Plots to take over the world

The topology of the puzzling shape I discussed in yesterday’s post is really fascinating. In some ways it acts like a square, but a very strange square.

Imagine that you lived in a house on a square plot of land. On all four sides you have neighbors, and when you step from your property onto an adjoining property, that neighboring yard seems like a perfectly normal square. So wherever you go, everything appears fine and undistorted.

But the rules for travel are a little interesting, because things act kind of funny at some of the corners. Your neighborhood to the southwest and northeast seems normal — each neighbor shares one other mutual neighbor between them.

But to the southeast things are a bit stranger. Your neighbors there share a border with each other — it takes only three border crossings and right-angle turns to get back into your own yard. And your neighbors to the northwest have two neighboring yards between them — around that shared corner, it takes five border crossings and right-angle turns to get back into your own yard:

As we already know from the posts of the last two days, the world you are living on consists of sixty plots of land, and the topology of that world is a sphere.

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