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edited 4/14/12
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I considered calling this essay, "Isomorphisms Made Easy," but understanding isomorphism is already very easy. The idea looks formidable because the term comes from mathematics, for many people a strange planet. But the idea itself is something we learned as little children and one which we use in many ways in our daily life without any consciousness of it.
We easily see similarities where experience tells us they don't really exist; for example, faces or animals in clouds or in shadows. And we tend to exaggerate the similarities we do see and imagine (hope?) that they are structural identities, that is, isomorphisms. (For more on this, look up an article on pareidolia.)
This essay deals with those familiar experiences in which we use the concept of isomorphism in ways which, interestingly enough, are far more sophisticated than, say a course or book introducing us to graph theory might bring us to believe. If you can recognize both a Timex and an Omega as watches; or, if you speak a language, you know how to use isomorphisms. But caution is necessary: not every similarity is an identity. In this article will learn how to prove if a similarity is really an isomorphism.
A. To begin (Get this straight!)
1. No single thing by itself [1] is isomorphic. Two things may be isomorphic, but there is no straightforward answer to the question "Is this house isomorphic?" The response will be "Isomorphic to what?" You will have to mention another structure for comparison.
2. Structures may be isomorphic. What are structures? A set of things, objects, whatever, together with their mutual relationships. For example, a structure would be either of the following:
a. the letters of the English alphabet and the relationship "follows in the alphabet;" to illustrate: ABCDEFGHIJ... but not ZMOAQUX... where the position of the letters in the letter-string indicates the relationship.
b. the cardinal numbers from 1 to 26 and the relationship "is preceded by;" to illustrate: 26,25,24,23,22, ... but not 1,2,3,4,5,...where the comma stands for the relationship.
3. Structures are anything we can conceive of as having "parts" such that the parts are somehow connected or related. They may be "hard objects", like a chair, patterns of written symbols, like this text, or processes, like a dance.
Some structures may seem to us to be more "natural" than others. But this is probably just familiarity influencing us. You are invited to use your imagination and "see" structures that others might overlook without your insight. Pay attention to the relationships between parts!
4. A single object may be thought of as having many possible structures. For example you might consider yourself as a group of connected bones. Or as a set of muscle tissues connected by bone and ligaments. Or as an endocrine system. (A system is a structure; but a structure is not necessarily a system.[2])
B. To prove an isometric relationship (Make a chart, please.)
1. An isomorphism, or, isomorphic relation, is a one-to-one correspondence between the parts and also a one-to-one correspondence relationships of two structures. It turns out that there is an isomorphic relation between the examples given above, that is, the letters of the alphabet in normal order and the whole numbers from 26 back to 1.
2. To demonstrate, if necessary, that two structures are isomorphic, use a table: for (partial) example
|
Part type |
Relation |
Element1 |
Element2 |
Element3 |
Element4 |
Element5 |
... |
|
Letter |
follows |
A |
B |
C |
D |
E |
... |
|
Number |
precedes |
26 |
25 |
24 |
23 |
22 |
... |
We see that as A corresponds to 26; B, to 25, C, to 24, etc.
Also A is to B as 26 is to 25; B is to C as 25 is to 24, etc. this shows that follows corresponds in the first structure with precedes in the second.
C. Some easy examples of isomorphic relations. (How is a turkey like your brother?)
Can you identify the parts and relations needed to make up the structures to be compared? (You need not use every recognizable part in the structure you construct.)
1. The spelling of a word with its location in a dictionary (use page, line numbers).
2. The alphabet with the alphabet in reverse order.
3. Feathers, beak, wattles, wings, tail, talons with mouth, hair, feet, arms, buttocks.
4. The even numbers with the odd numbers.
5. The months of the year with the seasons. (Group some months into single units.)
6. The pronunciation of caterpillar with the symbols ter, cat, lar, pil.
7. The meals you eat with the times of the day.
8. Your family members with their birthdays.
9. Any two limericks.
10. A tune you can hum with what a band that plays it produces.
D. Common ways to define sets of isomorphisms: with formats
The following items, call them formats,[3] define sets of isomorphic structures. In this exercise the structures are generally procedures or (sub)formats:
| 1. a business plan 2. a piece of sheet music 3. a rule book 4. a map 5. a recipe |
6. an equation with variables 7. a computer program 8. a poem printed in a book 9. a melody in your head 10. a dance you know |
Pick a particular format.[3] Choose a procedure (or subformat) that that format identifies. Can you identify the parts and relationships which make it isomorphic to another procedure (subformat) in its set?
[1] In graph theory, everything is isometric with itself. But the mathematical "things" are graphs, deliberately concocted simple structures, not the complex real-world objects or ideas we will start this discussion with. A person, a building, etc. may comprise a number of structures.
[2] See "What is a system?" at http://www.newfoundations.com/EGR/system.html
[3] Formats, clearly, require an interpreter through which they are mediated into procedures. None of "hard objects" in the charts generate procedures in and of themselves.