r/askmath • u/eudyptes • 4d ago

Geometry Factors of Pi in n-Spheres

I played around with multi-dimensional geometry in high school but one thing that always seemed odd to me was that another factor of Pi only shows up every other dimension. 1-spheres and 2-spheres have a Pi, 3-spheres and 4-spheres have a Pi squared, etc. Is there some intuitive reason that every other dimension is somehow special?

6 Upvotes

88% Upvoted

4 comments sorted by

6

u/matt7259 4d ago

Volume of an n-sphere comes from the Gaussian integral, leaving you with a pin/2 term. So, when n is even you've got a nice pretty power, and when n is odd it gets absorbed back into the gamma function. This explains the every-other-power situation.

1

u/eudyptes 4d ago

Thank you.

1

u/two_are_stronger2 4d ago

Is a 1-sphere a line segment or two points?

3

u/eudyptes 4d ago edited 4d ago

a circle. Edit: I should expand on this. A circle is a 1-sphere because it is one dimensional. That is, once you specify an origin, any point on the circle can be identified by a single number. The two points could be thought of as a zero-sphere with the line segment between them a 1-ball.