Chapter 1: The Geometry of Pizza Cutting

Mathematicians from the University of Liverpool have proposed a unique method for slicing pizzas that deviates from traditional straight-line cuts. This innovative approach arose from a long-standing mathematical curiosity: “Is it possible to cut at least one slice of a circular pizza into equal pieces without intersecting the center?”

Research has previously demonstrated that it is feasible to divide a pizza into 12 equal segments while ensuring at least one of the slices does not touch the center. The researchers achieved this by creating six equal sections and employing three curved lines that pass through the center, as opposed to conventional straight cuts at 30-degree angles. By subsequently halving these sections, they produced equal slices, as illustrated below.

The first video titled "The best ways to slice your pizza, according to math" explores the mathematical principles behind optimal pizza slicing.

Section 1.1: The Development of a New Slicing Technique

Dr. Joel Haddley and Ph.D. candidate Stephen Worsley devised this new slicing methodology using curves to create sections with an odd number of edges. After forming these uniquely shaped slices, they divided each one into two equal parts. For instance, the colored slices depicted in the images below demonstrate sections with 5, 7, and 9 edges, respectively. Each slice can be halved, resulting in equal portions.

Subsection 1.1.1: Understanding Tessellation

In mathematical terms, covering a plane with geometric shapes without any gaps is referred to as tessellation. For example, paving stones arranged in a specific pattern illustrate this concept. When the geometric shapes used for tessellation are identical, it is known as “monohedral tessellation.” The term "mono" refers to one, while "hedral" indicates shape. The arrangements formed by identical pizza slices serve as prime examples of monohedral tessellations.

Chapter 2: The Pizza Theorem and Its Implications

The second video titled "The Pizza Theorem" delves into the mathematical principles and implications of this theorem, providing a deeper understanding of the concepts discussed.