Sphere Packing Paradox

Quick Overview

There are many fascinating mathematical paradoxes, some easier to understand than others. I want to share one of my favorite paradoxes referred to as "The Sphere Packing Paradox" or "The Four Circle Paradox." This paradox is quite simple to understand and most interestingly exploits the lack of higher dimensional understanding in our human intelligence.

2-Dimensions

We begin in 2D space with a 4x4 square and we wish to fill each corner of the square with a unit circle (a circle of radius 1). We then wish to fill the center with a circle so that the center circle is touching each of the circles that are within each corner as pictured.

In order to compute the radius of the inner orange circle we do some simple arithmetic using geometric properties and arrive at the following using the Pythagorean theorem:

(1)^2 + (1)^2 = (r + 1)^2
            2 = r^2 + 2r + 1
            0 = r^2 + 2r - 1

--> r = sqrt(2) - 1

3-Dimensions

The 3D case is similar to the 2D case except this time we have a 4x4x4 cube that we wish to fill with unit spheres (i.e., spheres with radius 1) in each corner. We again fill the center gap with a sphere touching each of the 8 other spheres that are inside the cube.

When we compute the radius of the center sphere this time we end up with a greater value radius than in the 2D case. Do you see why? It's important to convince yourself of this, you can compute the value for the radius in a similar manner that we did in the 2D case:

(sqrt(2))^2 + (1)^2 = (r + 1)^2
                  3 = r^2 + 2r + 1
                  0 = r^2 + 2r - 2

--> r = sqrt(3) - 1

N Dimensions

Generalizing to N-Dimensions we have a hypercube of size 4^n with 2^n hyper-spheres placed into the corners. When we compute the radius of the center hyper-sphere it comes out to:

--> r = sqrt(n) - 1

The Interesting Part

If you haven't caught it yet, examine the generalized formula for the inner hyper-sphere's radius again. Notice anything? Let's examine the value in different dimensions:

2D:    sqrt(2) - 1 = 0.4142
3D:    sqrt(3) - 1 = 0.7320
4D:    sqrt(4) - 1 = 1.0000
5D:    sqrt(5) - 1 = 1.2360
6D:    sqrt(6) - 1 = 1.4494
7D:    sqrt(7) - 1 = 1.6457
8D:    sqrt(8) - 1 = 1.9294
9D:    sqrt(9) - 1 = 2.0000
10D:   sqrt(9) - 1 = 2.1622

The first thing we notice is that the radius is growing as we go into higher dimensions. Some very interesting dimensions in particular though are 4, 9, and 10.

• In the 4th dimension the center hyper-sphere becomes a unit sphere and is the same size as the hyper-spheres in the hypercube's corners.
• In the 9th dimension the center hyper-sphere has a radius of 2 and is at this point touching faces of the bounding hypercube (since the center hyper-sphere has a diameter is 4, the same width of the bounding hypercube)
• In the 10th dimension the center hyper-sphere has a radius larger than 2 and is actually going outside of the bounding cube!

Conclusions and Thoughts

The higher the dimension the more space there is between the packing spheres in the corners of the cube. After the 9th dimension the center hyper-sphere protrudes from the bounding hypercube, yet this hyper-sphere is still convex! An astonishing result to try to imagine.

This is very counter-intuitive and hard to comprehend given our limited spatial-dimension reasoning. Perhaps this is not very surprising in the grand scheme of things but only strange to humans since we live our daily lives in a much lower dimensional space.

"I am the wisest man alive, for I know one thing, and that is that I know nothing." - Plato

References:

1. Art of Doing Science and Engineering: Learning to Learn, Richard R. Hamming, CRC Press, 2003