We all know that the area of a disk — the interior of a circle — is where is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is .

The unit disk and ball have equations in two and three variables:

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These are special cases of an -ball, the region of -dimensional space comprising all points whose distance from the origin is less than 1:

.

or . For , we call a **hyperball**. A 1-ball is just a line segment. A 2-ball is a disk and a 3-ball is the interior of a sphere. The surface of the n-ball is the (n-1)-sphere

**Volumes and Hyper-volumes**

We denote the hyper-volume of the n-ball by and its surface area by . These quantities can be defined recursively, starting from or and iterating to higher dimensions. But there are also closed form solutions. The volume of the -ball is

where is the gamma function. We recall some properties of :

Using these properties, we can express without evaluation of integrals using Euler’s definition of the gamma function. We get

**Vanishing Hyper-volumes**

A table of the values of for is shown here. We note from this table that the volume of is proportional to . But the constant of proportionality behaves in a curious way. The coefficient increases with dimension for , but for higher dimensions it decreases. Indeed, as we shall see, this coefficient tends to zero as .

The expressions above allow us to relate the hyper-volumes for different dimensions:

It is a simple matter to confirm that the familiar values for low dimensions satisfy these relationships.

**Hyperballs and Hypercubes**

We may wonder how we can compare hyper-volumes in different dimensional spaces. But it is straightforward to compare the hyperball in dimensions with a fixed cube in the same space. The hyperball of unit radius, and diameter 2, fits within a cube of edge-length 2, touching it at the centre of each face. The volume of this cube is , and we may compare the volume of the n-ball to this.

With , the above equations become

Combining these, we see the effect of an increase in dimension by 2 is

It is abundantly clear that, as increases, becomes smaller, rapidly approaching zero for large . This is in marked contrast to the surrounding cube, whose volume doubles with every additional dimension. For growing , the hyperball occupies a smaller and smaller proportion of the associated cube.

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