Dan Piponi spotted an amazing pattern:

> You left out the row for d=1. It's a row of integers too. So the dimensions that give associative division algebras give integers. I bet there's an algebraic argument that works only for those cases.

Amazing! The unit spheres in \\(d = 1, 2, \\) and \\(4\\) dimensions are the only spheres that are also _groups_. They are unit real numbers, the unit complex numbers and the unit quaternions.

For some reason the even moments of the distance of two randomly chosen points on the sphere are _integers_ in these three cases! Why?

> You left out the row for d=1. It's a row of integers too. So the dimensions that give associative division algebras give integers. I bet there's an algebraic argument that works only for those cases.

Amazing! The unit spheres in \\(d = 1, 2, \\) and \\(4\\) dimensions are the only spheres that are also _groups_. They are unit real numbers, the unit complex numbers and the unit quaternions.

For some reason the even moments of the distance of two randomly chosen points on the sphere are _integers_ in these three cases! Why?