## Acute triangulations of pentagons.(English)Zbl 1240.52004

An “acute triangulation” is a triangulation consisting of triangles with all their angles less than $$\pi/2$$.
In this paper, the author considers acute triangulations of pentagons and doubly covered pentagons in the plane, where the last one can be seen as a degenerate convex polyhedral surface (homeomorphic to the 2-sphere) consisting of two isometric convex pentagonal (planar) sides glued along the boundaries in the obvious way. He shows that for every pentagon (respectively, double pentagon), there is an acute triangulation into, at most, 54 triangles (respectively, 76 triangles).

### MSC:

 52A15 Convex sets in $$3$$ dimensions (including convex surfaces) 53A05 Surfaces in Euclidean and related spaces