TITLE: Good examples of Everyday Isomorphisms
QUESTION [5 upvotes]: Whilst trying to explain the concept of an isomorphism to a non-mathematician, it didn't seem to suffice to me to just give a precise definition, or leave it at some vague statement like "structure preserving map, so that the objects in question are essentially the same without being identical"; I wanted to give a few examples we use every day without realising.  
I came up with the following:  
Simple counting: Every time we count something we are just simply setting up a bijection (being an isomorphism of sets) between the collection of objects we're counting and some set $\{1,2, ..., n\}$ for some $n \in \Bbb{N}$. No one would ever explain finger counting to a child as 'setting up a bijection between the objects in question and some number of fingers on your hand' but I suppose that is actually what's going on.  
Planar Euclidean Geometry: Slightly more mathematical, but not too hard to understand is the fact that given a point in $\Bbb{R}^2$ we associate an ordered pair $(a,b)$ and clearly whilst these $2$ things aren't identically the same (one is a geometric point, the other an ordered pair of numbers) they are clearly so closely related that we can think of as being essentially the same. In this case we don't just have a set isomorphism as we can think of adding two vectors and show the addition corresponds to adding the ordered pairs in the usual way.  
I was wondering if anyone had any other good examples of this; specifically to illustrate to a non-mathematical person? (Also I realise category theory tag is a stretch but I wasn't too sure where to put this)

REPLY [3 votes]: Ask them to play a game with you.  You will take it in turns to give a number between $1$ and $9$.  Nobody is allowed to say a number that the other player has said already.  At any point, if one of you has said three numbers that add up to $15$, then they win.  Otherwise, if we get through all $9$ numbers with nobody winning, then the game is a draw.  
Secretly, draw the following magic square on a piece of paper:

As you take it in turns, draw either an $\mathsf X$ or an $\mathsf O$ on the appropriate squares.  Since three numbers add up to $15$ if and only if they are in a line on the grid above (you can check this yourself), a player wins the game if and only if they get three in a row of their symbol.  
In other words, this game is isomorphic to noughts and crosses*! 
*tic-tac-toe for the Americans
(There is a further example given at http://www.j-paine.org/students/lectures/lect6/node12.html using words.)
This is intuitively an isomorphism because the games can be played 'in the same way'.  But it is also rigorously an isomorphism in various categories of games and strategies.  For example, it gives us an isomorphism between game trees in the category of partially ordered sets.  A more sophisticated approach is that we get an isomorphism in the Joyal category of games: loosely speaking, if I play both games against you but let you start in one game and insist on starting myself in the other, then I can ensure, by copying your moves against you, that I never lose both games.