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Right-Angled Polytope

A right-angled polytope is a convex polytope whose dihedral angles are all pi/2. In hyperbolic geometry, reflections in its facets generate a discrete reflection group (Potyagailo and Vinberg 2005).

Guo and Zheng (2026) proved that every compact right-angled hyperbolic 4-polytope has at least 600 vertices, 1200 edges, 720 two-dimensional faces, and 120 facets, and has hyperbolic volume at least 34pi^2/3. Equality in any of these bounds holds iff the polytope is isometric to the compact right-angled hyperbolic 120-cell. The statement concerns the hyperbolic realization, rather than the Euclidean regular polytope. Guo and Zheng (2026) report AI assistance with the mathematical proof.

See also

Hyperbolic Geometry, Polytope

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References

Guo, L. and Zheng, F. "On the Minimality of the Compact Right-Angled 120-Cell." 19 Oct 2026. https://arxiv.org/abs/2610.33053.Potyagailo, L. and Vinberg, E. "On Right-Angled Reflection Groups in Hyperbolic Spaces." Comment. Math. Helv. 80, 63-73, 2005. https://doi.org/10.4171/CMH/4.

Cite this as:

Weisstein, Eric W. "Right-Angled Polytope." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Right-AngledPolytope.html

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