Digital Jordan Curves and Surfaces with Respect to a Closure Operator

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

Knobbly but nice

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

On realizabilty of Gauss diagrams and constructions of meanders

The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.

Consider the less traveled path, because of the Jordan Curve Theorem

“Consider the less traveled path, because of the Jordan Curve Theorem” offers a basic introduction to simple, closed curves and explains why the theorem asserting that every simple closed curve in the plane separates the plane into an “inside” and an “outside” is best appreciated when considering pathological curves. A pathological curve, such as a space-filling curve or the Koch snowflake, is one that lacks features of so-called well-behaved curves. The discussion is enhanced by numerous hand-drawn sketches and a reference to “The Road Not Taken” by poet Robert Frost. Mathematics students and enthusiasts are encouraged to consider the less traveled path in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

A closure operator for the digital plane

We introduce and study a closure operator on the digital plane Z2. The closure operator is shown to provide connectedness that allows for a digital analogue of the Jordan curve theorem. This enables using the closure operator for structuring the digital plane in order to study and process digital images. An advantage of the closure operator over the Khalimsky topology on Z2 is demonstrated, too.