Tara Taylor

A NEW CLASSIFICATION OF NON-OVERLAPPING SYMMETRIC BINARY FRACTAL TREES USING EPSILON HULLS

This paper presents an analysis of non-overlapping symmetric binary fractal trees using methods of computational topology. In particular, we study the topological aspects of the epsilon-hulls of the trees as epsilon ranges over the non-negative real numbers. The complexity of a tree is defined in terms of the maximum range of the number of levels of holes that can be present for a given ∊. All self-contacting trees have infinite complexity, with the important exceptions of the self-contacting trees with branching angles 90° and 135° (which are space-filling). These two angles have been identified by Mandelbrot and Frame as being topologically critical, and our analysis provides further support to that claim. We will show that for trees with branching angles 90° or 135°, there is a finite upper bound to complexity. For all other branching angles there is no upper bound to complexity. Self-avoiding trees are topologically equivalent because they are all contractible. Our new definition of complexity provides a new way to compare self-avoiding trees.

Thin Sets with Fat Shadows: Projections of Cantor Sets

A Cantor set is a nonempty, compact, totally disconnected, perfect subset of IR. Now, the set being totally disconnected means that it is scattered about like a “dust”. If you shine light on a clump of dust floating in the air, the shadow of this dust will look like a bunch of spots on the wall. You would be very surprised if you saw that the shadow was a filled-in shape (like a rabbit, say!). That would be pretty unbelievable. So, is this possible? We can think of the projection of a Cantor set onto a subspace as the shadow on that subspace. Is it possible that a cloud of dust (a Cantor set) could have a shadow (projection) which is “filled-in” (homeomorphic to the n − 1 dimensional unit ball)? The answer is YES! In fact, it is possible to have the shadow in every direction be “filled-in”! In this note we give an example of a simple construction of a Cantor subset of the unit square whose projection in every direction is a line segment. This construction can easily be generalized to n dimensions.

Exploring Concepts from Abstract Algebra Using Variations of Generalized Woven Figure Eights.

Abstract Students often struggle with concepts from abstract algebra. Typical classes incorporate few ways to make the concepts concrete. Using a set of woven paper artifacts, this paper proposes a way to visualize and explore concepts (symmetries, groups, permutations, subgroups, etc.). The set of artifacts used to illustrate these concepts is derived from our investigation of open-work woven mats produced in several cultures in the South Pacific. The exemplars that will be shown present variations of the figure eight, and can be created using readily available materials and straightforward instructions.