Angels and Demons

by Burkard Polster and Marty Ross

The Age, 11 May 2009

Angels and Demons is about to hit our screens. Eagerly awaited by fans of Dan Brown’s thriller, the movie will also feature some stunning mathematical gems known as ambigrams. These ingenious symmetric writings can be read in different ways.

For example, the graphic above shows the title of the movie. If you (carefully!) turn your computer screen upside down the “Angels&” turns into “Demons”, and vice versa. This is known as a half-turn ambigram. Here is another one, conjured up by an ambigrammist maths master:

There are many other types of ambigrams. For example, the following ambigram has left-right symmetry and so is unchanged when viewed in a mirror:

You can play clever tricks with such an ambigram, as illustrated by the (untouched) photo below. The serious looking maths master seems to have wandering eyes, and six fingers on his right hand. We assure you, neither is the case.

The beautiful Angels & Demons ambigram, and all the ambigrams in the movie and the novel, were created by John Langdon. John is an American graphic designer, painter and professor. He is one of the inventors of this geometric wordplay. His artwork can be viewed at www.johnlangdon.net.

Incidently, John Langdon lent his name to Robert Langdon, the hero of Dan Brown’s novels. John and his ambigrams were introduced to Dan by Dan’s father, a former mathematics teacher. It was also Dan’s father who introduced him to the golden ratio and Fibonacci numbers, mathematical gems that feature prominently in The Da Vinci Code. Disappointingly, just about all of Dan Brown’s maths there is wrong.

We haven’t forgotten that this is a mathematics column. So we’ll end with a number ambigram. Notice that there are very natural half-turn ambigrams among the digits: the 6 and 9 swap; and 0 and 8 stay unchanged. The same is also true for 1, if you use a simple font. Now here is a stunning magic square, featuring only these special digits. We’ll leave it to you to locate all its symmetries and special properties.