Theorem (Hopf and Pannwitz, 1934): Let be a set of points in the plane in general position (no three points on a line) and consider line segments whose endpoints are in . Then there are two disjoint line segments.
Micha Perles’s proof by Lice:
Useful properties of lice: A louse lives on a head and wishes to lay an egg on a hair.
Think about the points in the plane as little heads, and think about each line segments between two points as a hair.
The proof goes as follows:
Step one: You take lice from your own head and put them on the points of $X$.
Step two: each louse examines the hairs coming from the head and lay eggs (on the hair near the head)
Step three (not strictly needed): You take back the lice and put them back on your head.
To make it work we need a special type of lice: spoiled-left-wing-louse.
A spoiled-left-wing louse lays an egg on a hair if and only if the area near the head, 180 degrees to the right of this hair is free from other hairs.
Lemma: Every louse lays at most one egg.
Proof of lemma: As you see from the picture, if the louse lays an egg on one hair, this hair disturbs every other hair.
Proof of theorem continued: since there are line segments and only at most eggs there is a hair X between heads A and B with no eggs.
We look at this hair and ask:
Why don’t we have an egg near head A: because there is a hair Y in the angle 180° to the right.
Why don’t we have an egg near head B: because there is a hair Z in the angle 180° to the right.
Y and Z must be disjoint. Q. E. D.
Remarks: We actually get a ZIG formed by Y, X, and Z
If we use right-wing-spoiled lice we will get a ZAG.
We can allow the points not to be in general position as long as one hair from a head does not contain another hair from the same head.