Ordinary planar and three-dimensional Euclidean geometry extends naturally to $n$ coordinates.

A point in $\mathbb R^n$ is

$$\mathbf x=(x_1,\ldots,x_n),$$

and the Euclidean distance between two points is

$$d(\mathbf x,\mathbf y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}.$$

Balls and spheres

The $n$-dimensional ball of radius $R$ centered at the origin is

$$B_n(R)=\lbrace\mathbf x\in\mathbb R^n:\lVert\mathbf x\rVert\le R\rbrace.$$

Its boundary,

$$\lVert\mathbf x\rVert=R,$$

is the $(n-1)$-dimensional sphere.

This terminology differs slightly from everyday language, where the solid ball and its surface are often both called a sphere.

Familiar low-dimensional cases

For $n=1$, the ball is the interval $[-R,R]$.

For $n=2$, it is a disk.

For $n=3$, it is the ordinary solid ball.

The same equations continue to make sense when $n$ is larger even though the geometry can no longer be visualized directly.

Higher-dimensional geometry is therefore not based on new distance rules; the surprising behavior comes from how familiar Euclidean rules interact with many independent directions at once.