Why is a soap bubble like a railway?

TL;DR

This paper explores the mathematical similarities between soap bubbles and railway networks, focusing on graph minimization principles and their applications in physics, chemistry, and geometry.

Contribution

It provides a self-contained introduction to graph minimization, connecting diverse scientific concepts through the analogy of soap bubbles and railway networks.

Findings

Both soap bubbles and railways minimize over graphs.

Introduction to minimal networks and structures in multiple dimensions.

Discussion of algorithms, complexity, and physical principles involved.

Abstract

At a certain infamous tea party, the Mad Hatter posed the following riddle: why is a raven like a writing-desk? We do not answer this question. Instead, we solve a related nonsense query: why is a soap bubble like a railway? The answer is that both minimize over graphs. We give a self-contained introduction to graphs and minimization, starting with minimal networks on the Euclidean plane and ending with close-packed structures for three-dimensional foams. Along the way, we touch on algorithms and complexity, the physics of computation, curvature, chemistry, space-filling polyhedra, and bees from other dimensions. The only prerequisites are high school geometry, some algebra, and a spirit of adventure. These notes should therefore be suitable for high school enrichment and bedside reading.