**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.

**Description**

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.

**Level**

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.

**Mechanics**

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.

**Appearances**

- Harvard University Graduate Student Seminar, November 1995
- Amherst College MAA Student Chapter, March 1996
- State College Area High School, January 1997
- Math Circle, Boston, February 1997
- Mathcamp, August 1997
- McMaster University Graduate Student Seminar, December 1999
- Radical Pi, Ohio State University, November 2001
- Math H294 Guest Lecture, Ohio State University, May 2001
- Ross Program, Ohio State University, July 2002
- Natural Sciences and Mathematics Colloquium, St. Mary’s College of Maryland, November 2005
- Mathematics Colloquium, Virginia Commonwealth University, April 2009

Online Flier for Radical Pi Talk

*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.*

Printed Flier for Ross Program Talk.