A newly reported mathematical construction has eight planar nine-sided faces, 24 vertices and 36 edges—not 26. Its surface has genus 3, meaning it has three handles. The corrected counts come from mathematician Ruslan Mizhaev’s 2026 arXiv preprint; the 26-edge figure appeared in a conflicting Popular Science headline and article text.
What is the shape, and how many edges does it have?
It is a polyhedral surface embedded in three-dimensional space. Mizhaev’s 2026 preprint describes eight flat, nine-sided faces (nonagons), joined at 24 vertices and 36 edges. The preprint gives the construction and its geometric data at arXiv.
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The 26-edge count in the Popular Science title and article body conflicts with the preprint and New Scientist’s report, both of which give 36. For the construction described by Mizhaev, 36 is the supported count.
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“Three holes” is an informal way to describe genus 3: the surface has three handles. It does not mean that the object is a solid with three drilled openings. Genus is a topological property of a surface, describing its handle structure rather than the shape’s precise dimensions or appearance.
How do the faces fit together?
Each vertex is incident with three faces. Every pair of faces shares at least one edge: among the 28 possible pairs of eight faces, 20 pairs share one edge and eight pairs share two, according to the preprint.
The surface is closed and orientable, according to Mizhaev’s construction. In plain terms, it has no boundary edge where the surface simply stops, and it has a consistent two-sided surface orientation.
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Can the construction be checked or recreated?
Mizhaev’s preprint supplies integer coordinates for the vertices, face walks describing which vertices form each face, and plane equations. It reports exact verification that the faces are planar, the surface is closed and orientable, there are no unintended intersections, and the construction has C4 symmetry. Those details make the result a specified geometric realization rather than only a sketch or a list of counts.
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No. A separate 2026 arXiv preprint by Gergely Röst and Viktor Vígh describes another genus-3 realization with eight faces, 24 vertices and 36 edges: their preprint. It is not combinatorially equivalent to Mizhaev’s construction, so the broad combination of counts does not identify a unique arrangement.
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| Construction | Faces | Vertices | Edges | Relationship |
|---|---|---|---|---|
| Mizhaev | 8 planar nonagons | 24 | 36 | Preprint reports C4 symmetry and gives coordinates and face data. |
| Röst and Vígh | 8 | 24 | 36 | Separate realization; not combinatorially equivalent to Mizhaev’s. |
What is established about the result?
The constructions are described in 2026 arXiv preprints. The sources cited here do not establish that either preprint has undergone journal peer review or independent peer review. The geometric claims should therefore be attributed to the authors’ preprints rather than presented as peer-reviewed findings.
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