The Discovery of Non-Euclidean Geometry

Sample Abstract
Euclid's Fifth Postulate (paraphrased): Given a line and a point not on that line, there is exactly one line through the given point parallel to the given line.

Euclid's Parallel Postulate spawned the longest-standing controversy in the history of mathematics. Could it really be that the fifth postulate does not follow from the other four? Why does Euclid prove the converse of his fifth postulate but not the fifth postulate itself? Only after two thousand years of failed attempts to demonstrate the Parallel Postulate was it finally established that the statement cannot be proven or disproven from Euclid's other postulates, and that assuming its opposite yields a new and radically different form of plane geometry.

This talk will give an overview of the events leading to this discovery, which heralded a revolution in modern mathematics, and a description of the properties of the non-Euclidean plane. The technical prerequisites are very basic Euclidean geometry (e.g., the sum of the angles in a triangle is 180 degrees) and a little bit of trigonometry, although the latter is not essential.

This talk evolved from a lecture I gave in graduate school motivated by the pedagogical complaint that we never tell our students the history of the math we teach them. With a mix of slides, posters, and models, this talk meanders through twenty-three centuries of mathematical history. The path is marked by a list of equivalents of Euclid's Parallel Postulate, and the talk concludes with a demonstration that each of these equivalents fails in the non-Euclidean plane.

Totally user-friendly.

Almost all of the mathematics can be followed with a modest awareness of Euclidean geometry, especially facts about angles in triangles and in parallelograms. Some ideas from trigonometry and calculus appear in the final section, but listeners can easily follow the intuition even if these concepts are unfamiliar.

The lecture requires a computer projector and a surface such as a blackboard to which signs may be taped and viewed while the projector screen is down. The signs and models are visible in a typical classroom for 150 people or fewer.


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If you click on "Non-Euclidean Geometry", you will find Joel Castellanos's package for making compass and straightedge constructions in the Poincare disk. If you click on the blue tiling, you will see an Escher print that I use to illustrate the Poincare disk in this talk.

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