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Integer Realization of an Equivelar Octahedron of Genus 3
Abstract
We present an integer-coordinate realization of a genus-3 polyhedral surface with eight planar simple nonagonal faces. It has 24 vertices and 36 edges, and each vertex is incident with three faces. Every pair of faces shares an edge: 20 pairs share one edge and 8 pairs share two. Vertex coordinates, face walks, and plane equations are given. Exact verification confirms that all faces are planar, the surface is closed and orientable, no unintended intersections occur, and the realization has C4 symmetry. The construction is based on an earlier construction and preserves its incidence structure.
Related
BibTeX
@misc{mizhaev2026integer,
title = {{Integer Realization of an Equivelar Octahedron of Genus 3}},
author = {Ruslan Mizhaev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17700},
eprint = {2609.17700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17700},
}
RIS
TY - UNPB
TI - Integer Realization of an Equivelar Octahedron of Genus 3
AU - Ruslan Mizhaev
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17700
UR - https://arxiv.org/abs/2609.17700
AB - We present an integer-coordinate realization of a genus-3 polyhedral surface with eight planar simple nonagonal faces. It has 24 vertices and 36 edges, and each vertex is incident with three faces. Every pair of faces shares an edge: 20 pairs share one edge and 8 pairs share two. Vertex coordinates, face walks, and plane equations are given. Exact verification confirms that all faces are planar, the surface is closed and orientable, no unintended intersections occur, and the realization has C_4 symmetry. The construction is based on an earlier construction and preserves its incidence structure.
ER -