Thoughts of mathematics are never far from my mind -- a hazard of the profession as mathematician, perhaps. Often these thoughts are inspired by current events, most recently the extraordinary and extraterrestrial successes of the New Horizons and Kepler missions, much of which was made possible by mathematics. But the truth is that you don't need to travel to Pluto to see or think about math. You can do it just by going out for breakfast in a bakery in Santa Fe.
The Chocolate Maven in Santa Fe is a bakery well-known for its delicious pastries, cookies, and breads. It's also a great place for breakfast that has the added attraction of a large glass wall at one end, allowing the diners to watch the baking in action. As I was sitting there slightly bleary-eyed with my wife, I was soon transfixed by the croissant-making going on just on the other side of the wall, enjoying and relaxing to the rhythm of the process and slowly, but inexorably, starting to think about mathematics.
What can I say? I can't help myself. Part of the danger of the profession I suppose. I like to believe that for most of us the events of the day, even the most quotidian, are filtered through the things that we think about most, so that pretty much every experience becomes some kind of odd Rorschach test. On the other side of the glass an economist might have seen an Adam Smith-like division of labor, a chemist or physicist could have reflected on the baking process, while a businessperson considered the profit margin or workflow. I'm a mathematician and I saw math.
Math? Math in bakery? Sure. Plenty of it.
Directly in front of me a large sheet of pastry dough is laid out, that the baker smooths and trims into a rectangle as the first step to making a tray of croissants. He then runs a knife lengthwise down the middle of the sheet, cutting it into two, he then makes 18 cuts in the opposite direction, creating a grid of 36 smaller rectangles. Finally, he slices each of these rectangles along the diagonal, to make 72 triangles, each of which would soon be rolled up from base to tip (after filling some with either a slab of chocolate or marzipan) before being placed in their trays. The sequence of multiplications that I did (2 times 18 times 2) allows me to count how many croissants were about to be produced with out actually counting them out individually, and I presume the baker did the same as he quickly wrote the number down on the board before rolling up the pastries. There are so many kinds of math, even in this little exercise: the implicit "factoring" of 72 as 2x18x2 that leads one to consider factoring in general and the wide world of number theory. Or the hints of "combinatorics" in the resolution of the question of "how many croissants did we get?"
Then of course there are the pleasures of geometry, excited as we watch the steady subdivision of the pastry sheet into smaller and smaller rectangles and then triangles, creating a mosaic of embryonic breakfast treats. What other ways are there to evenly divide the sheet? What about continuing this on, ad infinitum? Turning these triangles into even smaller triangles (can you think of a way to do this?). How many croissants would we get after each step? It's a quick leap from dreams of stomach-filling pastries to infinite space-filling "tessellations".
In the background the huge mixing machines go through their motions, steadily kneading the dough. Sorry, but here my thoughts turn to topology where we study something that is actually called "the baker's transformation," a mathematical model that describes the way in which a little pool of vanilla is, through successive kneadings, dispersed evenly through the entire doughy blob. Right here in front of my very eyes I'm watching one of the fundamental models of applied mathematics - a part of "dynamical systems" and "chaos theory" that shows how the repetition of simple mathematical transformations can quickly create systems with great mixing properties.
Number theory, combinatorics, geometry, topology, dynamics, all going on in front of me. From the practical to the abstract, but we can go back again. The number theory of factoring is at the heart of all secure digital communication. Combinatorics underlies much of real-time scheduling and operations research and database management. Those triangulations of the dough when done digitally are at the heart of all of computer graphics and creating the CGI worlds we enjoy on gaming platforms and in movie theatres. Understanding the mixing of dough is a first step to understanding the mixing of chemicals to create more fuel efficient engines or the atmospheric flows that give us the weather and the climate. There's all of that in my morning musings at The Chocolate Maven. You don't need to go to Mars to think about math. You just need a little marzipan.