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Quasigeodesics on the Cube

MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Thomas C. Hull, Anna Lubiw, Jayson Lynch, Klara Mundilova, Chie Nara, Joseph O'Rourke, Frederick Stock, Josef Tkadlec, Ryuhei Uehara

TL;DR

This work resolves the enumeration of simple closed quasigeodesics on the cube by proving there are exactly $15$ such curves beyond the three known simple closed geodesics. The authors develop a slope-based framework, proving only five slopes can occur for geodesic segments composing a quasigeodesic, via a seven-case unfolding and F-cone analysis, thereby ruling out spiraling on the cube. They then confirm the inventory with a computer-assisted DFS that respects symmetry and the five-slope constraint, yielding a complete list of 15 quasigeodesics. The results establish a finite upper bound for the cube and advance understanding of quasigeodesics on convex polyhedra, while outlining open questions about bounds for general polyhedra and box-like families.

Abstract

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le π$ surface to each side at every point. In contrast, a geodesic has exactly $π$ to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly $15$ simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail $15$ simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

Quasigeodesics on the Cube

TL;DR

This work resolves the enumeration of simple closed quasigeodesics on the cube by proving there are exactly such curves beyond the three known simple closed geodesics. The authors develop a slope-based framework, proving only five slopes can occur for geodesic segments composing a quasigeodesic, via a seven-case unfolding and F-cone analysis, thereby ruling out spiraling on the cube. They then confirm the inventory with a computer-assisted DFS that respects symmetry and the five-slope constraint, yielding a complete list of 15 quasigeodesics. The results establish a finite upper bound for the cube and advance understanding of quasigeodesics on convex polyhedra, while outlining open questions about bounds for general polyhedra and box-like families.

Abstract

A quasigeodesic is a curve on the surface of a convex polyhedron that has surface to each side at every point. In contrast, a geodesic has exactly to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

Paper Structure

This paper contains 12 sections, 2 theorems, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quasigeodesics
  3. Three simple closed geodesics
  4. Outline of Argument
  5. Fifteen Simple Closed Quasigeodesics
  6. Five Slopes
  7. Case 1.
  8. Case 4.
  9. Case 5.
  10. Case 7.
  11. Search for Quasigeos
  12. Discussion and Open Problems

Key Result

Lemma 1

A geodesic segment that is a component of a simple closed quasigeodesic on the cube can only have one of the five slopes shown in Fig. fig:FiveSlopes: $0/1,1/3,1/2,2/3,1/1$.

Figures (10)

  • Figure 1: F,R,T,K,L,B $=$ Front, Right, Top, bacK, Left, Bottom. B vertices indexed $1,2,3,4$; T vertices indexed $5,6,7,8$. $v_1$ is marked white.
  • Figure 2: The three simple closed geodesics on a cube. The first is an equatorial band. The other two are as depicted.
  • Figure 3: $(v_1,v_2,v_3,v_3,v_4)$ is a simple closed quasigeodesic. Based on Fig. 2 in demaine2020quasi.
  • Figure 4: The $15$ simple closed quasigeodesics.
  • Figure 5: The five possible distinct slopes.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Corollary 1
  • Claim 1
  • Claim 2