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Suppose that one tosses a coin three times. The result of tossing each time is either "heads" or
"tails". Two tosses leads to either heads-head, heads-tails, tails-head or tail-tail, that is, (2×2=4) possible
outsomes. Three tosses leads to (2×2×2=8) outcomes. Now suppose that the coin is tosses another three
times. Again, there are eight different combinations of head and tails. Each outcome from the first three
throws may be followed by any one of the eight results of the second set of throws. This means that there are
(8×8=64) possible pairs of outcomes of throwing three coins and then repeating them. Eight of these are when the
outcome of the second set of throws is identical to that of the first not only in numbers of heads and tails but
also *in the order* in which the faces of the coins turn up. The number of different pairing of eight
objects without regard for their ordering is 28. The 64 pairs of outcomes consist therefore of eight pairs when the
eight possible results of tossing the three coins merely repeat themselves, 28 pairs when the outcome for the
second set of throws changes and 28 pairs when the outcome of the second set of throws is the reverse of these,
e.g., instead of (say) head-tail-head being followed by tail-tail-head, the outcome is tail-tail-head followed by
head-tail-head.

This, essentially, is how the ancient Chinese system of divination known as I Ching is used to answer questions. Originally, yarrow stalks were used to generate a set of
six results indicating the answer to a question, but this method is a biased random generator, so that the
possible outcomes are not equi-probable. The yarrow stalk method was gradually replaced during the Han
Dynasty by the three coins method. Many methods have evolved over the years to generate the results of the six tosses, which are
called "hexagrams." Some people prefer using dice to coins because they avoid the problems of whether the coin
is tumbling when cupped in the hand and of it bouncing and scattering when flipped. In this case, if an odd
number of pips appears in the die, it counts as "heads;" if an even number of pips appears, it counts as
"tails." Outcomes of a randomly generated event like tossing a coin or die are represented by either
unbroken lines (—) representing the Yang aspect of the event or broken lines (− −) representing the Yin aspect.
Either type of line is characterized as either *stable* ("young") or *changing* ("old"), so that
there are four possibilities for each line. The set of three tosses of the coin generates three lines called a
"trigram." The eight possible outcomes of throwing three coins are represented by eight trigrams. The hexagram
is a pair of trigrams representing the outcome of the two sets of three throws. Once the hexagram has been
generated, one consults the oracle in the form of commentaries in the I Ching ("Book of Changes") for each of
the 64 hexagrams in order to determine the answer to one's question.

For the purpose of this website, no attempt needs to be made to answer the issue of whether I
Ching "works" and (if so) how or why it does. That is irrelevant in the present context. What is important
here is the mathematical nature and meaning(s) of the 8×8 matrix of hexagrams that grew out of Taoism and
their connection to the sacred geometries of other religions. This array is an abstract representation of the
Tao. Literally translated as the "path" or "way," although more accurately as the "*right* way," this
ancient word has no single meaning and is interpreted in many different ways even within Taoist sects. *The
Tao Te Ching* is the ancient Chinese classic text written according to tradition by the Chinese sage Lao Tzu in
about the 6th century BCE. But even then, the word "Tao" had no well-defined, single meaning. Indeed, the opening
of the *Tao Te Ching* illustrates its undefinable aspect: "The Tao that can be spoken of is not the real
Tao." The Tao is the driving principle behind the natural order, yet it not Nature. It signifies "the way things
are," yet it is not a force or object existing in the world of duality. It is the causeless Cause, the ultimate
source of all phenomena. In symbolically representing something as mundane as all possible outcomes of two sets of
three tosses of a coin, the table of hexagrams is expressing something far more profound and fundamental about the
mathematical nature of the Tao. At the most elementary level of space-time — the subatomic world of the superstring
— we shall discover in the section **Superstrings as sacred
geometry** how this nature manifests.

The stroke order for the Chinese character for "Tao".

For an online introduction to I Ching by Richard Wilhem, one of its greatest exponents, see here.

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