Memories Awakened by “The Farmer Was Replaced”

Memories Awakened by “The Farmer Was Replaced”

It seems like a very trivial question: “What should a drone do if it is on the lower (South) edge of the field and it has a move(South) command to perform?” But, strangely, this question leads to quite an interesting set of consequences.

In The Farmer Was Replaced the drone teleports to the other end of the field. It means that the upper edge of the field is glued to the lower edge. As I drew it, a square with upper and lower edges glued together becomes a cylinder from the topological point of view. Then the same goes for the left and right boundaries, and we have a torus.

There are other options. For instance, if the drone disappears on the lower edge and then appears on the left one (and similarly with upper and right), we have a sphere.

But that’s not all. You can play with the orientation of edges. If the drone appears at the beginning of the lower edge and then at the end of the upper edge, it means that we reversed the orientation. After we finish gluing edges, we get a Klein bottle or the projective plane. Unfortunately, you can’t embed these figures in 3D without self-intersection.

About memories. I first read about this in Anatoly Fomenko’s book. He also wrote “History: Fiction or Science?” (New Chronology) and, while he’s quite famous for that, he’s also a mathematician; his “Visual Geometry and Topology” is an interesting read. It even contains illustrations for “The Master and Margarita.” What I explained about gluing the edges of a square together is a simplified version of the introductory section of that book.