# A high-dimensional sphere spilling out of a high-dimensional cube despite exponentially many constraints

Make a square, split each side into two halves, producing four cells. Put a circle into each cell such that it fills it completely. There is a small gap right in the middle of the square. Put a circle there again such that it touches the other four circles. The central circle is *obviously inside* the square, right? Yes, but only if the dimension you are in is $D\le9$. Above that, the central cicle actually *spills out* from the cube, despite the $2^D$ spheres in their cells keeping it in. In this post I present this simple-to-compute yet utterly counter-intuitive result.

**Stanislav Fort** (Twitter, Scholar and GitHub)

Let’s do it step by step:

- Make a square of an edge length $4a$ (the 4 will make our life easier later).
- Divide each side into two halves. This will produce 4 equal cells in two dimesions, or $2^D$ cells in $D$-dimensions.
- Put a circle (two dimensions) / sphere ($D$-dimensions) to each cell to fill it up completely. Each sphere will touch the walls of the cell and will therefore have a radius $\rho=a$. There will be $2^D$ such spheres in $2^D$ cells.
- Put a circle as big as possible to the exact center of the square / cube such that it touches all $2^D$ spheres. This will uniquelly determine its radius. In two dimensions, this circle is
*obviously inside*the square, right? - The distance from the center of the square to the center of each cell is $\sqrt{2}$a in two dimensions, or $\sqrt{D}a$ in $D$-dimensions. The reason for this is that it is a $1/4$ of the full diagonal of the cube, which is generically $\sqrt{D} \times 4a$.
- The distance from the center of the cube to the center of each cell must be the sum of the radius of the inner sphere $r$ and the radius of the cell-filling sphere $\rho=a$, since these two touch by definition. Therefore $\sqrt{D}a = a + r(D)$.
- This implies that $\boxed{r(D) = \left ( \sqrt{D} - 1 \right ) a}$ .

So far, nothing seems to be intuitively wrong. For $D=2$, we get the radius of the central circle to be $r(D=2) = (\sqrt{2}-1) a$, which is smaller than $2a$, the half-size of the square. Therefore the central circle is *indeed* nicely contained within the square. All is good.

Notice, however, that $r(D)$ is monotonic and unbounded from above in $D$. This means that as the dimension grows, the central sphere will grow in radius, while the linear size of the cube stays the same. At some point, **the sphere must stick out of the cube**! And it isn’t even such a high dimension you need for that to be the case: $r(D) > 2a \implies D > 9$.

For dimension $D>9$, the simplified, three-dimensional-turned-two-dimensional mental picture is of a cube from which parts of the inner sphere are sticking out through the middles of the cube walls.

This is a pretty intuitively unnerving result for me! It also goes against an intuitely strong principle that an exponential number of constraints (here the $2^D$ spheres in their respective cells) should be sufficient to constrain things even in high dimensions. On a vague level, this reminds of the recent Basins with tentacles paper by Yuanzhao Zhang and Steven H. Strogatz.