Examples

Gluing shapes

Gluing shapes

• Glue a pair of triangles along an edge.
• Glue a line segment to a triangle, endpoint to vertex.

Real projective space RPn

Real projective space ${\mathbb{RP}}^n$

Consider the unit sphere ${\mathbb{S}}^2\subset {\mathbb{R}}^3$. Antipodal points of ${\mathbb{S}}^2$ are those determining a straight line passing through the origin.

Define ${\mathbb{RP}}^2$ as the quotient ${\mathbb{S}}^2/\sim$ where equivalence classes of $\sim$ are the pairs of antipodal points.

Define ${\mathbb{RP}}^n$ similarly by gluing together antipodal points in ${\mathbb{S}}^n$.

We can also describe ${\mathbb{RP}}^2$ using a disk with antipodally glued boundary:

This surface is also derivable by gluing a disk to a Mobius band: