[Paper Review] Surface-area-minimizing n-hedral Tiles

Whan Ghang, Zane Martin+1 more|arXiv (Cornell University)|May 6, 2013

Computational Geometry and Mesh Generation9 references3 citations

TL;DR

This paper proposes conjectured surface-area-minimizing n-hedral tiles for space-filling polyhedra with 4 to 14 faces, providing rigorous proofs for the optimal 4-hedral (orientation-preserving tetrahedron) and 5-hedral (right equilateral-triangular prism) tiles. It establishes that the right equilateral-triangular prism circumscribed about a sphere minimizes surface area among unit-volume 5-hedra, confirming its optimality as a tiling tile.

ABSTRACT

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 or 6. We find the optimal "orientation-preserving" tetrahedral tile (n=4), and we give a nice proof for the optimal 5-hedron (a triangular prism).

Motivation & Objective

• To identify the surface-area-minimizing n-hedral tiles of unit volume in R³ for n from 4 to 14.
• To extend known results beyond n = 5 and n = 6, where the triangular prism and cube are known to minimize surface area.
• To prove that the right equilateral-triangular prism circumscribed about a sphere is the surface-area-minimizing 5-hedron and 5-hedral tile.
• To provide a new geometric proof for the optimality of the 5-hedron using symmetry, centroid properties, and vector geometry.
• To conjecture optimal tiles for higher n, including Goldberg polyhedra and Kelvin’s truncated octahedron for n ≥ 14.

Proposed method

• Uses geometric symmetry and centroid properties to analyze the configuration of tangent points between an inscribed sphere and the faces of a 5-hedron.
• Applies linear algebra to show that midpoints of edges and centroids of faces satisfy parallelogram and centroid relations, implying equilateral symmetry.
• Employs vector dot product analysis to prove that the triangle formed by face-touching points of the inscribed sphere is equilateral.
• Imposes a coordinate system with the sphere’s center at the origin and projects points to the xy-plane to analyze planar symmetry.
• Considers two cases for edge directions: parallel or concurrent; eliminates the concurrent case via contradiction based on unequal distances from the origin.
• Uses the fact that equilateral triangles minimize perimeter among 3-gons to support the optimality of the prism’s base shape.

Experimental results

Research questions

• RQ1 What is the surface-area-minimizing 4-hedral tile of unit volume that tiles R³, under the orientation-preserving constraint?
• RQ2 Is the right equilateral-triangular prism the unique surface-area-minimizing 5-hedron that tiles R³ among all convex polyhedra with five faces?
• RQ3 Can a geometric proof be constructed to show that the right equilateral-triangular prism circumscribed about a sphere minimizes surface area among 5-hedra?
• RQ4 How do the properties of face-touching points of an inscribed sphere constrain the shape of the optimal 5-hedron?
• RQ5 What are the optimal surface-area-minimizing n-hedral tiles for n = 7 to 14, and how do they relate to known polyhedral families like Goldberg’s or Kelvin’s?

Key findings

• The surface-area-minimizing 4-hedral tile under orientation-preserving constraints is a tetrahedron formed by cutting a triangular prism into three congruent tetrahedra with sides √3, √3, and 2.
• The right equilateral-triangular prism circumscribed about a sphere is proven to minimize surface area among all unit-volume 5-hedra.
• The optimal 5-hedron has a base-length of 4^(1/3) and a height of 4^(1/3) / √3, derived from the sphere-tangency condition.
• The proof relies on showing that the triangle formed by the sphere’s contact points on the three lateral faces is equilateral, forcing the base to be equilateral.
• The case where the three lateral edges of the 5-hedron concur at a point is ruled out by contradiction, confirming only the parallel-edge configuration is valid.
• The conjectured optimal tiles for n = 7 to 14 include specific Goldberg polyhedra and Kelvin’s truncated octahedron for n ≥ 14, all with vertices of degree three.