The Strangely Serious Implications of Math's 'Ham Sandwich Theorem'

Abstraction: Ham sandwich theorem bisection proof and gerrymandering implications

Key points:

• The ham sandwich theorem: for any n objects in n-dimensional space, a single (n-1)-dimensional cut can simultaneously bisect all of them.
• Proven via the intermediate value theorem — continuously rotating a bisecting line through 100% to 0% of a shape must pass through 50%.
• The theorem holds even when objects are broken into many disconnected pieces (e.g., croutons, scattered points).
• Applied to gerrymandering: repeated ham sandwich subdivision can produce convex polygon districts that still give one party every seat, showing shape restrictions alone cannot prevent gerrymandering.
• Example: 60 Purple / 20 Yellow voters split so every district is 15/5, giving Purple 100% of seats despite 75% of votes.

Connections: Ham Sandwich Theorem · Gerrymandering · Intermediate Value Theorem · Combinatorial Geometry

Source: https://www.scientificamerican.com/article/the-strangely-serious-implications-of-maths-ham-sandwich-theorem/