Abstract
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.
Keywords
Tai Chi Symbol. Yin Yang, geometry
of the circle, circular inversion. straightification
of
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
enclosing circle.
2. Different philosophies of splitting a circle
The problem of splitting the area
of the circle into equal parts has been handled in
art and
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
splitting
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
more har-
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.
Fig.12
3. Geometry of the circle in class - yy diameters
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
its
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
its circle.
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
metric
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
diameter.
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.
Fig.7
So we see that t (=r*PAI) is not
just a term or half of the perimeter, but
characterizes many
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
of the
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
and compass
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).
Fig.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.
4. Extension from a circle to the plane
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).
Fig.16
5. "Straightening" of the yin yang diameter and of the perimeter
Euclidean geometry knows many ways
of approximating the length of a perimeter
by a
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.
Mathematically this
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.
Fig.18
Fig.19
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
right order.
Another ordering clearly has combinatorial
character. It starts with one hexagram
without
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.
References
[1] Graf, K.D.:
Powerful Means in Mathematics
and Computer Science Education:
Mathematical,
Logical, Mechanical and other Roots of Computer
Science in History.
Jounal
of the Cultural History of Mathematics,
1, Mathematics Education Society of
Japan,
1991
[2] Needham, J.:
Science and Civilization in China, Vol.
I-IV, 1+2. Cambridge 1954-
1965
[3] Zacher, H.J.:
Die Hauptschriften zur Dyadik von G.W.
Leibniz. Frankfurt am Main
1973
[4] Mall, Ram
A. and Hiilsmann: Die drei Geburtsorte
der Philosophic. Bouvier-VerIaLy.
Bonn 1989
Communication with the author:
Institut für Informatik
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FAX: ++49-30-838-75-109
e-mai: graf@inf.fu-berlin.de