Extending mathematics Book

The book Excursions into Mathematics goes into a lot of detail into specific topics in mathematics. It provides rigorous proofs of various theorems in these topics. The book is divided into 6 chapters and each chapter has about 10 sections. One very interesting chapter in my opinion was the chapter called Games. It defined two types of games: "tree" games and "matrix" games. "Tree" games, as said on page 319, are 2-player games where each player takes turns, the number of possibilities per turn and the number of turns is limited, and each player is aware of the other player's moves. Examples include chess, tic-tac-toe, and checkers. The book provided an interesting proof why Player 2 (Black) cannot win tic-tac-toe with perfect play from both players using something it calls the "Answering Strategy" (Beck, Bleicher, Crowe 325). Matrix games are where each of 2 players chooses an outcome and Player 1 or Player 2 pays the other player based on the combination of the two outcomes. Even though some of the proofs were boring, the chapter was great.

Another interesting chapter to talk about is Some Exotic Geometries. It discusses axioms that hold in regular flat geometry, especially the Parallel Postulate, on pages 214-217. It then proceeds to ditch that specific axiom and talk about what happens with no parallel lines per line and point not on line (spherical geometry) and with multiple parallel lines (per line and…) (hyperbolic geometry). It discussed, for example, how the sum of the angles in a triangle is not 180 degrees but less than that in a hyperbolic triangle. The chapter then proceeded to talk about finite geometries, where there are only a certain number of points and the geometry wraps around. I am surprised and disappointed that it didn't talk about flat torus geometry. One thing that sticks out in the chapter is the result of Exercise 71 Problem 3 on page 280. An "equilateral triangle" on a 9-point square of finite geometry is a loop that has all sides equal (1) and all the angles equal (180 degrees (!)), but to 180 degrees because of the wrapping around! That was a perfect opportunity to talk about the oddities of flat torus geometry.

Other chapters the book had were chapters about the properties of numbers (including proofs of easy divisibility tests like how if the sum of digits of a number is divisible by 9, the number is divisible by 9), perfect numbers, Euler's "vertices + faces - edges = 2" formula, and the concept of area. It was interesting to know that even though it takes 4 colors to color a map on a flat piece of paper so that no two touching regions have the same color, it takes 7 colors to do that on a torus. There is a polyhedron consisting of 7 vertices that are connected to each other vertex. It is mentioned on pages 36-37 and it is called the Császár polyhedron. Overall, I would say that a lot of the proofs got boring and some of the sections were boring, but the book was great.