How many bullets would a four dimensional revolver hold?

A three-dimension revolver contains six chambers for bullets. How many bullets would a four-dimensional revolver hold? Hint: this relates to a disagreement between Isaac Newton and David Gregory (it turns out Newton was right).

The answer is derived from kissing numbers. A kissing number is the maximum number of circles/spheres/hyperspheres that can be fitted around a single circle/sphere/hypersphere. They can touch, but they cannot overlap. The kissing number for two dimensions is 6. This is the maximum number of circles that can be fitted around a central circle.

This is why a three dimensional revolver has six bullets. (It could have fewer, but people who buy guns tend to want as many bullets as possible.) So the cross section of the chamber of a four dimensional revolver would be a three dimensional sphere enclosed by spheres through which the four dimensional bullets pass.

So the question is, what is the kissing number for three dimensions?

This is quite hard to work out. Isaac Newton thought it was 12, but that leaves quite a lot of space between the enclosing spheres, which led David Gregory to believe that you could fit an extra one to get 13. It was not until 1953 that mathematicians proved that the answer is indeed 12.

So the answer is that the chamber of a four dimensional revolver can hold no more than 12 bullets. Some of the kissing numbers for higher dimensions have also been calculated, but there is uncertainty about many of them. The kissing number for 24 dimensions is known. So a 25 dimensional revolver would hold 196,560 bullets.

For the full gory details about kissing numbers, please see the Wikipedia article.

Full gory details about revolvers, please see this gun website.

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