The amazing thing about the number 6

What's the saddest thing about being a mathematician?  Seeing worlds of soul-shattering beauty - but being unable to share this beauty with most people.

I actually used to have dreams about this.  In my dream, I would be hiking with friends.  Then, in the distance, I'd see some beautiful snow-capped mountains.  They were surreal, astoundingly tall - and not even very far away!   I wanted nothing more than to run over and start exploring them. 

But my friends weren't interested.  I had to either persuade them to go with me, stay with them and leave the beautiful mountains unexplored - or leave them and go climbing all alone.

At this point I'd always wake up, stuck in the dilemma.

Maybe this explains why I spend a lot of time explaining math here on G+.  The sad part is: you can take a lot of people a short distance toward the beautiful mountain peaks... or take a few people all the way up into the peaks.  You can't get everyone up to the top.

For example, I know the picture in this post is too complicated, and not flashy enough, for most people to enjoy.  But to me it's more beautiful than other pictures that will get a lot more +1s.

Any sort of mathematical gadget has a symmetry group.  The simplest sort of gadget is a finite set, like this:

{1,2,3,4}

The symmetries of a finite set are called permutations: they're the ways of rearranging its elements.   Here's a permutation of the set {1,2,3,4}:

1 |→ 4
2 |→ 1
3 |→ 3
4 |→ 2

The permutations form a 'group'.  This means we can 'multiply' two permutations, say f and g, by doing first f, and then g - and the result is another permutation, called fg.   Also, for every way of permuting things, there's some other permutation that undoes it.  For my example, this is

4 |→ 1
1 |→ 2
3 |→ 3
2 |→ 4

So, the permutations of a set form a group, called the permutation group of that set.

Now, I said every mathematical gadget has a symmetry group.  This is also is also true for permutation groups!   What's a symmetry of a permutation group? 

(This is where I may lose you.  This is where it gets interesting.  This is where I can see the mountain peaks and want to start climbing.)

A symmetry of a permutation group is a way of sending each permutation f to a new permutation F(f), obeying

F(f) F(g) = F(fg)

So, it's a way of permuting permutations - a way that is compatible with multiplying them. 

How do we get such a thing?  We can get it from a permutation of our set.  Any permutation lets us take any other permutation, and permute the numbers in the description of that permutation, and get a new permutation.  And that turns out to work!

Now for the cool part.  Every symmetry of the permutation group of the set {1,2,3,4} actually arises this way.

And this is also true for {1,2}, and {1,2,3}, and {1,2,3,4,5}, and so on.  In every case, all symmetries of the permutation group of the set come from permutations of the set.

Except for the exception.  The only exception is the number 6.

There are symmetries of the permutation group of the set {1,2,3,4,5,6} that don't come from permutations of this set! 

To understand this, you need to ponder the picture here, drawn by Greg Egan.

If you look carefully, there are 15 red dots and 15 blue ones.  Each red dot has a pair of the numbers 1,2,3,4,5,6 in it.  There are 15 ways to choose such a pair.  Each blue dot has all 6 numbers in it, partitioned into 3 pairs.  There are 15 ways to do this, too.  We draw an edge from a red dot to a blue dot if the pair of numbers in the red dot is one of the pairs in the blue dot. 

The resulting picture has a symmetry that switches the red dots and the blue dots!  And this symmetry is - somehow or other - the symmetry of the permutation group of {1,2,3,4,5,6} that's not a permutation of {1,2,3,4,5,6}.

"Somehow or other"?  That's not a very good explanation!  I've only shown that there's something special about the number 6, which gives a surprising symmetry.  It takes more work to see how this does the job.

For the full explanation, try my blog article:

http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/

The trail gets a bit steeper at this point... but the view is great.

You may be wondering: So, what's this all good for?

And the answer is: nobody knows yet.  But this amazing fact about the number 6 is connected to many other amazing things in mathematics, like the group E8 and the Leech lattice, both of which show up in string theory.  I don't know if string theory is on the right track.  But I hope that someday, when we understand the universe better than we do now, these mysterious and beautiful mathematical structures will turn out to be important - not just curiosities, but part of why things are the way they are.

That is my hope, anyway.  So, I'm glad we have some people thinking about these things.  And besides, they're beautiful.

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