# Bottema’s Theorem

This seemingly magical result from Futility Closet defies proof at first.  Go to the Wolfram demo by Jay Warendorff and then …

“Grab point B above and drag it to a new location. Surprisingly, M, the midpoint of RS, doesn’t move.

This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.

It was discovered by Dutch mathematician Oene Bottema.”

As we shall see, Bottema’s Theorem has shown up in other guises as well.

# Mathematics, And The Excellence Of The Life It Brings

I am a regular reader of Ash Jogalekar’s blog Curious Wavefunction, but I found my way to his latest via the eclectic website 3 Quarks Daily, also highly recommended. I could not resist the title, “Mathematics, And The Excellence Of The Life It Brings”. The entirety of the post was about the mathematician Shing-Tung Yau’s recent memoir, The Shape of a Life, but Jogalekar’s introductory remarks about his personal involvement with mathematics stirred so many personal recollections of my own, that I thought I would provide an excerpt, followed by my own comments. Furthermore, he also addresses in passing the perennial question of whether math is invented or discovered.

(Update 8/9/2021) Jogalekar’s story about his embracing math and the effect Simmon’s topology book had on him is even more amazing than I thought.  Throughout his younger years he had always been labeled “bad at math” and did poorly in school.  But a teacher and Simmon’s book changed all that.  He explains in a recent article in 3QuarksDaily, which I also provide here

I can’t help singling out a section where he, too, extols the significance and importance of high school geometry (see my post “Down with Geometry”):

“… Purely through accident at this time, I had gotten my hands on a book on topology, a subject that I had become mildly interested in because of its deep connections to geometry; interestingly, while I was rather abysmal at algebra in school, I always did well with geometry because I was good at visualization.  …

The topology book and the professor completely changed my outlook and saved me. I started doing well and tackling advanced topics and started to love math. I also got interested in physics and did well. Most importantly, I started appreciating the beauty of math. Over time I found that people interested in mathematics are generally of two kinds, although there’s some overlap: there are those who really enjoy mathematical puzzles and puzzle-like problems, relishing the raw process of problem-solving. Then there are others who simply enjoy the abstract nature of proofs and the connections between different topics: I am definitely part of this second group. In fact, another revelation I had was that most of the high school curriculum needed the students to be good at the former skill and had no appreciation of the latter, thus simply weeding out students like myself who wanted to understand the big picture and see the connections rather than just become adept at problem-solving.”

I confess I share this view and find it somewhat ironic that my website has devolved into a problem-solving source.  I have tried to show the wider picture of fascinating connections, but that often takes more skill and time than I currently possess.