Small triangulations of real projective five-space
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How few vertices can a triangulation of real projective five-space use, and can a small triangulation be realized as the antipodal quotient of a centrally symmetric simplicial polytope? A 24-vertex combinatorial triangulation is already known.
Why it matters and what counts as progress
Why it matters: A polytopal realization adds geometric structure without by itself improving the known vertex count or proving minimality.
What counts as progress: A new small triangulation, a stronger lower bound than 22, or a proved antipodal polytopal realization at 24 vertices; these are distinct achievements.
How the problem developed
- 8 Mar 2026 · Construction [1]
Dan Guyer, Stefan Steinerberger and Yirong Yang Constructed a 24-vertex triangulation of RP^5 and conjectured it has the fewest vertices among all triangulations.
- 18 Oct 2026 · Minimum proof [2]
Xinrui Chang Gave a computer-assisted proof that every finite simplicial triangulation of RP^5 has at least 24 vertices, matching the construction; the argument does not assume the triangulation is combinatorial.
AI contribution
An Efficient Triangulation of ℝP⁵
A centrally symmetric simplicial six-dimensional polytope with 48 vertices has an antipodal boundary quotient that triangulates real projective five-space on 24 vertices.
Sources
Website timeline
Publication dates and editorial changes.
AlphaEvolve seed yields a geometric 24-vertex RP⁵ triangulation