Non-Euclidean Geometry

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. 
-Stephen Hawking

The world is not flat, why does geometry only study planes? In this seminar we go off the beaten path and explore surfaces that are mostly flat. Some of the ideas will extend nicely, but more often we need to explore and discuss situations before proposing analogous definitions and theorems. Despite the small changes made to the surface, we will discover a cohesive intuition for what happens in curved spaces.  

Am I ready for this? 

This course builds on many of the central topics in Euclidean Geometry. This document contains examples of questions you should be able to understand to ensure you are prepared for the explorations in this seminar.

What are some examples of what we will see?

This document is an introduction to our first expedition, and you are encouraged to think about the questions before our first meeting. It also contains some additional expeditions that we may undertake throughout the year.

What are some applications of this geometry?

Any time something is made from flat materials, this geometry describes the result.

  • Clothing is made from flat textiles, but by sewing along seams designers create beautiful shapes.  
  • Surgeons slice into skin that lays flat, but closing up that incision when the area changes in a way that it heals flat is well modeled by this form of geometry.   
  • Have you ever looked closely at the seams on new soccer balls? It makes you wonder how baseballs are made from two flat pieces of leather (???)!  
Where do people usually learn this?

Short answer: they don’t. Amazingly, what Euclid articulated created the territorial boundaries of geometry for nearly 2000 years, with every innovation limited by his five postulates. In the 1800s there was a rapid development of Non-Euclidean geometry, focusing entirely on smooth surfaces and soon after it became the foundation for Einstein’s theory of relativity. With little reason to go back and fill in the gaps, formal treatments of this modern geometry are only accessible at advanced undergraduate or graduate institutions.

Summery for Non-Experts

This seminar picks up where Euclid left off. It uses many of the techniques and ideas from a typical Geometry course to explore the strange consequences of a cone point or seam. How would an ant or ‘circle creature’ discover this geometry *intrinsically* (by wandering around the surface) and not *extrinsically* (by leaving the surface and looking back to it). Accomplishing this requires participants to reinterrogate all of the foundational aspects of geometry, deepening their understanding and improving their fluency. The material lends itself to exploration, making it the perfect place for someone who has become disillusioned with the drudgery of algebra to rekindle their love of mathematics.

Summery for Experts

This seminar extends the Euclidean theorems on triangle congruence, parallel lines, and chords/tangents/secants of circles, building up analogous results on piecewise flat surfaces. The bigon (2-sided polygon) makes an early appearance as we discover two different “straight” paths between a pair of points. While this is upsetting for some, the discovery that equilateral and equiangular bigons are not the same thing is sure to perplex and excite the curious adventurer.

Participants should be expected to develop strong intuition for point- and seam-curvature. They will understand geometric and topological constraints on convex and concave polyhedra. Additional conversations will introduce how these tangible constructions lead to notions like Gaussian Curvature, Spherical Trigonometry, and fractional dimensional spaces. 

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