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David Richeson

Publications and source records attributed to David Richeson.

14 recordsLinked to original sources
arXiv2024

How Much String to String a Cardioid?

A residue design is an artistic geometric construction in which we have $n$ equally-spaced points on a circle numbered 0 through $n-1$ and we join with a line segment each point $k$ to $ak$ modulo $n$ for some fixed $a\ge 2.$ The envelopes of these lines are epicycloids, like cardioids. In this note, we prove that the sum of the lengths of these line segments has a surprisingly simple closed form. In particular, if one wants to make one of these designs with string, it is easy to calculate how much string is required.

math.HO↗
arXiv2024

What's the Best Seat in the Game Left, Center, Right?

Left, Center, Right is a popular dice game. We analyze the game using Markov chain and Monte Carlo methods. We compute the expected game length for two to eight players and determine the probability of winning for each player in the game. We discuss the surprising conclusions of which players have the highest and lowest chance of winning, and we propose a small rule change that makes the game a little more fair.

math.HO↗
arXiv2016

A Trisectrix from a Carpenter's Square

In 1928 Henry Scudder described how to use a carpenter's square to trisect an angle. We use the ideas behind Scudder's technique to define a trisectrix---a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.

math.HO↗
arXiv2014

Gabriel's Paper Horn

Gabriel's horn is the famous mathematical object that has finite volume and infinite surface area. This article gives a template for making Gabriel's horn out of paper cones. It also describes the mathematics behind the construction of the template.

math.HO↗
arXiv2013

Circular reasoning: who first proved that $C/d$ is a constant?

We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid's \emph{Elements}; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes's work coexisted with the 2000-year belief -- championed by scholars from Aristotle to Descartes -- that it is impossible to find the ratio of a curved line to a straight line.

math.HO↗
arXiv2011

Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

We introduce topological definitions of expansivity, shadowing, and chain recurrence for homeomorphisms. They generalize the usual definitions for metric spaces. We prove various theorems about topologically Anosov homeomorphisms (maps that are expansive and have the shadowing property) on noncompact and non-metrizable spaces that generalize theorems for such homeomorphisms on compact metric spaces. The main result is a generalization of Smale's spectral decomposition theorem to topologically Anosov homeomorphisms on first countable locally compact paracompact Hausdorff spaces.

math.DS↗
arXiv2010

Entropy for symbolic dynamics with overlapping alphabets

We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of entropies of (standard) full shifts. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.

math.DS↗
arXiv2009

Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of $0\in S^{1}=\R/\Z$ under the rotation by $α\in\R\bs\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^{1}$, a finite union of closed intervals, can we recover $α$ and $I$? We prove that the itineraries do determine $α$ and $I$ up to certain equivalences. Then we present elementary methods for finding $α$ and $I$. Moreover, if $g:S^{1}\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.

math.DS↗
arXiv2009

Symbolic dynamics for nonhyperbolic systems

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.

math.DS↗
arXiv2008

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.

math.DS↗
arXiv2002

A fixed point theorem for bounded dynamical systems

We show that a continuous map or a continuous flow on $\R^{n}$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in $\R^{n}$ intersects W then there is a fixed point in W. Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point.

math.DS↗
arXiv2002

Bounded homeomorphisms of the open annulus

We prove a generalization of the Poincaré-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the study of bounded homeomorphisms of the open annulus. We prove a fixed point theorem for bounded homeomorphisms and study the special case of those homeomorphisms possessing at most one fixed point. Lastly we use the existence of rational rotation numbers to prove the existence of periodic orbits.

math.DS↗
arXiv1999

Shift Equivalence and the Conley index

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

math.DS↗