Introduction
This week we played games on different surfaces. We played tic-tac-toe and chess on a torus and on a Klein bottle!
A torus is the surface of a doughnut (the plural is tori):
To make a torus, take a square and glue the top and bottom edges together to make a tube, and then the side edges together to make a doughnut shape. If we make the identifications of the edges of the square abstractly without actually gluing them together we get a flat torus (match the arrows):
A Klein bottle is created in a similar way to a torus with one change. Instead of gluing the sides of a square together above to make a torus, we can glue the top and bottom edges to form a tube. Then glue the two ends of the tube with a twist (match the arrows):
which results in the following picture of a Klein bottle:
A "transparent" view of a Klein bottle:
Inspiration for this week's session was taken from The Shape of Space, by Jeffrey R. Weeks (2002).