VIII - On Separating Dots

Summary

The Intermediate Value Theorem shows that for any planar region (with a well-defined meaning of “area”) there is a straight line that separates it into two parts with equal area. The Two-Pancake Theorem states that two planar regions can be simultaneously so divided with a single straight line cut. (See [TANTON] for details.) As we saw in Appendix III, for two continuous functions on the surface of a sphere there exist two antipodal points at which their values match.

In the Spring of 2010 students of the St. Mark's Institute research class explored discrete versions of these three classic results. Here is what they discovered.

RESULT 1: One PancakeMade Discrete

Suppose an even number of dots are scattered about a page. Then there is a straight line that

1. i) passes through no dot,

2. ii) separates the dots into two groups of equal number on either side of the line.

Proof. Let the number of dots be 2N.

Sweep a horizontal line down from the top of the page across the dots towards the bottom of the page. Label the dots 1, 2, 3, …, 2N according to the order in which the line encounters them. If the line encounters several dots simultaneously, label the numbers from left to right.

If the dots labeled N and N + 1 have different heights, then any horizontal line between them does the trick and separates the 2N dots.

If dots labeled N and N + 1 have the same height, then draw the horizontal line that passes through them and rotate it about the midpoint of the line segment connecting them.