This video [mp4] shows a path in the moduli space of pairs of plane curves. The black curve has degree 2 and the blue curve has degree 3. Bézout's theorem predicts that these curves should intersect in 6 points. The video illustrates a proof of Bézout's theorem by degenerating both curves to parallel lines. The essential observation is that the number of intersection points does not change as we traverse the path.

Michael Artin supposedly said that to understand stacks you really only need to understand the moduli space of triangles. A metric triangle is a triple of positive real numbers a, b, and c that satisfy the triangle inequalities

a + b ≤ c

a + c ≤ b

b + c ≤ a.

Video 2 [mp4] shows a path in the moduli space of metric triangles. The next video [mp4] shows a loop in the same moduli space.

The video on the left below shows a trivial loop in which all of the triangles are equilateral.

Any triangle can be continuously deformed to an equilateral triangle, and any path in the moduli space of triangles can be continuously deformed until every triangle in the family is equilateral. The last video [mp4] shows what happens if the loop in Video 2 is deformed in this way.

Notice that both loops consist entirely of equilateral triangles, so they correspond to maps from the circle to the moduli space of triangles that send the entire circle to the single point corresponding to an equilateral triangle. Yet one can see that the two loops are not homotopic to each other! Therefore there are several loops in the moduli space of triangles concentrated at a single "point"!

The videos here were created with Sage.