High-Dimensional Spheres in Cubes

How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle in each. Now, draw a circle whose center is at the center of the big square and whose radius is just big enough to touch the four circles you just drew. We can perform an equivalent operation in a cube, inscribing spheres in each of its eight octants and then placing a sphere in the middle, just large enough to touch the other spheres.

It’s easy to see that the central circle is much smaller than the square and that the central sphere is much smaller than the cube. What happens if we keep doing this in higher dimensions? Will the central sphere grow or shrink in diameter, relative to the side-length of the cube, as you change the dimension?

Our intuition suggests that the central sphere doesn’t grow. After all, its boundary is determined by spheres which lie inside of the cube, hugging each of the corners.

It turns out though, that in the ninth dimension the central sphere is tangent to the cube, and in much higher dimensions the volume of the sphere is actually larger than the cube!

Presentation Suggestions:
Draw the case for the square (and if you’re artistic enough, the cube as well) on the board. Ask the students what they think happens in higher dimensions. Maybe the diameter of the central sphere is always less than some constant ‘a’ (a<1) times the side of the cube, or maybe the central sphere keeps growing and gets arbitrarily close to touching the cube, or maybe it gets much bigger. Take a vote.

The Math Behind the Fact:
We’ll make the numbers easy by using a cube (in dimension N) with side length 2. Then when we cut the cube into orthants, each sub-cube has side length 1. Let’s just look at one of those sub-cubes. The sphere S inscribed in that sub-cube has diameter 1. When we draw the central sphere, its center is on a corner of that subcube. Draw the diagonal from that corner to the opposite corner of the sub-cube. That diagonal has length Sqrt[N]. The part of the diagonal going through the sphere S has length 1 because that is the diameter of S. Of the part that is left over, half of it is in the central sphere, and in fact forms the radius of that central sphere. So the central sphere has radius (Sqrt[N]-1)/2.

If N=9, the radius of the central sphere is 1, so it is just tangent to the cube. If N>9, then part of the central sphere bulges outside the cube! And, eventually the volume of the central sphere is actually larger than the cube.

For a related surprise, see Volume of a Ball in N Dimensions.

Su, Francis E., et al. “High-Dimensional Spheres in Cubes.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Joel Miller

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