A new minimal hypersurface, called

Content shortened for clarity.

The surface is the zero set of a simple trigonometric function in \u{B7}4\(\mathbb R^4\). It is smooth everywhere except at the vertices of the regular 24‑cell honeycomb. At each vertex the hypersurface looks exactly like the Clifford cone, a classic non‑stable minimal cone in four dimensions. The surface is complete, properly embedded, and extends infinitely in all four directions, yet it has an infinite, discrete set of singular points.

The construction uses the lemniscatic sine \(\sigma(t)=\operatorname{sl}(\varpi t)\) with period two. That function appears in the defining equation \[S(x)=\sigma(x_{1})\sigma(x_{3})-\sigma(x_{2})\sigma(x_{4})=0. \] Solving this equation gives a hypersurface that repeats itself by translations of the root lattice \(D_{4}\). The symmetry group also includes a hyperoctahedral group of order 384, making the set of singular vertices transitively symmetric.

Unlike most known minimal hypersurfaces, which are area‑minimizing or stable and have only finitely many singularities, this example is not stable. In fact, a compact quotient of the surface has infinite Morse index, meaning it can be destabilized in infinitely many independent ways around each singular point. The singular set is not merely isolated points; near any such point the hypersurface cannot be described as a topological manifold, because its tangent cone is the unstable Clifford cone.

Although the hypersurface settles into a smooth shape away from singularities, it never becomes a topological manifold at any singular point, and the density at each singularity is \(\pi/2\). This is a rare counter‑example showing that complete embedded minimal hypersurfaces in Euclidean space can have a dense, infinite set of non‑stable singularities.