Well, the thing is that you can define π in ways that don't have anything (at least on the surface) to do with geometry. For instance, the normal (Gaussian) distribution is:
and you could use that define π if you really wanted to.
(Now that I think of it, I think that the book about the guy who finds out he's God also has him change the normal distribution at one point, with the result that people start suffocating when all of the air in a room suddenly collects in one corner, etc.)
Alternatively (if somewhat artificially) you could define pi as follows:
π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
You can also take the route of defining the sine function from looking at ex for complex x, then finding its period (2π), which again has nothing obvious to do with geometry. (If some alien civilization did this, I imagine that some mathematician might have been surprised to discover that half the period of the sine function turns out to be exactly the ratio of a circle's circumference to its diameter!)
It's unclear to me whether these things are also supposed to change when people talk about changing π. I would expect not and that what's being talked about is the ratio of a circle's circumference to its diameter. Things like Brunner's mentioning that even if π has a different value it has to be irrational make it clear to me, though, that in at least some cases there's some confusion about what's what.
Re: different golden ratio
and you could use that define π if you really wanted to.
(Now that I think of it, I think that the book about the guy who finds out he's God also has him change the normal distribution at one point, with the result that people start suffocating when all of the air in a room suddenly collects in one corner, etc.)
Alternatively (if somewhat artificially) you could define pi as follows:
π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
You can also take the route of defining the sine function from looking at ex for complex x, then finding its period (2π), which again has nothing obvious to do with geometry. (If some alien civilization did this, I imagine that some mathematician might have been surprised to discover that half the period of the sine function turns out to be exactly the ratio of a circle's circumference to its diameter!)
It's unclear to me whether these things are also supposed to change when people talk about changing π. I would expect not and that what's being talked about is the ratio of a circle's circumference to its diameter. Things like Brunner's mentioning that even if π has a different value it has to be irrational make it clear to me, though, that in at least some cases there's some confusion about what's what.
I'm interested in reading that Egan story.