A homomorphism whose inverse is also a homomorphism.

If an isomorphism exists between two structures, they are said to be isomorphic (literally, 'identically shaped').

For a fantastic explanation of isomorphism, I recommend the book Gödel, Escher, Bach by Douglas Hofstadter. The book _Godel,_Escher,_Bach_ is a Pulitzer Prize winning novel written in the 1970s that looks at the fundamentals of how humans think and what constitutes thought.

The subtitle on the book is "A single golden strand in the spirit of Lewis Carroll."

An isomorphism is a map between one situation and another. It is similar to a mathematical function. For each input or set of inputs there is a unique output or set of outputs. The difference is that an isomorphism is always reversible -- an output or set of outputs always yields the original input or set of inputs. It is a one-to-one mapping between two sets.

Traditionally, examples of isomorphisms are the writing or symbology we use to emulate some aspect of reality. The numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are isomorphic to the ideas of one-ness, two-ness, three-ness, etc. When we see two circles atop one another, there is nothing in the structure that implies eight-ness (the concept of the value eight). But in our minds, we have built up an isomorphism, that is a mapping, between one idea (the visual image) and another (the mathematical idea of eight).

V + V = X is true if your isomorphism is for Roman Numerals. But a different isomorphism, that of string concatenation, yeilds V + V = VV

Other nodes:

Being the type of person who likes to relate mathematics to everything, I think of an isomorphism as being isomorphic, so to speak, to the literary or linguistic concept of an analogy, metaphor or simile.

One of the primary purposes of an isomorphism in math is being able to study the structure of something in a different context, and this is also the purpose of things like analogies. We can compare or "map" the attributes of an intangible concept, say, to a physical object to try to get a better understanding of its nature by studying something that is similar in ways but easier to grasp.

Of course, the isomorphism will eventually break down. This will always happen when one tries to find an analogy between the perfect, crystalline mathematical world, where everything is ordered and idealized, and the real world, where things are random and uncertain. Similes and their like are not symmetric; if A is a good simile for B, that won't mean that B is a good simile for A. They are also not transitive. If A is a good analogy for B, and B seems to be analogous to C, it doesn't mean that A will be a good analogy for C.

The basic concept is the same, though, no matter how imperfect the translation. It's often helpful to try to find isomorphisms between mathematical concepts or other abstract ideas and more familiar and tangible things, because it's easier to understand something when it looks like something you already know.

Thanks to Gritchka for advice on the mathematical properties of similes etc.

Let's look at one interesting isomorphism.

But first, let's play a game. The rules are very simple: we take turns picking numbers between 1 through 9, without replacement (that is, if one of us already picked say 2, nobody can pick 2 again). The first person to use three numbers to add up to 15 wins! If we go through all 9 numbers without any one of us being able to add up to 15, it's a tie.

Here's the thing: you're probably already very good at playing this game but don't know it yet. ;-)

Here's a sample game:

Player 1's numbers  |  Player 2's numbers  | Available numbers
    5               |                      | 1 2 3 4 6 7 8 9
    5               |        9             | 1 2 3 4 6 7 8
    5 7             |        9             | 1 2 3 4 6 8
    5 7             |        9 3           | 1 2 4 6 8
    5 7 6           |        9 3           | 1 2 4 8
    5 7 6           |        9 3 2         | 1 4 8
    5 7 6 4         | PLAYER 1 WINS (with 6 + 4 + 5 = 15).

Notice that when player 1 grabbed 6, player 2 grabbed 2 in order to block 6 + 7 + 2 = 15 but then couldn't block Player 1 grabbing 4 and winning.

Play a few more rounds... see if you can figure out what this game is. Believe me, this is a game you know very, very well. In fact, you know this game so well, that you would normally only play it during times of intense boredom, or you haven't played it at all much since childhood.

Still haven't figured it out yet? Explanation below.

Right, the isomorphism between the 1-9 game and tic-tac-toe goes like this. Draw a 3x3 magic square:

     8| 1| 6
     3| 5| 7
     4| 9| 2

Then clearly choosing numbers between 1-9 is equivalent to playing X's and O's in the tic-tac-toe magic square. It is further easy to see that all 3x3 magic squares are the same as the one given here except for rotations and reflections, which does not alter the structure of the corresponding tic-tac-toe game either. To finally prove the isomorphism, we must show that all 8 winning combinations of tic-tac-toe correspond to all possible ways to add 15 with three different numbers chosen from 1-9. This can be seen by systematically enumerating all such possible sums:

     9 5 1
     9 4 2

     8 6 1
     8 5 2
     8 4 3
     7 6 2
     7 5 3
     6 5 4

I enumerated and grouped them in such a way that should be clear that these are the only eight possible combinations of winning moves in the 1-9 game. Thus, the 1-9 game is isomorphic with tic-tac-toe. Amaze your friends with your impressive finesse at playing the 1-9 game.

There are a few lessons here. First, isomorphism is a concept that is much more general than "a bijection or a homomorphism such that" blah blah blah... Serge Lang liked to categorise with extreme generality the notion of isomorphism through category theory: an isomorphism is an arrow with an inverse arrow. That's cute (and it was more cute to watch him ask a bunch of mathematics undergraduates what an isomorphism is and respond that all of their function- or set-based answers were incomplete), but it still fails to capture in my opinion what isomorphism is all about: same shape, same structure, same basic concept, same mathematics. Isomorphism is at the root of mathematics; indeed, I'm tempted to say that isomorphism is the birth of mathematics; the recognition that adding two apples is the same as adding two shekels is quite profound.

Next, a good isomorphism allows us to understand an unfamiliar concept in terms of the familiar. This we have already seen above. The last point here is that one needn't too advanced or esoteric mathematics in order to understand and appreciate interesting isomorphisms. :-)

I`so*mor"phism (?), n. [Cf. F. isomorphisme.] Crystallog.

A similarity of crystalline form between substances of similar composition, as between the sulphates of barium (BaSO4) and strontium (SrSO4). It is sometimes extended to include similarity of form between substances of unlike composition, which is more properly called homeomorphism.

<-- (math): see isomorphic -->


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