# Shredding the torus

We have considered several examples of cutting the torus along the lines parallel to the fundamental square of its plane model. Here I would like to discuss a more interesting problem where the cut is made along a slanted line. The problem of shredding a torus along such a line has two distinct solutions depending on whether the slope of the cut is rational or not.

First we'll look into the case of a rational slope which, perhaps surprisingly, under a general umbrella of the puzzles on graphs, is related to the Three Glass puzzle.

It's perhaps less surprising that the irrational case appears to be deeper, and the mathematics involved is less trivial. The case draws on a beautiful fact from the theory of rational approximation.

## Rational Slope

Thus, let's first assume the slope of a cut is a rational number $r = p/q$ $(3/4$ on the diagram). Imagine the model square whose sides we identified to obtain a torus, embedded into a 2-dimensional integer grid (lattice). Previously, when the cut reached a side of the square we used identification of the opposite sides to skip to another side. Now, instead, let's continue the line beyond the square. Observe, that we can start the cut at the point $(0,0).$ Indeed, if we proceed backwards from a torus to the square, we'll have to cut the torus along two perpendicular lines that will become the sides of the square. Therefore we may position the square any way we want.

There are two ways to visualize the cut of the torus using its plane model. One, as before: continue the slanted line until it reaches one of the sides. Then follow the blue lines to the opposite side, and, from there, continue the line with the same slope, and so on. The second one is to cut the plane along the grid lines and stack the squares on top of each other. Assuming the plane transparent and looking from the top of the stuck, we'll see the resulting line segments inside a single square. Convince yourself that both ways lead to the same configuration of lines.

## Reference

1. R.J.Wilson, Graphs And Their Uses, MAA, New Math Library, 1990.

Copyright © 1996-2017 Alexander Bogomolny

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