# Two Is the Oddest Number

One of the biggest problems facing anyone in a creative field is:  How do I create something new?  For obvious reasons, you can’t be taught how to be original.  But there are some ways you can make it easier for your creativity to emerge.  Here’s one strategy:  Two is the oddest number.

What does that mean?  One of my thesis advisors (and the original source of the saying) explained it like this.  To a mathematician, there are only three numbers:  zero, one, or infinity.  Either something doesn’t exist at all (zero); it’s unique (one); or it happens infinitely often (infinity).

For example, suppose you have a line and a point.  In Euclidean geometry, there is a unique straight line through the point that is parallel to the given line.  In spherical geometry, there are zero straight lines through the point that are parallel.  In hyperbolic geometry, there are an infinite number of straight lines through the point that are parallel.  Thus zero, one, or infinity.

Or consider primes.  There is a unique even prime:  2.  All other primes are odd.

So how does this help you create something new?  One way is to try and find a second example of something.  Here’s an example, that requires some background in permutation groups.  Here’s the short version, though if you really want to delve into the topic, take a look at my (in progress) videos on abstract algebra.

Suppose I have a set of distinct symbols $a, b, c, d, \ldots$.  A permutation occurs when I rearrange the symbols.  It’s best to think of the permutation as what happens when you do a “replace all” in a document.  A compact way to represent these permutations is cycle notation, where an expression like $(abc)$ indicates you’re going to replace all $a$s with $b$s, all $b$s with $c$s, and all $c$s with $a$s.  Because this cycle has three elements, it’s called a 3-cycle.

We can also have 2-cycles:  $(ab)$.  These occur so often that we have a special name for them:  they are transpositions.

We can juxtapose two cycles (of any length) and form a composition.  For example, $(ac)(ad)$.  For somewhat technical reasons, we read these from right to left.  Thus first we replace all $a$s with \$latex \$d\$s, and all $d$s with latex \$a\$s; then we replace all $a$s with $c$s and $c$s with $a$s.  The net effect is that all $a$s have been replaced with $d$s; all $d$s with $c$s (because the first cycle replaced them with a $a$, and the second cycle replaced the $a$ with a $c$); and all $c$s with $a$s.  So $(ac)(ad)$ is the same as $(adc)$.

In a similar manner, we can write $(ac)(bc)$ as $(acb)$.

At this point, the mathematician says “Hmmm…two times we’ve managed to replace a composition of two transpositions with a 3-cycle.  But two is an odd number…maybe we can always replace a composition of two transpositions with a 3-cycle?”

So now we try it on $(ab)(cd)$.  And we find…the first component of a rather important theorem in the study of permutations.