4th Saturday Math Meetups: Mathematical Illustrations - Shapes through Formulas

4th Saturday Math has now worked its way through Poyla and chapter 11 in Induction and Analogy in Mathematics over the past 4 months. Pólya is known for his work in problem solving, and his textbook does not disappoint in this regard. It gives a lot of "intuitive" ways of looking at larger concepts, but the book is also a single variable calculus textbook from the 1950's. It's a very interesting approach to the discipline and early on in the book he shows us interesting ways to abstract shapes mathematically, like breaking down shapes according to their Euler number. This sets the reader up for a lot of problems in 2D geometry, especially problems involving series of variables derived from concepts illustrated by geometry. Pólya himself likes to throw in problems towards the end of his problem sets to get the reader to think about spacial reasoning and then in the final chapter brings out a whole chapter based on deriving functions from a sphere. His final 3 problems (problems 21, 22, and 23) from chapter 11 really shine a light on how he was asking the reader to think about the formulas with multiple variables.

His method was created in a time when calculations were done with a slide rule, making his last question about rearranging proportions an interesting thought experiment. The reason why any of this is relevant to the Bill Cassleman book, Mathematical Illustrations, is that both books have an interest in higher level problem solving through geometery. Mathematical Illustrations approaches its formulas with matrices, due to the books focus on drawing shapes with the PostScript programming language. Last year, the group looked at chapter 1 and 2 of this free online book. This month, the 4th Saturday meetup will do a review of chapter 1 and 2 before working on chapters 3 and 4 in June and July, respectively. Chapter 1 worked as a way to get the reader to install ghostscript on their computer, but also familiarized the reader with the mathematical concepts needed for going forward. The two notable examples of problems in this chapter are to build a function to calculate Sin^-1 and to use PostScript to draw Proposition 1.47 from Euclid's Elements. Chapter 2 hits the reader with the linear algebra and introduces the idea of skew and how to deform shapes while maintaining their volume.

To get ourselves ready for the next 2 chapters while also coming down from Induction and Analogy in Mathematics the group will be looking at problems 21 to 23 in chapter 11 of Induction and Analogy, and problems 1.7, 1.10, 2.1 and 2.6 in Mathematical Illustrations.