Conway's doughnuts

TL;DR

The paper explores Conway's doughnuts diagrams, a class of geometric configurations related to Morley's Theorem, and discusses their existence proofs using holonomy methods for specific cases.

Contribution

It introduces Conway's doughnuts diagrams as a generalization of Morley's Theorem and applies holonomy techniques to prove their existence for certain cases.

Findings

Existence of Conway's doughnuts diagrams for n=2, 3, 4

Application of holonomy method for diagram existence proofs

Connection to Morley's Theorem and angle trisectors

Abstract

Morley's Theorem about angle trisectors can be viewed as the statement that a certain diagram `exists', meaning that triangles of prescribed shapes meet in a prescribed pattern. This diagram is the case n=3 of a class of diagrams we call `Conway's doughnuts'. These diagrams can be proven to exist using John Smillie's holonomy method, recently championed by Eric Braude: `Guess the shapes; check the holonomy.' For n = 2, 3, 4 the existence of the doughnut happens to be easy to prove because the hole is absent or triangular.