 Two-triangle disc and Möbius-band model of the projective plane

In a triangle whose two sides at an apex are glued with that apex fixed, cross-sections become circles and the apex becomes their cone point. This quotient is a disc. A second triangle with boundary word $aa{b}^{-1}$ becomes a Möbius band after the $a$ sides are identified. Gluing the remaining $b$ boundary to the first disc caps the band and gives the real projective plane. The resulting two-face cell structure has two vertices, three edges and Euler characteristic one.