4 ideas
Cut a wedge out of a flat paper disk and glue the edges to make a cone. Everywhere except the apex the paper is still flat, so all of the curvature is concentrated at that single point and equals the angle removed. Curved surfaces can then be approximated as flat facets joined at such conical points, which turns smooth curvature into something countable.
Anselme Lanturlu draws straight lines on a paper cone, meaning lines that stay straight when the paper is unrolled flat. A triangle made of such lines that does not enclose the apex has angles summing to exactly 180°. A triangle that encloses the apex has angles summing to 180° plus the angle of the wedge that was cut out.
A creature confined to a surface, unable to see it from outside, can still measure its curvature. It does so by walking straight paths and checking whether triangle angles sum to more or less than 180°, or whether a vector carried around a loop returns rotated. Curvature is an intrinsic property of the space, not a bending into some extra dimension, and this is what lets physicists speak of curved spacetime without an outside view.
If a surface is allowed to pass through itself but not to tear or crease, a sphere can be continuously deformed until its inside becomes its outside. Boy's surface, a closed self-intersecting surface, serves as a key halfway stage in drawing this process. The albums show the eversion step by step in drawings, arguing that topology can be taught through careful pictures rather than formalism.