Tag Archives: constructing any polygon

Book Review: Tales of Impossibility

Here is a book for anyone who has had high school geometry and likes math.  It is “Tales of Impossibility” by David Richeson.  It’s full of diagrams and is extremely well written.  A bright high school student could go all the way to the end, and would learn a lot of abstract algebra, up to and including complex numbers, irrational numbers and transcendental numbers.  It describes the 2000+ year search for ways to trisect an angle, double the cube, construct any polygon using a compass and straight edge,  and find the area of a circle (squaring the circle), or prove that it was impossible using basic methods.

It took until the late 1800s to finish the job.  Proving that something is impossible is subtle and difficult.    The book is 368 pages long and contains 40 pages of notes and references, but it is definitely not turgid.

There is a huge amount of historical detail about each of the great figures who worked on the problems starting with Euclid and going on through the the Greek geometers, Fermat, Descartes.

The battles about what could be considered kosher in math occurred every step of the way and is well covered.  Could algebra be used to solve a geometric problem?  Was a negative number a number. What about an imaginary number, or an irrational one?   Was something you could draw using a marked ruler (neusis) really a geometric figure?

If you look at nothing else, have a look at how Descartes was able to multiply and divide the length of various lines, using nothing more that Euclid’s geometry (but apparently no one had figured it out before).

The ultimate impossibility proofs involved abstract algebra, so we meet Viete and Descartes, Galois, Hermite etc. etc.  So it might help if some high school algebra was on board.

For the right smart high school kid, this book is perfect.  For the cognoscenti or even for nonCognoscenti with a lifelong interest in math (such as me) there is a lot to learn.  The proofs of all the geometric statements are all well laid out, and now it’s time for me to go through the book a second time and follow closely.