Sphere spilling out

Center a small blue sphere on every corner of an n-dimensional unit hypercube. These are the points in ℝⁿ for which every coordinate is either a 0 or a 1. Now inflate each of these small spheres at the same time until they touch. Each sphere will have radius 1/2. For example, the spheres centered at

(0, 0, 0, …, 0)

and

(0, 0, 0, …, 1)

can’t have a radius larger than 1/2 because they’ll intersect at (0, 0, 0, …, 1/2).

Now center a little red sphere at center, the point where every coordinate equals 1/2, and blow it up until it touches the blue spheres. Which is larger, the blue spheres or the red one?

The answer depends on n. In low dimensions, the blue spheres are larger. But in higher dimensions, the red sphere is larger.

The distance of from the center of the hypercube to any of the vertices is √n / 2 and so the radius of the red sphere is (√n − 1) / 2. When n = 4, the radius of the red sphere equals the radius of the blue spheres.

The hypercube and blue spheres will fit inside a hypercube whose side is length 2. But if n > 9 the red sphere will spill out of this larger hypercube.

This post was motivate by this post by Aryeh Kontorovich on Twitter.

More on high-dimensional spheres

• Nearly all the volume is in a thin shell
• Nearly all the area is near the equator
• Willie Sutton and the multivariate normal
• Random projections

The post Sphere spilling out first appeared on John D. Cook.