Real projective space 
is the space of one-dimensional linear subspaces
of 
. Equivalently, it is the quotient of the sphere 
that identifies each point with its
antipodal point. Its first two positive-dimensional
examples are the real projective line, homeomorphic to a circle,
and the real projective plane.
Small triangulations provide finite combinatorial models of these spaces. Guyer et al. (2026) constructed a 24-vertex triangulation
of 
and a 45-vertex triangulation of 
. Chang (2026) proved that every finite simplicial
triangulation of 
has at least 24 vertices, establishing
the minimality of the former construction. The lower bound
does not require the triangulation to be combinatorial.
Chang (2026) reports Codex assistance with proof exploration
and verification code.
Kolosov (2026) subsequently reported a 44-vertex triangulation of 
with face vector
 |
Its alternating sum is 1, as required by the Euler characteristic of 
. The construction is an antipodal
quotient of the boundary of a centrally symmetric simplicial 7-polytope with 88 vertices. Exact verification programs were replayed by VibeMathed
(2026). ChatGPT assisted the construction search and verification code, but independent
expert review had not been reported as of Sep. 7, 2026. The construction supplies
an upper bound of 44 vertices, not a proof
of minimality.
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18 Oct 2026. 
." 8 Mar 2026. 
." Sep. 6, 2026.