Higher Dimensional Spheres are not spiky
A viral math paradox about 10D spheres is solved by visualizing cube diagonals, not spikes.
A viral Numberphile video presented a paradox where a sphere placed inside a hypercube appeared to grow through the cube's faces in higher dimensions, humorously suggesting spheres become 'spiky'. TerriLeaf's detailed post on LessWrong systematically debunks this by focusing on the geometry of the cube's diagonal.
By visualizing 2D cross-sections of the 3D, 4D, and 9D constructs, the author demonstrates that the central sphere's radius mathematically increases with dimension. In a 9D unit hypercube, the face diagonal grows to nearly 3 times the side length. This expansion creates space for the central sphere, whose radius reaches approximately 2.16 in 10 dimensions, allowing it to contact—and even extend beyond—the cube's boundaries without requiring any 'spiky' deformation. The solution lies in the counterintuitive but purely geometric scaling of polytope diagonals versus their sides.
- Debunks Numberphile's 'spiky spheres' paradox by analyzing hypercube face diagonals in up to 10 dimensions.
- Shows central sphere radius grows to ~2.16 in a 10D unit cube, contacting faces due to geometry, not spikes.
- Uses 2D cross-sectional visualizations to make high-dimensional geometry intuitive and accessible.
Why It Matters
Clarifies foundational geometry concepts crucial for AI research in high-dimensional data spaces and vector embeddings.