A 2026 mathematical preprint describes a new polyhedral surface with eight nine-sided faces, 24 vertices and 36 edges. The “26 edges” figure in the original headline is incorrect. The surface has genus 3—often described as having three handles, or “holes”—and every face shares an edge with each of the other seven.
What is the shape, and what are its correct counts?
Mathematician Ruslan Mizhaev’s 2026 arXiv preprint presents a polyhedral surface embedded in three-dimensional space. It has eight planar nonagons (nine-sided faces), 24 vertices and 36 edges. New Scientist also reports 36 edges; Popular Science gives 26, the conflicting figure that should not be used.
These counts describe the construction in the preprint, not a physical object known to have been manufactured. The paper supplies integer coordinates, face walks and plane equations for the realization.
What does “three holes” mean?
The surface has genus 3. Genus is a topological property: for an orientable surface, it counts the number of handles. A familiar comparison is a doughnut-shaped surface, which has genus 1; this construction has three handles. “Holes” is an informal way of describing that topology, not a count of openings punched through individual faces.
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The reported object is a closed orientable surface. Its eight faces enclose a connected surface with that genus, rather than eight separate polygons.
How do the eight faces fit together?
Each vertex is incident with three faces. Every pair of faces shares at least one edge: of the 28 possible pairs, 20 share one edge and eight share two, according to Mizhaev’s preprint.
The preprint reports exact verification of the planar faces, closed orientable surface, absence of unintended intersections and C4 symmetry. Those are claims about the mathematical construction and its specified coordinates; they do not mean the object has been independently peer-reviewed or physically built.
Is this the only eight-faced genus-3 construction?
No. A separate 2026 arXiv preprint by Gergely Röst and Viktor Vígh describes another realization with eight faces, 24 vertices and 36 edges. The authors report similar face-adjacency counts, but say their construction is not combinatorially equivalent to Mizhaev’s. The shared headline counts therefore do not make the two surfaces the same arrangement.
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The available source descriptions establish this difference in combinatorial structure, but do not provide enough comparable detail to make a broader judgment about which realization is simpler or preferable.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What can readers verify or reproduce?
Mizhaev’s preprint includes the coordinate and face data needed to reconstruct the realization and reports exact geometric checks. This makes it possible for readers with suitable mathematical or modeling tools to examine the specified surface. The preprint status matters: the sources cited here do not establish that the work has appeared in a peer-reviewed journal.
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Sources: Ruslan Mizhaev’s arXiv preprint; New Scientist report; Popular Science report; Gergely Röst and Viktor Vígh’s arXiv preprint.
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