A Program For Surface Classification
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Authors
Dennis, Sarah
Issue Date
2020-05-01
Type
senior_thesis_campus
Language
Keywords
topology , theorem , Mobius , JavaScript , polygon , computers , Computer Sciences
Physical Sciences and Mathematics
Alternative Title
Senior Theses
Abstract
This thesis presents the classification theorem of compact connected surfaces, its proof, and a computer implementation. The theorem, first stated by Mobius in 1861 and more formally by Jordan in 1866, states that every compact, connected surface is homeomorphic to either a 2-sphere, a connected sum of tori, or a connected sum of projective planes. The proof we present consists of an algorithm to take a polygon representation to normal form. A polygon representation of a surface is essentially the result of cutting the surface up until it lies flat, where we identify the cut edges such that we could paste it back together again. A polygon representation in normal form has distinguishable tori or projective plane components and can be easily classified from this state. The computer implementation is written in JavaScript and includes a graphical interface that allows the user to progress through the algorithm by cut and paste operations. The program is intended as a learning tool for students in introductory topology courses.
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