this post was submitted on 09 May 2026
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Hairy ball theorem applies to even-dimensional spheres (the ordinary sphere is the 2D surface of the 3D solid), but a cube in four-dimensional space is a three-dimensional surface, so it doesn't apply.
This is a question about graph theory, not topology; it's asking for a Hamiltonian path on the surface of 4D cube (where faces are vertices, which is different than the normal polytope graph).
You are right.
However most proofs of the hairy ball theorem also prove the converse, so that there is a continous non vanishing tangent vector field on uneven dimensional sphere surfaces.
This can be extended to all 3 dimensional surfaces in 4 dimensions homomorphic to the sphere. The ant walking can follow the vector field and solve this problem topologically.
My point being that the HR goon following the expected leet code solution might not understand this because they might expect the "approved" graph theory solution rather than an alternative approach.
Why does following a tangent vector field visit all faces of the hypercube? Surely it's not going to visit something like a dense subset of the hypersphere's surface? (Or is it? My intuition comes from thinking about the torus)
I'm more interested in the maths ;)
You're hired 🤝
Yaayyy, where's my hypercubicle?