A periodic minimal hypersurface in Euclidean 4-space with Clifford cone singularities

Let σ(t)=sl( t) be the lemniscatic sine, normalized to have period $2$. We study the zero set M of S(x)=σ(x₁)σ(x₃)-σ(x₂)σ(x₄) in R⁴. We show that M is a complete embedded minimal hypersurface with infinitely many isolated singularities, which are exactly the vertices of the regular honeycomb of R⁴ by $24$-cells. Near every singular point M is an analytic image of the Clifford cone, and M is a stationary integral varifold with density $π/2$ at each singularity. The isometry group of M is generated by the translations of the root lattice D₄ and a hyperoctahedral group of order $384$ fixing the centre of a $24$-cell; it has index three in the symmetry group of the honeycomb, acts transitively on the singular points, and contains a symmetry which is not visible from the defining equation. Seen from the centre of a $24$-cell, M is the zero set of a theta series of a translate of the Hurwitz quaternions, which makes the whole symmetry group manifest; the sections of M by a family of parallel hyperplanes are triply periodic surfaces with cubic symmetry which change topology through conical points. Finally, M is obtained from a union of $2$-planes by attaching cones over $2$-tori, and its quotient by the lattice of periods is a compact minimal hypersurface in a flat $4$-torus with exactly three singular points and infinite Morse index; we show that its regular part is one-sided and compute its Euler characteristic and integral homology.