks right, it is right.

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