Year

1994

Paper Type

Master's Thesis

College

College of Arts and Sciences

Degree Name

Master of Science in Mathematical Sciences (MS)

Department

Mathematics & Statistics

Committee Chairperson

Dr. Jingcheng Tong

Second Advisor

Dr. Scott Hochwald

Rights Statement

http://rightsstatements.org/vocab/InC/1.0/

Third Advisor

Dr. Peter Braza

Department Chair

Dr. Donna Mohr

Abstract

The Fibonacci sequence and Cantor's ternary set are two objects of study in mathematics. There is much theory published about these two objects, individually. This paper provides a fascinating relationship between the Fibonacci sequence and Cantor's ternary set. It is a fact that every natural number can be expressed as the sum of distinct Fibonacci numbers. This expression is unique if and only if no two consecutive Fibonacci numbers are used in the expression--this is known as Zekendorf's representation of natural numbers. By Zekendorf's representation, a function F from the natural numbers into [0,0.603] will be defined which has the property that the closure of F(N) is homeomorphic to Cantor's ternary set. To accomplish this, it is shown that the closure of F(N) is a perfect, compact, totally disconnected metric space. This then shows that the closure of F(N) is homeomorphic to Cantor's ternary set and thereby establishing a relationship between the Fibonacci sequence and Cantor's ternary set.

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