BOTW: Visualizing Quaternions

After a couple weeks of delay I finally finished reading “Visualizing Quaternions” by Andrew Hanson. Part of the delay was self-imposed and part of it was a result of the book’s density. Not only is it over five-hundred pages in length, the author does not squander his words or your time.

The beginning of Chapter Four has a more accurate description of the book than I could write myself:

If this book were a musical composition rather than a mathematical one, perhaps it would begin with a single melodic theme carried by a solo trumpet, followed by other instruments entering in harmony. It would then move to a full ensemble—echoing, developing, and exploring dozens of variations on the original theme, some barely recognizable—and finally reach a climactic crescendo, trailing with a lone trumpet reprising the original perfect melody. […] we present that mathematical melody: the four quaternion equations from which every theme in the entire book is in some way derived.

Not All Equations

‘Visualizing Quaternions’ is a math book in every sense of the phrase. But I found it to be more entertaining than most, and I am someone who tends to enjoy the boring ones. Author Andrew Hanson does not simply fill his text with copious amounts of mathematical description. He balances the book nicely with discussion on the practical application of quaternions and their position in various circles, both historic and contemporary.

The first chapter is a great example of what I mean: an interesting and fun telling of how Sir William Rowan Hamilton discovered quaternions in 1843 while on a walk with his wife. Hamilton was so excited by his flash of genius that he immediately took out a knife and carved the fundamental formula for quaternion multiplication into the Broome Bridge in Dublin, Ireland. He did it so that it would at least be written somewhere in case he died before having the chance to tell anyone—seriously.

Hanson’s book contains a number of anecdotes like that, along with entertaining explorations into how quaternions manifest in 3D gaming, the Apollo 10 mission, and a wild bug in an F-16 flight simulator that caused the jet to flip upside-down whenever it crossed the Earth’s equator.

‘Visualizing Quaternions’ is filled with diagrams, which feels not only appropriate but necessary. There are even chunks of C code here and there for the programmers in the audience. They all do justice to the first word in the book’s title: Hanson exerts a tremendous amount of effort teaching and demonstrating the why’s and how’s of mentally visualizing quaternions, as well as how to spot their physical applications. These diagrams and demonstrations—he describes simple tests you can do at home that reveal properties of quaternions—flow together seamlessly with the inevitable math itself, and the two play off each other harmoniously to better reinforce every point the book sets out to make.

Next Book of the Week

I honestly have no serious complaints about ‘Visualizing Quaternions’. It is exemplary in its breadth and depth. I am certainly no mathematician, and so there may be valid academic criticisms which are simply beyond my comprehension. But as a computer programmer who only knew a little about quaternions beforehand I found the book to be wonderful and immensely informative.

I must admit, however, that I’ve had enough math for the past weeks. So next I am reading “Ada 95: The Craft of Object-Oriented Programming” by John English. I have long considered Ada an interesting programming language but have never spent an extended amount of time using it. I know that Ada 95 does not reflect the latest language standard, but nonetheless I look forward to having fun comparing and contrasting it to other languages, as well as becoming more familiar with Ada in general.


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