“When I use a word, it means just what I choose it to mean — neither more nor less.” - Humpty Dumpty

Creating new words can clarify mathematical communication and logic, especially when forging new pathways. In this session, a few new words will help us approximate rationality in a purely geometric context. These proofs will not use high-powered tools, so leave your slithy toves at home. Instead, we will rely on geometric constructions, proportional reasoning, tiling, dissection, and proof by descent.

We will determine whether 3, 4, 5, 6, and 9 are “grational,”as defined below. We will use the stealth of geometry to sneak up on number theory, such as proving all perfect squares are grational.

Define a nice-gon to be a regular polygon with integer side lengths in Euclidean geometry. A nice n-gon is a nice-gon with n sides. An integer n is grational if and only if there exists a nice n-gon such that its area equals the sum of areas of n congruent nice n-gons.

Open questions about grationality will follow you home and may inspire your next mathventure. What would grationality look like with higher values or with higher dimensions? How is grationality related to rationality? Might grationality be related to the work of Hippasus of Metapontum, described by Plato?