Date of Award
1998
Degree Name
Mathematics
College
College of Science
Type of Degree
M.A.
Document Type
Thesis
First Advisor
John Lancaster
Second Advisor
Evelyn Pupplo-Cody
Third Advisor
John Drost
Abstract
Much of what we call fractal geometry today can be traced to the critical years around the turn of the 19th/20th century. One of the cracks in what some had considered to be a virtually complete mathematical edifice may have been the publication in 1883 of Georg Cantor's famous "perfect and dense-in-itself' set. This set, a set of points on the real line everywhere dense in itself but nowhere dense on the line, was considered at best paradoxical and at worst monstrous. It challenged standard notions of continuity since Cantor was able to prove that there existed a one-to-one and onto mapping between his set and the unit line segment, for example. Yet the Cantor middle-thirds set, as it is called, is completely separated. Further developments along these lines resulted in a proof for the existence of curves that could be continuous everywhere but differentiable nowhere, again flouting received wisdom. The first example of such a curve was published by Giuseppe Peano in 1890. The Peano curve was soon followed by other examples, notably those of David Hilbert and Helge von Koch and the interesting constructions of Sierpinski and Menger. Work in this area is also related to the beautiful creations of Gaston Julia and Pierre Fatou.
Subject(s)
Random measures.
Multifractals.
Fractals.
Recommended Citation
Sunderland, Kenneth E., "Random fractals and multifractal measures" (1998). Theses, Dissertations and Capstones. 2156.
https://mds.marshall.edu/etd/2156
