High-dimensional geometry is full of surprises. For example, nearly all the area of a high-dimensional sphere is near the equator, and by symmetry it doesn’t matter which equator you take.

Here’s another surprise: **corners stick out more in high dimensions**. Hypercubes, for example, become pointier as dimension increases.

How might we quantify this? Think of a pyramid and a flag pole. If you imagine a ball centered at the top of a pyramid, a fair proportion of the volume of the ball contains part of the pyramid. But if you do the same for a flag pole, only a small proportion of the ball contains pole; nearly all the volume of the ball is air.

So one way to quantify how pointy a corner is would be to look at a neighborhood of the corner and measure how much of the neighborhood intersects the solid that the corner is part of. The less volume, the pointier the corner.

Consider a unit square. Put a disk of radius *r* at a corner, with *r* < 1. One quarter of that disk will be inside the square. So the proportion of the square near a particular corner is π*r*²/4, and the proportion of the square near any corner is π*r*².

Now do the analogous exercise for a unit cube. Look at a ball of radius *r* < 1 centered at a corner. One eighth of the volume of that ball contains part of the cube. The proportion of cube’s volume located within a distance *r* of a particular corner is π*r*³/6, and the proportion located within a distance *r* of any corner is 4π*r*³/3.

The corner of a cube sticks out a little more than the corner of a square. 79% of a square is within a distance 0.5 of a corner, while the proportion is 52% for a cube. In that sense, the corners of a cube stick out a little more than the corners of a square.

Now let’s look at a hypercube of dimension *n*. Let *V* be the volume of an *n*-dimensional ball of radius *r* < 1. The proportion of the hypercube’s volume located within a distance *r* of a particular corner is *V* / 2^{n} and the proportion located with a distance *r* of any corner is simply *V*.

The equation for the volume *V* is

If we fix *r* and let *n* vary, this function decreases rapidly as *n* increases.

Saying that corners stick out more in high dimensions is a corollary of the more widely known fact that a ball in a box takes up less and less volume as the dimension of the ball and the box increase.

Let’s set *r* = 1/2 and plot how the volume of a ball varies with dimension *n*.

You could think of this as the volume of a ball sitting inside a unit hypercube, or more relevant to the topic of this post, the proportion of the volume of the hypercube located with a distance 1/2 of a corner.