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Oriented and Non-oriented Cubical Surfaces in The Penteract

Manuel Estevez, Erika Roldan, Henry Segerman

Abstract

Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to isomorphism in the 5-cube. They give a classification in terms of their genus $g$ for closed orientable cubical surfaces and their demigenus $k$ for a closed non-orientable cubical surface. In this paper, we explain the main idea behind the exhaustive search and we visualize the projection to $\mathbb{R}^3$ of a torus, a genus two torus, the projective plane, and the Klein bottle. We use reinforcement learning techniques to obtain configurations optimized for 3D printing.

Oriented and Non-oriented Cubical Surfaces in The Penteract

Abstract

Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to isomorphism in the 5-cube. They give a classification in terms of their genus for closed orientable cubical surfaces and their demigenus for a closed non-orientable cubical surface. In this paper, we explain the main idea behind the exhaustive search and we visualize the projection to of a torus, a genus two torus, the projective plane, and the Klein bottle. We use reinforcement learning techniques to obtain configurations optimized for 3D printing.
Paper Structure (8 sections, 2 equations, 3 figures)

This paper contains 8 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cubical Surfaces in the Penteract
  3. Embeddings of Cubical Surfaces
  4. Orientable Surfaces
  5. Non-orientable Surfaces
  6. Using Reinforcement Learning to Minimize Face Intersections and Edge Overlaps
  7. Minimizing the number of intersecting faces
  8. Minimizing edge overlaps for 3D-printing

Figures (3)

  • Figure 1: Orientable cubical surfaces
  • Figure 2: A cubical surface homeomorphic to the projective plane.
  • Figure 3: A cubical surface homeomorphic to the Klein Bottle