Oriented and Non-oriented Cubical Surfaces in The Penteract
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909142436806656 |
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| author | Estevez, Manuel Roldan, Erika Segerman, Henry |
| author_facet | Estevez, Manuel Roldan, Erika Segerman, Henry |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_12825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oriented and Non-oriented Cubical Surfaces in The Penteract Estevez, Manuel Roldan, Erika Segerman, Henry Geometric Topology Combinatorics History and Overview 0501, 0506, 05A99, 05B25, 05C10, 05C35 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. |
| title | Oriented and Non-oriented Cubical Surfaces in The Penteract |
| topic | Geometric Topology Combinatorics History and Overview 0501, 0506, 05A99, 05B25, 05C10, 05C35 |
| url | https://arxiv.org/abs/2403.12825 |