Spheres cause contradictions in dimensions $10$ and more?

According to this Numberphile video, if you tightly pack hyper-spheres into a hyper-box and then find the radius of the largest hyper-sphere that could possibly fit in the remaining space, the resulting hyper-sphere would somehow exceed the confines of the box that contained all of the hyper-spheres (where the number of dimensions are greater or equal to 10).

Isn't a logical contradiction generally considered a disproof of something?

Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?

Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

Solution 1:

No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $\frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

Solution 2:

In the video, they get a sphere too large to fit in the cube because they did not look for a sphere that fit inside the cube, they just looked for a sphere that fit between the other spheres.

If you require that the central sphere also has to fit inside the original cube, as well as fit between the spheres in the corners, then once you get past dimension $4$ the central sphere will no longer touch the spheres in the corners. There will be gaps between the central sphere and the others.

That is, in $5$ or more dimensions, if the central sphere is small enough to fit inside the box it is too small to touch all the corner spheres at once; if you make it large enough to touch all the corner spheres at once then of course it is now too large to fit between the sides of the box.

It's all a result of the (possibly counterintuitive) fact that in $5$ or more dimensions, if you start at the center of a hypercube you have to travel farther to reach one of the "corner spheres" than you do to reach one of the sides of the hypercube.

Solution 3:

Think of it another way. The hyperspheres are the size you expect, but the hyperbox is much, much bigger than you'd expect.

A 1m diameter circle in a square has 4 pyramids (well, triangles), each with a height of about 0.207m.

A 1m sphere in a cube has 8 pyramids, each with a height of about 0.366m

A 1m 10-sphere in a 10-cube has 1024 pyramids, each with a height of 1.081m - that's now longer than the cube's edge.

Now imagine a 2m side 10-cube packed with 1024 of those 10-spheres. In between the 10-spheres there is a 2.162m wide void that can fit a 2.162m diameter hypersphere. It feels wrong, but it follows naturally as you increase in dimensions.

Solution 4:

Just take an $n$-dimensional hypercube of edge size 2 centered at the origin. Take furthermore hyperballs with unit radius centered at each of its vertices. Then those evidently will be touching by construction.

By Pythagoras the distance from the origin, i.e. from the body center of that hypercube, towards either of its vertices happens to be $$\sqrt{1^2+1^2+...+1^2}=\sqrt{n}$$

Accordingly an hyperball, which is centered at the origin and is touching those corner-centered unit hyperballs, should have a radius of $$\sqrt{n}-1$$ But that number clearly gets larger than $1$, which in turn clearly is the radius of the in-hyperball of that hypercube, as soon as $n>4$. - That is all of that counterintuitive magic.

--- rk