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* pi = 3.14159… appears in analysis and by extension statistics, independent of geometry. So aliens in these other universes would know this value, they’d just

* pi = 3.14159… appears in analysis and by extension statistics, independent of geometry. So aliens in these other universes would know this value, they’d just have a different constant for circles. Since they wouldn’t use Greek letters anyway, we’d have to translate, and it would be a bit silly to equate their 3.757… with “pi” instead of their 3.14159… * Personal aside: Of course, whether 3.14… (pi), 6.28… (2pi) or even 0.785… (pi/4) should be the fundamental constant is debatable, and aliens might have different ideas about that. * The article introduces the concept of metrics to explain that there could be different circle constants in other universes. But arbitrary metrics don’t necessarily have linear scaling or translation invariance. You need stronger assumptions than a metric to meaningfully define a circle constant at all, like a normed vector space. AFAICT, all of the given examples are in fact normed vector spaces, not just metric spaces.

dylukes 3y ago
I don't find the first point surprising. (Our) pi is the one tied to the only metric where the unit circle is perfectly continuous, differentiable, etc. The 2-norm is very special for many reasons I won't enumerate... and it seems apropos that its corresponding constant (pi)... for relating a distance from a point (wlog 0,0) to the result of integrating a constant around the path those points occupy/form/consist in... would itself tend to be found more than others. Perhaps this is simply because without that continuity and differentiability everywhere of the corresponding path generated by the metric's unit circle, many other pieces would fall like dominoes. There is something uniquely central about a concise relation between a point, a distance, and a path.
PennRobotics 3y ago
https://tauday.com/tau-manifesto#table-quadratic_forms https://tauday.com/tau-manifesto#table-quadratic_forms (Not to sound all Buzzfeed-y, but Table 3 makes a lot of sense)
BlueTemplar 3y ago
Yes, and they actually keep using 2pi over and over in their examples.
aesch 3y ago
I wonder about the first point. As you explain in another comment the value of pi, 3.14159, can be derived from number theory alone but magically it plays a huge role in shaping the physical world we know. Would a different universe have a different number theory or is number theory something that is True regardless of the universe? What would an alternate number theory even look like?

dylukes 3y ago

I don't find the first point surprising. (Our) pi is the one tied to the only metric where the unit circle is perfectly continuous, differentiable, etc. The 2-norm is very special for many reasons I won't enumerate... and it seems apropos that its corresponding constant (pi)... for relating a distance from a point (wlog 0,0) to the result of integrating a constant around the path those points occupy/form/consist in... would itself tend to be found more than others. Perhaps this is simply because without that continuity and differentiability everywhere of the corresponding path generated by the metric's unit circle, many other pieces would fall like dominoes. There is something uniquely central about a concise relation between a point, a distance, and a path.

PennRobotics 3y ago

https://tauday.com/tau-manifesto#table-quadratic_forms https://tauday.com/tau-manifesto#table-quadratic_forms (Not to sound all Buzzfeed-y, but Table 3 makes a lot of sense)

BlueTemplar 3y ago

Yes, and they actually keep using 2pi over and over in their examples.

aesch 3y ago

I wonder about the first point. As you explain in another comment the value of pi, 3.14159, can be derived from number theory alone but magically it plays a huge role in shaping the physical world we know. Would a different universe have a different number theory or is number theory something that is True regardless of the universe? What would an alternate number theory even look like?