Peter E. Francis

Publications

• [2021] Opponent Hand Estimation in the Game of Gin Rummy (Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 35 No. 17)

  Abstract: In this article, we describe various approaches to opponent hand estimation in the card game Gin Rummy. We use an application of Bayes' rule, as well as both simple and convolutional neural networks, to recognize patterns in simulated game play and predict the opponent's hand. ...

• [2021] Secrets and Quantifiers (Math Horizons, MAA)

  American writer, teacher, and comedian Sam Levenson gives us the following warning. "Somewhere on this globe, every ten seconds, there is a woman giving birth to a child. She must be found and stopped."...

• [2020] Maximal Sizes of Weak (2,1)-Sum-Free Sets in Finite Abelian Groups (Gettysburg College Research Papers In Mathematics)

  Abstract: The finite abelian group G is type I if |G| has a prime divisor congruent to 2(3), type II if |G| is divisible by 3 but has no divisors congruent to 2(3), and type III if all divisors of |G| are congruent to 1(3).... (Presentation)

• [2019] The Maximum Size of Weak (k, l)-Sum-Free Sets (Gettysburg College Research Papers In Mathematics)

  Abstract: A subset A of a given finite abelian group G is called weakly (k, l)-sum-free if the set of all sums of k distinct elements of A is disjoint with set of all sums of l distinct elements of A.... (Presentation)

Other Writing

• [2022] Cαlculus

  This text is meant to serve as a condensed resource for a student taking their first or second semester of calculus

  
• [2021] The Beautiful Mathematics of IFS Fractals

  Abstract: Just a few random numbers and 2-by-2 matrices can be used in unison to create spectacularly beautiful images of planar fractals. These iterated function systems (IFS) are easy to understand, but lead to deep constructions...

• [2021] The Beauty of Schottky Groups

  Abstract: Möbius transformations are a convenient group of maps to study. We will explore a specific kind of Möbius transformation used to generate particular discrete subgroups of Möbius transformations called Schottky Groups. We will then use Schottky Groups to explain and make fractal-like images out of circles.

• [2020] On Moments and Centroids

  While serving as a PLA for a Calculus 2 class, I realized that the students were not understanding the origin of the formulas they were trying to memorize and use, which motivated me to write this short tutorial.

  
• [2020] Algebra in the Game "Lights Out"

  (In Progress) "Lights Out" is a game using binary-state cellular automota. This paper explores the algebraic structures the game admits.

Web Tools

• ifs-Fractals
  

  A showcase and collection of the Mathematics and beauty of fractals generated with iterated function systems. This web app allows for the creation and sensitivity investigations of IFS fractals.

• AddComb.js
  

  is a javascript library for additive combinatorics calculation. This web app allows has four Additive combinatorics tools free for use: a function calculator, sumset calculator, table generator, and playground Jupyter-style notebook.

• Complex Function Plot
  

  This web app plots functions f from the complex plane to itself by first coloring the plane and then recoloring the plane, assigning to z the color at f(z). I believe this is an improvement on software written in MatLab: with this tool, you can upload your own image to be used as domain coloring.

• Color Pixels as Points
  

  This tool is an interface for some javascript that colors pixels on an HTML canvas element.

• Table Editor
  

  This is a web app that can create, edit, convert, export, and save table files, such as CSV, TSV, and others.

• Autostereogram Generator
  

  A tool to generate custom autostereograms from black and white images.

• Random Generators
  

  Here is an assortment of random generators that I have been requested to make.

Python Modules

• graphPlot
  

  graphPlot is a Python module (released on PyPi) that uses physical simulation to find ideal plotting coordinates for graphs.

• ifsFractals-py
  

  ifsFractals is a Python module (released on PyPi) that can be used to generate fractals that are fixed point sets of contraction mappings.

Games

• Canvas Games
  

  This is a collection of games that are all made using the HTML5 canvas element and javascript. They are intended to be fun as well as instructional for somone wanting to make thier own.

• Online Texas Hold 'em
  

  This is a Texas Holdem' web application that allows users to create and join games. My intention is that players use Zoom (or equivalent) simultaneously.

• Bingo Caller
  

  I was annoyed when I struggled to find a simple web-based Bingo Caller that would be good to project on a screen, so I made one.

• Play Quadegories!™
  

  Quadegories!™ is an extremely challenging, clue-based trivia game that works the brain's executive funtion. It was developed by Brook Zelcer.

Other

• New Projects and Problems
  

  (Always in Progess) This CoCalc folder contains some of my "backburner" projects and ideas. Some subfolders have unanswered questions and probelems, so feel free to take a look!

• Lorentz Attractor Animation
  

  I made this animation as a demonstration of the way to animate solutions to three-dimmensional systems of ODEs.

• MathWeather.live
  

  After hearing the chair of the Math Department at Gettysburg College (Dr. Ben Kennedy) asses the "math weather" at the beginning of each colloquium, I decided to make a web app to do it everyday!

• Sample LaTeX Files
  

  Everyone who has done it knows that learning LaTeX can be a struggle at first, but the rewards are huge! I made these templates so that people who have never learned a markup language before can get started with typesetting.

• Desmos Graphs
  

  Building and using Desmos Graphs can give great intuitive and visual understaing of various topics. These are some of the graphs that I have made that I either find helpful or nice to look at.

• Python Course
  

  (In progress) This is the homepage for an introduction to Python Programming course (and textbook) I wrote and have taught a few times upon request.

Philosophical Writing

• What is the Scientific Method?
  

  Although 'The Scientific Method' is a fairly ubiquitous phrase, we will begin to see that the poseter flowchart that we recall adorning the wall of our fifth grade classroom does not do it justice. What is the Scientific Method? aims to take the reader through notable episodes of the history of particle physics while examining the ongoing philosophical discussion on the scientific method.

• The Life and Comedy of Tom Lehrer

  In 1967, the heart of the Cold War, a lanky Russian Jewish mathematician from the United States sat at a piano in front of a large audience in Copenhagen, singing a satire about mutually assured destruction and nuclear holocaust, that was televised to viewers around the world. Should we be that surprised?...

• Ironic Power: Analyzing Gender Roles in Transition from Phase I to Phase II of Jewish Comedy

  World War II induced a fear in American-Jewish communities, which instigated a sharp turn from the chaotic energy that had been driving the subversive Jewish comedy in America. However, the common and blatant sexism of the time affected Jewish male and female comedians in opposite ways.... (Photo Slides)

Fiction Writing

• The Year That Everyone Forgot

  (In progress) I have always enjoyed writing short stories, and I have begun collecting and organizing them into a book.

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• A Short Melody to The End

  (In progress) A Short Melody to the End is a one Act Play that I am writing. I hope to be able to announce the debut soon!

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Music

• Original Compositions and Arrangements on Noteflight

  I have arranged and composed for various groups including marching band, choir, piano, and small chamber ensembles. Here you can access my Noteflight Marketplace as well as view all of my public scores.

Photography

• Flickr

  I started photography in high school, and I started working professionally for the office of Communications and Marketing at Gettysburg College in 2018. I enjoy shooting portaits and events, but I have also done wildlife, landscape, and travel photography.