College of Arts and Sciences
Master of Science in Mathematical Sciences (MS)
Mathematics & Statistics
Dr. Jingcheng Tong
Dr. Scott Hochwald
Dr. Peter Braza
Dr. Donna Mohr
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.
Samons, John David, "A Relationship Between the Fibonacci Sequence and Cantor's Ternary Set" (1994). UNF Graduate Theses and Dissertations. 285.