A high-dimensional sphere spilling out of a high-dimensional cube

# · 🔥 299 · 💬 135 · 2 years ago · stanislavfort.github.io · EvgeniyZh · 📷
Put a circle there again such that it touches the other four circles. Put a circle / sphere to each cell to fill it up completely. Each sphere will touch the walls of the cell and will therefore have a radius $rho=a$. 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. Notice 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 should be sufficient to constrain things even in high dimensions.
A high-dimensional sphere spilling out of a high-dimensional cube



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