{"id":33008,"date":"2019-12-27T13:15:38","date_gmt":"2019-12-27T18:15:38","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=33008"},"modified":"2019-12-27T13:15:38","modified_gmt":"2019-12-27T18:15:38","slug":"mathematics-from-arts","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2019\/12\/27\/mathematics-from-arts\/","title":{"rendered":"Mathematics from arts?"},"content":{"rendered":"

Last Halloween I found myself the lone math student at a party, wearing a Springer Yellow Book costume. While I do not claim to be good at making costume decisions, to my relief people enjoyed a good \u201ctextbook costume\u201d pun. Most of all, I was happy that nobody found my costume scary1<\/a><\/sup>.<\/p>\n

After a while I started a conversation with a couple of people about our jobs and what we enjoy about it. I told them about the research and teaching aspects of my graduate program. This semester I was a teaching assistant for a lower-division Linear Algebra and Differential Equations<\/em>\u00a0course. I find this course to be quite fun to teach, because I get to help students develop a geometric intuition for abstract mathematics and point to wonderful\u00a0applications<\/a> of that abstraction.<\/p>\n

As it turned out, one of the people in our group was a graphics design student. He told me about a project involving linear algebra, and how he wished that he had taken more math courses. He also mentioned using the\u00a0B\u00e9zier curves<\/em>\u00a0in his classes. I had never heard of that name, so I wrote a note to look into it later. This conversation reminded me of something I had read in Jordan Ellenberg pitch for\u00a0Outward-Facing Mathematics<\/a>:<\/p>\n

\u201cThose of us who teach spend a lot of hours talking about math in front of students who have been forced to be there. That makes it easy to forget that people out in the world generally admire math and are excited to learn about it, if we give them a way in!\u201d<\/p><\/blockquote>\n

Back at home, I looked up\u00a0B\u00e9zier curves<\/em>, which lead me down a delightful rabbit hole of computer fonts and automobile design2<\/a><\/sup>, and in the process I learned new math. In this post (and hopefully others) I am going to write about the wonderful mathematics that I learn inspired by people in other professions.<\/p>\n

<\/p>\n

What does graphic design have to do with math?<\/h3>\n

Geometry. Even the most basic illustrations involve lines and areas. To design the fancy $\\LaTeX$ fonts used for mathematical symbols, for instance, each glyph is pieced together by many curves enclosing a shaded region.<\/p>\n

\"\"<\/a>

\u201cAnd since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art.\u201d \u2013 Albrecht D\u00fcrer (1471\u20131528)<\/em><\/p><\/div>\n

This might sound like a trivial fact, like answering a toddler who asks \u201chow are words written?\u201d or at best something that typographers, not mathematicians, would find interesting. In that case, you might be surprised to hear that in the late 70\u2019s and 80\u2019s the AMS formed an advisory Standing Committee on Composition Technology3<\/a><\/sup>\u00a0and helped work on a then up-and-coming software by Donald Knuth called $\\TeX$.<\/p>\n

\"The<\/a>

This is an\u00a0initial<\/a>\u00a0type<\/a>\u00a0stored in a wooden\u00a0type case<\/a>\u00a0at the\u00a0Minnesota Center for Book Arts<\/a>. Look at those beautiful plant-form spirals.<\/em><\/p><\/div>\n

Stay with me and I will explain why I find this mathematically interesting.<\/p>\n

How does interpolating curves work?<\/h3>\n

Euclid postulated that given any two points, we can draw a straight line passing through them.<\/p>\n

Question:<\/strong> in how many different ways can the statement above be generalized?<\/p>\n

Here are a few I can think of:<\/p>\n