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 $Dle9$. 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:

1. Make a square of an edge length $4a$ (the 4 will make our life easier later).
2. Divide each side into two halves. This will produce 4 equal cells in two dimesions, or $2^D$ cells in $D$-dimensions.
3. Put a circle (two dimensions) / sphere ($D$-dimensions) to each cell to fill it up completely. Each sph