# Good examples of Everyday Isomorphisms

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)

• An isomorphism is like an analogy ... May 23, 2017 at 13:04
• Actually, in your simple finger counting problem, we may not use the word "bijection" but the concept is nonetheless exactly what we teach when children learn to count objects using natural numbers. "Oops, you skipped over that one", in other words "That was not surjective". "Oh wait, you counted that one twice", in other words "That was not injective". May 23, 2017 at 13:04
• May 23, 2017 at 13:05
• How about the group isomorphism between the shuffles of a 3-card deck and the symmetries of an equilateral triangle ($S_3$)? Or the ways you can flip a nonsquare rectangular mattress and the transformations of the states of a pair of light switches (the Klein $4$-group)? May 23, 2017 at 13:24
• @Travis I really like those, they're great! May 23, 2017 at 13:33

A simple arithmetic isomorphism: in many (older) video games, points are given out in multiples of $1000$ (say) to create a sense of excitement. We could scale down the points by a factor of $1000$ and preserve the essential structure of the point system: for example, if you finish the game and get the highest score, you would still get the highest score in the scaled-down version. Specifically, the scaled-down final score is the sum of the scaled-down points during the game.

In planar geometry, consider that diagrams and proofs on a piece of paper don't change when you rotate that paper through any angle in three-dimensional space or carry it around the room.

There are strategic situations that we refer to as rock/paper/scissors because they can effectively simulated with that game, the only difference being the labels of the strategies.

The Richter scale is an example of the isomorphism between the reals under addition and the positive reals under multiplication.

• Very nice, especially the logarithmic/exponential scale example! May 23, 2017 at 18:43

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.

• Very nice indeed; I may in fact do this! May 23, 2017 at 18:55

This might be mildly more sophisticated than you're looking for, but I would imagine you could convince anyone who remains vaguely aware of their high school algebra of a (vector space) isomorphism between the collection of quadratic polynomials and three-dimensional space.

You could also suggest some homeomorphisms: the torus with the coffee cup is rather famous, but such simpler examples as the cube to the sphere are also nice.

A third category to look in could be posets. For instance, a small company with a manager and two employees is anti-isomorphic to a family with two parents and their single child. In preordered sets, the collection of times of year is isomorphic to the collection of angles around a table (or a more abstract circle.)

Suggestions:

• n civil servants have to be assigned to n job positions
• 10 kids choose a pencil each from a colorbox
• next week in some hospital, each of the 7 surgeons will be on duty one night.
• You're coaching the soccer team and have to assign a position to each of the 11 players.
• These are good examples of bijections, but I don't see how they constitute isomorphisms. May 23, 2017 at 15:02
• Well, these all set up isomorphisms between, say, the set of kids and the set of pencils chosen, but they're are all iterations of the same idea, which fall within the "counting" section of the OP. May 23, 2017 at 17:11
• @KevinCarlson I see. But in these examples, there is effectively no "structure" being preserved, so in my opinion they don't really convey what isomorphism means or why it matters. For anyone trying to grasp the concept, I would venture to say that these examples are misleading. May 23, 2017 at 18:28
• I would tend to agree-isomorphisms of sets are not a sufficient range of examples. May 23, 2017 at 18:30
• I generally agree with your remarks. My examples should be considered as "concrete life implementation of the counting analogy" May 24, 2017 at 9:24