9: Selected Topics

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Throughout this book, we have presented various practical and useful topics in which mathematics plays a crucial role in solving problems. These include logical fallacies, abuse of percentages, compound interest, chances, statistics, exponential growth, perspective in art, harmony in music, and political applications. We have also seen the inspiring side of mathematics in topics like the Golden Ratio and fractals. Unfortunately, many of these “crowning achievements” in mathematics stay hidden from students because of the traditional emphasis on algebra-and calculus-based college mathematics curriculum. In this chapter, we will present a few selected topics featuring some of the “cool” topics that you can appreciate without taking advanced mathematics. We hope to add more topics in our future editions of the book.

9.1: Four Color Theorem
The Four-Color Theorem states that in any plane surface with regions in it (people think of them as maps), the regions can be colored with no more than four colors in such a way that two regions that have a common border do not get the same color. They are called adjacent (next to each other) if they share a segment of the border, not just a point.
9.2: How Big is Infinity? Or Is it “Infinities”?
There are infinitely many levels of infinities.
9.3: “Seven Bridges of Konigsberg”
Through the city of Königsberg in Russia flowed the Pregel River. In this river were two large islands, which were part of the city. Joining the mainland either side of the river and those two islands there stood seven bridges. Was it possible to cross each bridge once and once only during the course of a single walk? More to the point: could this be done such that the walkers end up back at their starting point?
9.4: Russell’s Paradox
Russell's Paradox is a well-known logical paradox involving self-reference. It is a little tricky, so you may want to read this carefully and slowly. If you have a list of lists that do not list themselves, then that list must list itself, because it doesn't contain itself. However, if it lists itself, it then contains itself, meaning it cannot list itself. This makes logical usages of lists of lists that don't contain themselves somewhat difficult.
9.5: Non-Euclidean Geometry
In your geometry class, you probably learned that the sum of the three angles in any triangle is 180 degrees. This is a well-known theorem in geometry—more specifically, “plane” or “Euclidean” geometry, which has served as the foundation for all learning since the time of the Greek civilization. But it turns out that if you go away from the “plane,” then plane geometry may not work. This gives rise to non-Euclidean geometry.

Thumbnail: Königsberg graph (CC BY-SA 3.0; Booyabazooka via Wikipedia)

Contributors and Attributions

Saburo Matsumoto
CC-BY-4.0