Graduation Year

Spring 2013

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science



Reader 1

Francis Edward Su

Reader 2

Michael E. Orrison

Reader 3

© 2013

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© 2013 Connor Ahlbach


The Poincare-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincare-Miranda Theorem that are discrete in nature - that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. Finally, we extend these discrete ideas to attempt to prove the existence of a zero with the boundary condition of Morales.