by Edward G. Rozycki, Ed. D.
C. Some easy examples of isomorphic relations.
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).
Take an arbitrary list of words, put them in alphabetical order. Call this structure A.The words are the vertices (variables, islands) and the relationship is "follows".
Take a dictionary and look up the page number of each word. If there is no other word from your list on the page, write down the page number followed by a period an a zero, ".0", as, for example, 3.0 or 234.0. If two words are on the same page, write down their line number after the period; for example, 235.15 and 235.24. The page numbers are the variables and the relation that connects them into a structure is "follows".
Map each word to its page.line number in the dictionary.
2. The alphabet with the alphabet in reverse order.
Write out the alphabet. This is structure A. Each letter is a vertex. The relationship needed is "is adjacent to."
Write out the alphabet backwards. This is structure B. Again, each letter is a vertex. The relationship needed is "is adjacent to."
Map Structure A vertices to Structure B as A--Z, B--Y, C--X, ..., Y--B, Z--A.
3. Feathers, beak, wattles, wings, tail, talons with mouth, hair, feet, arms, buttocks.
Group the bird body parts and rename the groups as "head parts" = beak & wattles; "extremities" = wings&tail&talons; and "body covering " = feathers. For the human body parts use almost the same new terms but associate them with the appropriate parts: "head parts" = mouth& hair; "extremities" = feet&arms; and "body and body covering " = buttocks.
The relationship is "named starting at the head and going down to the ground." This gives us the following isomorphism:
|Mapped Variables||head parts||extremities||(body and) body covering|
|Bird Body Parts||beak & wattles||wings&tail&talons||feathers|
|Human Body Parts||mouth& hair||feet&arms||hair&buttocks|
This is very awkward. It forces us to treat the body structure of a human as though it were identical to that of a bird. It is probably a misleading and useless isomorphism. Nonetheless it is an isomorphic relation as defined.
4. The even numbers with the odd numbers.
Separate out the evens and the odds taking care to keep them in order. The order structures them with the relationship "the present number +2 =". Begin with 0, which is an even number and map it into the odd number which is 0+1. Go on then to the next even number which is 0+2 and map it, 2, to 2+1, or 3. And so on. Stop at some point to eat, to sleep and to get a life.
5. The months of the year with the seasons. (Group some months into single units.)
The seasons are Winter, Spring, Summer and Fall. These begin, respectively, in December, March, June, and September. Group each of these months with the two that follow and map the groups into their respective seasons. The relationship which structures the seasons is "follows". The relationships which structures the months within groups is "is in the same season." The relationship which structures the months-groups to each other is "follows."
6. The pronunciation of caterpillar with the symbols ter, cat, lar, pil.
Rearrange the symbols into the sequence cat-ter-pil-lar. The now are in the same order as they are pronounced. The relationship both set share is "follows".
7. The meals you eat with the times of the day.
Write down the approximate times of your meals; examples, breakfast, lunch, dinner, supper. Baptize the approximate times you have these meals as "breakfast time," "lunch time," "dinner time," and "supper time." A structuring relationship might be, for example, "follows" or "is bigger than" or "is later than."
8. Your family members with their birthdays.
This is similar to problem 1 above. The only difficulty happens if two have the same birthday. In that case use birthdate.
9. Any two limericks.
A limerick is recognized by its structure, called "meter" or "rhythm." Here is an example:
There was an old man from Calcutta
Who coated his tonsils with butta
Thus converting his snore
From a thunderous roar
To a soft, oleaginous mutta.
Look up other limericks on the web. You will see that you can generally find a syllable-by-syllable correspondence with the example given.
10. A tune you can hum with what a band that plays it produces.
A melody, too, is a structure that is contained in many other complex structures such as an band or choral piece. The trick here is to ignore variables in a band rendition that you could not possible supply by merely humming it. I will leave this example for later discussion. (Hint: A simple one voice musical score provides a common structure.)