"Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of a sculpture..."

I recently found this drawing I did in 1st grade.

In 2002 I completed my master's thesis on Desargues' Triangle Theorem under the direction of Dr. Elena Anne Marchisotto at California State University, Northridge.

Thesis dedications and special thanks

Girard Desargues

AN EXPLORATION OF DESARGUES’ TRIANGLE THEOREM

Introduction

The triangle theorem of Girard Desargues (1591-1661) is the focal point for this thesis.  We begin with a history of its creator and his work, briefly tracing the development of the branch of mathematics where it resides:  projective geometry.  Our examination of the triangle theorem will involve interpretation and analysis of Desargues’ original proof.  Next we trace the evolution of different proofs of the theorem constructed by noted geometers of the nineteenth and twentieth centuries.  We analyze these proofs in comparison to one another and to Desargues’ original demonstration.  Within the context of these proofs, we construct a new proof using the postulates of the Italian geometer Mario Pieri (1860-1913).  Our goal in examining different proofs will be to identify distinct threads in the evolution of proofs of the theorem, as well as to demonstrate (with our proof from Pieri’s postulates) a proof based on a minimal set of axioms.  We will also focus on the assumptions different geometers made in constructing their proofs and show in several cases the equivalence of their axioms to those of Pieri.  

            Mathematician Gian-Carlo Rota (1932-1999) claimed Desargues’ theorem has “many more applications both in geometry and beyond than any theorem in number theory, maybe even more than all the theorems in analytic number theory combined.”  Nathan Altshiller Court (1881-1968), a well-known geometer, expressed a similar opinion, noting that the most striking thing about this theorem is its “fertility,” and describing the consequences of the theorem as “rich and far-reaching.” We will explore this richness in some of its applications.  We will consider special characteristics of certain geometric structures, distinguishing between planes in which the theorem holds and those where it fails.  We will also examine what is known as the “Desargues configuration,” and the finite geometry constructed from it.

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Me and Dr. Marchisotto

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