The paper describes how the
geometric properties of the Tai Chi
Symbol can be exploited
in geometry education to intensify the studies on circles and to introduce some views which
are different from the standard views on elementary notions such as diameter, splitting of
areas or straightening, of arcs. Also, due to the philosophical background of the Tai Chi
Symbol, interdisciplinary aspects as well as intercultural aspects can be integrated into
mathematics education. Similar reasons can be given for teaching the hexagrams from the I
Ging, in computer science education. They reveal very interesting aspects of binary coding.
Tai Chi Symbol. Yin Yang, geometry
of the circle, circular inversion. straightification
circular arcs. hexagrams, binary codinly.
The Buddhist Tai
Chi Symbol (TC Symbol), also called
Yin Yang Symbol, has a very
strong graphical appeal. It is very often used for decoration. In modem representations the
separation line is formed bv two semicircles, each having half the radius of the enclosing
circle. (fig. 1). In the following we will call this line a yin yang diameter (yy diameter). Old
representations are based on different construction modes as shown in figures 2 and 3.
Fig.1 Fig.2 Fig.3
In the modern as
well as in the old symbols you
can discover mathematical symmetries
which possibly give a formal background for the aesthetic appeal of the symbol. The two
areas created bv the separation line. e.a., are congruent and they have a central symmetry
related to the center of the circle. Their circumferences are also equal to the perimeter of the
The problem of splitting the area
of the circle into equal parts has been handled in
mathematics since ancient times. I found solutions of a very high artistic standard in a
temple at the old city of Nara, Japan (fig. 9) and on a door in the old German castle
Wartburg (early 15th century, fig. 10). There are obvious similarities in the yin yang
solution of the problem both in Japan and Germany, and there are generalizations of
splittings with three, four and more areas. In European decorative arts the shape of a fish-
bladder is taken as a basic element of symmetry in the circle (fig. 11).
Fig.9 Fig.10 Fig.11
From the point of view of pure mathematics
or geometry, the most simple way of
a circle into two equal parts is by using a straight diameter (see fig. 4). To get n equal areas
you use n/2 diameters, if n is even. For an odd n the splitting can be done by using radii. It
is well known, however, that ruler and compass can help in this construction for some n
only. n has to be a product of powers of 2 and Format primes such as 3, 4, 5, 6, 8, 10, 15,
16, ... 65537. The same rules apply if yy diameters are used instead of straight diameters
(fig. 6) or semicircles instead of radii.
Fig.4 Fig.5 Fig.6
Using, straight lines seems to express
more efficiency, using semicircles reveals
mony in the splitting. The mode which is chosen possibly reveals an Eastern or Western
philosophical attitude. Figure 12 shows how this idea is underlined by a symbol on the
cover of a book about the three main philosophical streams in China, India and Europe.
When teaching about
the circle in geometry education in
schools, its interactions with
straight lines or sections of a line play the most important role. You look at diameters,
chords, secants, tangents etc. Interactions with other circles or circular arcs are studied later
and less intensively. At the same time splitting of the area of the circle are constructed and
discussed using straight lines.
Certainly one reason for this is
the importance of this interaction for the solution of practical
problems related to areas and circumferences, for example. From a systematic point of
view the idea is close, however, to examining the interactions of circular arcs early as well
and to finding out mathematical relations and properties.
The Pythagorean number PAI is a
symbol for the very special relation
between circular arcs
and straight line sections. PAI describes the metric relation between the diameter d of a circle
and its perimeter u, u = d*PAI. At the same time, for the area f of the circle we have
f = (1/4)*d*d*PAI. The mathematical nature of PAI forbids the construction of a straight
section with a length equal to the perimeter of a circle using only ruler and compass. as well
as the "squaring of the circle".
These relations can be described
in a different way using the yy diameter of a circle. In
modern representation this splitting, line of the TC symbol is composed of two semicircles.
and it has some properties which are similar to those of the classical straight diameter. The
line runs through the center of the circle, its endpoints have maximum distance on the
perimeter of the circle and it has a central symmetry related to the center. The line does not.
of course. generate an axial symmetry like the straight diameter.
It is obvious that the length of
a yy diameter equals half of the perimeter of
Through rotation around the center we can get such yy diameters infinitely.
Naming the radius r, we have these formulae, avoiding PAI:
u = 2*t f = r*t.
These formulae are not really "practical",
since t, unlike r, is not measurable with a
ruler, but they contain something of the harmony which the splitting of a circle by a yy
diameter shows, in comparison with the coolness of a straight section with a standard
Figure 7 shows that there are
more lines which are composed of two
semicircles and which
split a circle in a wave similar to the yy diameters. Their endpoints halve the perimeter and
their length equals t (=r*PAI). The diameters of the two semicircles are d' and d-d', where
0<d'<d. These "pseudo yy diameters" split the area r*t of the circle into two areas (r-r')*t
and r'*t, where r'=d'/2. Of course if r'=r/2 we get the yy diameter. The circumference of
the two areas is the same as the perimeter of the surrounding circle.
So we see that t (=r*PAI) is not
just a term or half of the perimeter, but
circle splitting, sections which are composed of semicircles.
Choosing a sequence r' = r/n (with
n natural > 1), r" = 2*r/n, etc. you get a splitting
area of a circle with radius r in (n+l) equal areas as shown in figure 7. This method goes
back to L. Collatz. This is easily seen, since if fi = i*r/n*t and f(i+1) = (i+1)*r/n*t are two
of the areas generated with ri = i*r/n and r(i+l) = (i+l)*r/n, their difference is r*t/n.
This method of splitting the circle
into n equal parts can be realized with ruler
for any natural n>l. So the pseudo yy diameters are a better means to solve the problem
than standard diameters. On the other hand, we must admit that straight diameters are easier
to construct for practical purposes.
Iterating the method of splitting
the diameter d of a circle to get pseudo yy diameters brings
you to partitions rl+r2+...+rn = r which also generate pseudo yy diameters of length
t = (r*PAI). Putting- ri=r/n for all 1, with the resulting pseudo yy diameter you can approach
the straight diameter as closely as you want by just choosing n large enough. The differ-
ence in length is always t - d or PAI*r-2*r, however (figure 8).
Another interesting generalization of the idea of the yy diameter starts with intersection a
circle with a straight line and studying the arc and secant section thus generated. There is
always a line similar to a yy diameter which has equal length with the arc.
Putting the Tai Chi Symbol on the
unit circle of a complex plane and
applying a reciproque
complex mapplg of type z -> 1/z you extend the mathematical idea and properties of this
symbol to the whole plane (figure 13). The image of a Collatz splitting is particularly
fascinating (figure 16). If you change the semicircles from the yy diameters to different
arcs, you get very interesting separation lines for the plane which have a strong graphical or
even artistic appeal (figures 14 and 15).
Euclidean geometry knows many ways
of approximating the length of a perimeter
section of a straight line, or rather a sequence of such sections as Given by a regular
polygon. Looking at Collatz' figure and its image under circular inversion, you can
discover a method which gives an impression of how a semicircle is really bent towards a
straight line. To discuss this method we do not need the background of the Collatz splitting
and its circular inversion.
We start with a TC symbol in a unit
circle with radius 1. Then t=u/2=PAI.
Next we imbed the
unit circle in a circle with radius 2 and perimeter 4*PAI as shown in figure 17. Now t equals
one fourth of this perimeter. Next we imbed in a circle with radius 4 and t equals one eighth
of its perimeter. Iterating this procedure we get a sequence of circular arcs of length t.
which are getting "straighter and straighter". 1/4, 1/8, 1/16 etc. of a circle's perimeter can
be constructed easily with ruler and compass; so we have a geometrically effective way of
finding an arc of length t finally with the chord related to it coming close enough to t. This
is the "straightening" of the yy diameter we started with and it is half of the "straightening"
of the surrounding circle.
method corresponds to the method of
Archimedes, who approximates
the perimeter of a circle with regular polygons with numbers 4, 8, 16, 32, ... or 6, 12,
24, ... of vertices.
The hexagrams from the
I Ging became public in Europe through
an exchange of letters
between G. W. Leibniz and the Jesuit missionary J. Bouvet, who stayed in Beijing around
1700 (figure 18). Leibniz was mostly interested in the interpretation of the hexagrams as
binary numbers. From the point of view of modern informatics it is quite clear, however,
that the creators of the hexagrams more than 5,000 years ago were developing a binary
coding system. The purpose was to describe and classify all phenomena of the universe
growing out of Tai Chi, the universal power, by the interaction of yin and yang.
The system should also describe
the changes between the phenomena, such
as from winter
to spring to summer to fall and so on. This is one reason why they selected orderings for
the hexagrams which are different from the numerical ordering. Since the 9th century A.D.
or earlier, the Chinese have known the ordering following the binary number system.
Again, for them the meaning was different. The ordering arises from a binary coding tree
(figure 19) of depth 6, which "by accident" yields the binary numbers from 0 to 63 in their
Another ordering clearly has combinatorial
character. It starts with one hexagram
yins (0 yin), followed by six hexagrams with one yin, 15 with two yins, 20 with 3 yins
etc. Research in mathematical education at Rutgers State University in the USA (Bob
Davis) has shown that many primary school children intuitively choose this way of
ordering when the following problem is put to them: You have red and blue cubes.
Construct all different towers with 3 (or 4 or 5) cubes.
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 Needham, J.: Science and Civilization in China, Vol. I-IV, 1+2. Cambridge 1954-
 Zacher, H.J.: Die Hauptschriften zur Dyadik von G.W. Leibniz. Frankfurt am Main
 Mall, Ram A. and Hiilsmann: Die drei Geburtsorte der Philosophic. Bouvier-VerIaLy.
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