Sphere

Definition

As a subset of Euclidean space

The unit $n$-sphere $S^n$ is defined as the subset of Euclidean space ${\mathbb{R}}^{n+1}$ comprising those points whose distance from the origin is $1$.

$S^n=\{(x_0,x_1,\dots ,x_n)\mid x_0^2+x_1^2+\dots +x_n^2=1\}$

Particular cases

$n$	sphere $S^n$
0	$S^0$ -- discrete two-point space
1	circle
2	2-sphere
3	3-sphere