An acute triangulation subdivides a polytope or a space into simplices forming a face-to-face simplicial complex in which every dihedral angle is strictly less than π/2\pi/2. Kopczyński, Pak and Przytycki (2012) proved that the nn-cube admits no acute triangulation for n≥4n \ge 4, that Rn\mathbb{R}^n admits none for n≥5n \ge 5, and that R4\mathbb{R}^4 admits none when all dihedral angles are uniformly bounded away from π/2\pi/2, leaving dimension four as the unresolved boundary of the whole program. The open questions are (1) does R4\mathbb{R}^4 admit an acute triangulation in full generality, a question on which Krížek conjectures a negative and Kopczyński–Pak–Przytycki a positive answer, and (2) do the regular 4-polytopes 16-cell, 24-cell, and 120-cell admit acute triangulations (conjectured not)?

Background

An acute triangulation of a polytope or of a Euclidean space is a subdivision into simplices that form a face-to-face simplicial complex — any two simplices intersect in a common face, so a vertex of one simplex cannot lie in the interior of a face of another — in which every dihedral angle is strictly less than π/2\pi/2. The dihedral angle at a codimension-2 face is the interior angle, measured in the two-dimensional plane orthogonal to that face, between the two facets (codimension-1 faces) that meet there; a 4-simplex has ten triangular 2-faces and hence ten dihedral angles, each between two of its five tetrahedral facets.

The question whether space can be dissected into acute simplices goes back to Aristotle's (false) claim that regular tetrahedra tile space. In the plane the problem is essentially solved: every polygon, however non-convex, admits an acute triangulation, with linear bounds on the number of triangles. In dimension three the picture is partial: Eppstein, Sullivan and Üngör (2004) constructed the first acute triangulation of R3\mathbb{R}^3 by adapting a classical space-filling tiling; VanderZee, Hirani, Zharnitsky and Guoy (2010) built an acute triangulation of the cube using 1370 tetrahedra; and Kopczyński, Pak and Przytycki (2012) constructed acute triangulations of the regular octahedron and the regular tetrahedron (543 tetrahedra) and showed that all five Platonic solids admit them. In the negative direction, the same authors proved that the nn-cube admits no acute triangulation for n≥4n \ge 4 and that Rn\mathbb{R}^n admits none for n≥5n \ge 5. The latter proof rests on a theorem of Kalai (1990) that every polytope of dimension at least 5 has a 2-dimensional face that is a triangle or a quadrilateral: in a triangulation, at most four dihedral angles around such a low-degree codimension-2 face must sum to 2π2\pi, which forces at least one of them to be non-acute.

Dimension four is the unresolved boundary case of the program and carries two opposing conjectures. Krížek (with Brandts, Korotov and Šolc, 2009) conjectured that R4\mathbb{R}^4 admits no acute triangulation at all, and that locally such a triangulation would need at least 600 simplices around each vertex; the proof given there for the related higher-dimensional claims contained a gap, corrected by Krížek for dimension 5, and Kopczyński–Pak–Przytycki gave an independent proof for all n≥5n \ge 5. Kopczyński, Pak and Przytycki themselves conjecture the opposite: that an acute triangulation of R4\mathbb{R}^4 does exist, while noting that constructing it should first require mastering acute triangulations in dimension 3 and in the spherical case. For individual convex 4-polytopes the situation is similarly split. The 600-cell has an easy acute triangulation by cones from its center over its tetrahedral facets, while the 4-cube and the standard (cube-corner) 4-simplex admit none, because each tiles R4\mathbb{R}^4 by reflections and an acute triangulation of either would extend by reflection to an acute triangulation of R4\mathbb{R}^4 itself. The remaining regular 4-polytopes — the 16-cell (the regular cross-polytope in R4\mathbb{R}^4, dual to the 4-cube), the 24-cell, and the 120-cell — are conjectured by Kopczyński, Pak and Przytycki not to admit acute triangulations; the tempting tessellation argument against the 16-cell fails because the 16-cell tiling of R4\mathbb{R}^4 has an odd number of cells around each codimension-2 simplex, so subdivisions of adjacent cells cannot be made orientation-consistent. The spherical analogue raised by the same authors was resolved by Kim and Walsh (2016): the sphere SdS^d admits an acute spherical triangulation (into geodesic simplices with all dihedral angles acute) if and only if d≤3d \le 3, so S4S^4 admits none, while acute triangulations of S3S^3 — which would arise as vertex links (the intersection of the triangulation with a small sphere around a vertex) of any acute triangulation of R4\mathbb{R}^4 — do exist, the boundary of the 600-cell being one, but are not combinatorially classified.

Problem Statement

The problem is to settle acute triangulability in dimension four, in both of its forms.

(1) Does R4\mathbb{R}^4 admit an acute triangulation? That is, does there exist a face-to-face simplicial complex covering R4\mathbb{R}^4 whose 4-simplices all have all ten dihedral angles strictly less than π/2\pi/2 — with no requirement that the dihedral angles be uniformly bounded away from π/2\pi/2, and no restriction on the geometry, shape distribution, or number of the simplices? This is the still-open case of Krížek's conjecture (no such triangulation exists) versus the conjecture of Kopczyński, Pak and Przytycki (one does exist).

(2) Does each of the three remaining regular 4-polytopes — the 16-cell, the 24-cell, and the 120-cell — admit an acute triangulation into 4-simplices, i.e. a face-to-face simplicial complex subdividing the polytope in which every dihedral angle of every 4-simplex is strictly less than π/2\pi/2? Kopczyński, Pak and Przytycki conjecture that none of the three does.

This task concerns the four existence questions — R4\mathbb{R}^4, the 16-cell, the 24-cell, and the 120-cell. The same source list poses further open problems in dimension three (whether every convex 3-polytope admits an acute triangulation, and the minimal numbers of tetrahedra for acute triangulations of the regular tetrahedron and the cube) and in spherical and hyperbolic geometry; these are related open problems from the same source and are not part of this task.

References

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