The Strangely Serious Implications of Math’s ‘Ham Sandwich Theorem’

Geometry offers a surprising insight into both everyday mishaps and political quandaries through the “ham sandwich theorem.” This mathematical principle asserts that for any three objects, no matter how irregularly shaped or oriented, there exists a single straight cut that can bisect all three simultaneously. This theorem extends to higher dimensions, suggesting that n objects in n-dimensional space can be bisected by an (n – 1)–dimensional cut.

The theorem’s proof involves a simplified scenario with a circle and a blob in two dimensions, using a windmill-like rotation to find a bisecting line. This concept scales up to more complex shapes and dimensions, and even applies when objects are fragmented, such as ham cut into snowmen or bread into croutons.

Beyond its quirky applications, the theorem has serious implications for gerrymandering, the manipulation of electoral district boundaries for political gain. It demonstrates that even with restrictions on district shapes, gerrymandering can still occur through strategic subdivisions, as the theorem can be used to ensure a political majority in every district, regardless of the overall population’s preference. Thus, the ham sandwich theorem reveals that simple geometric solutions may not suffice to address the complexities of political gerrymandering.
Read more at Scientific American…