Brian Mintz
Dartmouth college mathematics
6188 kemeny hall

brian.a.mintz.gr(at)dartmouth(dot)edu

Math Art

George Hart once said that "Math is the most abstract art," and indeed few people see the creativity and beauty in it that many mathematicians do. By combining mathematical ideas with traditional media like sculpture and music, the broader public can appreciate this artistic side of math. In this talk, we'll explore the uncountable ways art has been infused with math, from using DeBrujin's theorem to count the patterns in traditional Japanese braiding, to edible polyhedra and much more!

slides

Below is some of the mathematical art I have made:
Pie of the aztec diamond. Pie of a pentagonal tiling. Pie of the aperiodic monotile.

These pies are inspired by Lauren Ko, author of Piometry, whose book was helpful in making these. They are joint work with Elizabeth Buchanan, Beth-Anne Castellano, and Alex Wilson. Each represents a different mathematical concept.

The first is a random tiling of the "Aztec diamond" by dominos, color coded by their orientation. One of the artic circle theorems says that a randomly selected tiling will have a circle of randomness, surrounded by "frozen" sections at the corners, where all pieces have the same orientation.

The second is one of the pentagonal tilings discovered by Marjorie Rice, an amateur mathematician that made a massive contribution to a longstanding problem about the possible tilings by irregular pentagons. It reminds us that everyone can have great ideas, and that math is something for everyone.

The last pie is the aperiodic monotile discovered by Smith, Myers, Kaplan, and Goodman-Strauss. This tile solves the "Einstein" problem, namely it is a single tile that can only cover the plane without any symmetries, which had been a longstanding question that ended up admitting a unexpectedly simple solution. This too was solved by an amateur mathematician.

Temari 1. Temari 3. Temari 2.

These are part of a series of temari I have been making. Each one starts as a styrofoam ball, is wrapped in yarn, then thread, then embroidered with a pattern. This is a craft from Japan (traditionally with a bundle of cloth instead of the styrofoam core) which originated in China. In addition to being quite stunning, they are a great avenue to play around with spherical symmetries.

The first two are different Brunnian links, extensions of the Borrommean rings, where if one band is removed, they all can separate. These designs are based on polyhedra, where each face has a modular structure that combines to form the link. Specifically, the first has a swirl of three bands for each face of an octahedron, and the second has a swirl of four bands for each face of the cube. This allows the missing band to propagate, untangling the rest of the link. I'm planning on also laser cutting wood with these patterns into tiles that you can assemble on a plane, or latch together into polyhedra. This systems can produce a unique Brunnian link for any polyhedron. I've also color coded the bands to help evoke the underlying symmetry.

The last ball is a an "orderly tangle" of sorts, a link where each component is a polygon. There are some great models of these by George Hart and also discussed in a book by Alan Holden. I've found making models with plastic straws made into triangles and square is a good way to play around with these, and have a few as well. This design uses the duality of the color scheme to draw attention to the dual nature of the octahedron and cube, every vertex of one corresponds to a face of the other.

Probability Paradoxes - Middle and Highschool Summer Math Camp

Along with other graduate students David Freeman, Alina Glaubitz, and Jinman Park, I developed material for and instructed the Exploring Mathematics Summer camp for middle and high school children in Summer 2022. One week was on paradoxical results in probability, which I summarized in a presentation.

Directed Reading Program

Along with Ben Adenbaum, I co-ran the Dartmouth mathematics DRP in the below terms. We pair undergraduate students with graduate students and post docs to learn math beyond the standard curriculum.

Hackenbush - Science day 2023

For science day 2023, graduate students Grant Molnar, Kelly Cantwell, and I, along with post doc Longmei Shu, ran a session on the game Hackenbush for middle and high school students. We made a handout for students to complete which explained how to play the game and some of the math behind it. Further references are:

- A website where you can play Hackenbush interactively and make your own games.
- Two sets of notes about the math behind this game.
- A short story explaining another way to construct the Surreal numbers:
- A Youtube video explaining this game.

Non-Transitive Dice - Sonia Kovalevsky Day

Most springs Dartmouth runs a fun-filled day of mathematics with hands-on workshops and talks for middle and high school female students and their teachers, both women and men. Originally started and funded by the Association for Women in Mathematics, The purpose of the day is to encourage young women to continue their study of mathematics and to assist the teachers of female mathematics students.

In the Spring of 2022, I assisted fellow graduate Yanbing Gu in a presentation on the four color theorem and its generalizations to higher genus surfaces realized through crochet.

In the Spring of 2023, I co-led a workshop on mathematical games, presenting on non-transitive dice. Students collected data on rolling pairs of dice, seeing there was no best die, then saw the math to prove this.

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