Exploring Discrete Geometry (MATH 46800 Directed Reading)

Reading: Exploring Discrete Geometry by Thomas Q. Sibley

Discrete geometry, like all of mathematics, has developed piecemeal. It is still new enough to have unsolved problems that seem just a few steps from introductory ones. The problems of [Chapter 1] seek to whet your appetite. Think about them—draw pictures for small cases, look for patterns, make conjectures, and try to solve them. Complete answers are great, but so are partial answers, guesses, and even mistakes. Recall: mathematics is not a spectator sport; so enjoy engaging with these problems. We’ll revisit these problems in later chapters, providing answers, relevant mathematics, and especially new questions probing these areas more deeply.

For a preview of the book, click here.

 

Expectations:

  • 1 one hour meeting per week (1 credit for MATH 46800)
  • 1-3 hours of homework (assigned readings or problems)

 

Prerequisites:

  • High school algebra and geometry
Name of research group, project, or lab
Mulholland Directed Reading Group
Logistics Information:
Project categories
Mathematics
Student ranks applicable
First Year
Sophomore
Student qualifications

Familiarity with common geometrical and algebraic concepts.

Hours per week
1 credit / 3-6 hours
Compensation
Research for Credit
Number of openings
2
Techniques learned

Discrete geometry can improve valuable visualization skills because it asks different kinds of questions from those asked in traditional geometry classes. The problems in discrete geometry, like any area of mathematics, will also improve your problem solving skills, a valuable asset for anybody. The process of “questioning the answer” and the variations it spawns raises problem solving to a new level. From a mathematician’s point of view, the variations provide more important benefits: they help deepen mathematical reasoning and geometrical intuition, and they introduce one way mathematicians find new research questions. Indeed, some variations of problems we consider lead to open research questions or recent results. Mathematical research often develops from seeing patterns and making conjectures about why those patterns happen. I hope you will grow mathematically by wrestling with the variations in this book, seeing patterns in them, making conjectures about them, solving them, and even making up your own variations to explore! But the main motivation for writing this book is that I found these problems and variations fun to play with—and I hope you find much pleasure in pondering them as well.

from the Introduction of the text.

Project start
Spring 2025
Contact Information:
Mentor
Kathryn Mulholland
kburton@nd.edu
Assistant Professor of the Practice
Name of project director or principal investigator
Kathryn Mulholland
Email address of project director or principal investigator
kmulholland@nd.edu
2 sp. | 0 appl.
Hours per week
1 credit / 3-6 hours
Project categories
Mathematics