Mathematicians have a simple explanation for turning big worlds into small worlds: "six degrees of separation" -- that any person can link to any other person on earth merely by a chain (indirect network) of six people BETWEEN.

For example, if your (1) neighbor knows a (2) fellow office-worker who knows (3) a teacher who has (4) a cousin who went to school with (5) Madeline Albright, our Secretary State (friendly with world celebrities and dignitaries), then there exist just six degrees of separation between you and (say) (6) Queen Elizabeth of Great Britain!

This thesis was popularized in 1990 by John Guare's broadway play, Six Degrees of Separation, involving a conman who convinced wealthy gullibles that he was Sidney Poitier's son -- filmed in 1993 with Stockard Channing, Will Smith, and Donald Sutherland.

Then college students developed a game, "Six Degrees of Kevin Bacon," to link the actor Bacon to other actors through as few links as possible. The "Oracle of Bacon at Virginia" Web site of Brett Tjaden liinked (in July, 1998) 322,095 actors to Bacon, assigning each a Bacon score.

The original idea began in 1967 when Harvard social psychologist Stanley Milgram suggested that Nebraskans and Kansans write two Boston people in Boston they did not know, asking them to forward the letter to acquaintances who might know the specified Boston people. The average number of links between Midwesterners and Bostonians was six, hence, ....

Social scientists call these "self-organizing systems", such as support networks, Internet and electric power grids, and the process by which fertilized eggs create complex animals from protoplasm.

In the June 4, 1998, Nature journal, mathematical models confirmed the "6-degree" thesis when mathematicians Steven Strogatz and Duncan Watts introduced "short-cuts" between "nodes" of the models (analogues of persons in a support network) in a manner that looks "random". These short-cuts explain how strangers find acquaintances in common and why epidemics spread the world, and could accelerate communication in cellular phone and Internet system. In fact, very few short cuts between nodes and clusters turn a big world into a small world!

Mathematicians Strogatz and Watts applied shortcuts to the three connected systems: the power transmission grid of the western United States; the 282 neurons composing the nervous system of a simple worm called Caenorhabditis elegans; the nearly 250,000 actors listed in the Internet Movie Database. Each of these small worlds has elements interconnected to neighboring elements: one power station to the next; one cell to another; one actor to another actor who tends to be cast in the same kind of films.

The short-cut of jet travel make the world seem smaller. Experts believe that AIDS first spread to the United States from Africa through an infected airline attendant who introduced the virus into gay bathhouses in New York. (This is described in Randy Shilt's historical book,And the Band Played On, filmed by HBO in 1993.) Says Strogatz, "Infectious diseases spread more quickly and easily in a small world, and it is alarming how few short cuts are needed to make the world small."

In MATHEMATICAL language, "6 degrees" is an existing or constructible PATTERN in social systems, technological systems, biosystems, etc.. Strogatz and Watts use mathematics to show this in these systems. But what about the system of MATHEMATICS? I'explain via THREE LITTLE WORDS -- RELATION, FUNCTION, OPERATION -- THREE LITTLE WORDS which COMPREHEND "the mostest" of MATHEMATICS.

In fact, I'll show you how to show yourself that a chain of mathematical CONCEPTS -- namely,

YOU Þ MEASURES Þ TYPING/ORDERING Þ SETS Þ RELATIONS Þ  FUNCTIONS Þ
 OPERATIONS

-- can shrink the big world of Mathematics to a small world of Math Appreciation just your size!

All of these 6 links-from YOU are explicated in YOU-language for interactive learning of PATTERNS, STRATEGIES, PROSTHETICS. (For example, the mathematical structure of RELATION is explained in terms of "Bill and Hillary Clinton, Bob and Elizabeth Dole, Newt Ginsberg and Monica Lewinski").

Let's pass from first link (YOU) to second link in this chain:     YOU Þ MEASURES.