the system of Tao

The symbol traditionally associated to Tao or, more specifically, to the Teh del Tao - the knowable manifestation of it, is the Taijitu, commonly known as the Ying-Yang symbol, tied to the male/female duality but - more generally- representative of any opposite duality.
The historical association between the symbol and the Taoism, and to its main text - the Tao-Teh-Ching - is not clear, however it represents then completely in its simplest and most synthetic form.
Drawing the symbol can be made by drawing an external circle and two internal circles of half radius, erase the two semicircles on the opposite sides and coloring - usually by white and black - the two remaining parts:

Graphic drawing of the Taijitu symbol.

The symbol represents a polar duality between opposite elements, different and distinct, and describes in general the eastern traditional vision of the polar duality opposed to the western logical one:

Symbolic representation of polar dualities
according to logic western vision (left), intermediate (center) an eastern (right).

In the western vision of classical logic of greek origin, symbolized by the circle to the left, it is drawn a distinction through two opposite and symmetrical elements/processes which, by logical definition, are not mixable and that together describe the totality where the dual distinction is drawn. For example dualities like day/night, negative/positive, war/peace, femminine/masculine and so are composed by opposite elements or processes and completely described the reference context where they apply. The resulting vision is completely static ande binary.
A further improvement starts from the consideration that these dualities are processes, and as such are dynamical; the circle to the middle illustrates a more dynamical vision between the polar opposite processes of the duality.
In the Taijitu symbol the dynamical vision of the polar duality processes and elements reaches at the same time its maximum simplicity and dynamical complexity of representation. Not only the polar processes have a recursive but there's also an interaction between polar processes and elements, shown by the two circular dots internal to the process of opposite sign.
The Taijitu symbol may be considered as a system, and therefore analyzed in its systemic characteristics:

• System elements

They are symbolized by the two internal polar circles placed at the center of the symbol two semicircles, where the corresponding process of opposite sign reaches its maximum amplitude.

• System processes

They are symbolized by the two polar symmetrical recursive polar shapes which together divide the external circle of the symbol. The reppresented dynamic is  è both of process and between processes.

• Processes-elements interaction: system dynamics

The most complex feature of the symbol is the contemporary representation both of polar elements and processes, linked together through a specific dynamic.
At the point where an increasing process reaches its maximum and starts to decrease there's the presence of an element of opposite sign. This type of dynamics may be understood in different ways:
at the peak of its growth a process generates an element of opposite sign;
the presence of an element breaks the opposite sign process growth and makes it decreasing till reset;
lthe presence of an element induces an increasing process of the same sign which reduces the one of opposite sign;

• Process amplitude

To determine the process dynamic is necessary to compute its amplitude as a function of some evolution variable. To define them the following model is used:

Model to compute process amplitude for the Tao symbol.

within the external circle Cext of radius R one draws the red C1 and blue C2 circles with radius R/2. To define a coordinate which is always in the middle of the process the green C3 circle is used with the center moved to the left of R/4 and 3/4R radius. The upper green semicircle defines a radial coordinate which remains always central to the process and that may be assumed as its evolution coordinate s for the upper semicircle of the symbol. For the lower semicircle the central coordinate is a vertical line segment of length R/2 perpendicular to the horizontal axis from the circle C2 center to its border.
The process amplitude may be defined, for any coordinate s=angle*3/4R with angle that varies from 0 to 180 degrees, as the distance between the P1 intersection point of the C3 radius extension withe external circle Cext and the intersection point P2 of the C3 radius with the C1 circle. For any s value in the upper semicircle the P1-P2 segment is perpendicular to C3 and to compute it the sine and cosine theorems are applied on the triangles defined by P1 and P2, where two sides and one angle are known.
For the lower semicircle the amplitude calculation as a function of s is immediate and coincides with a decreasing quarter arc of circle function with radius R/2.
The result for a symbol circle with radius R=1 is:

Process amplitude as a function of evolution for a unitary radius circle.

The process starts from zero, reaches its maximum value R for a s value of 3/4πR and decreases as a quarter of circle to  zero at the value 3/4πR+R/2.
The progress for the two opposite symmetrical processes is:

Polar two-processes amplitude as a function of evolution for a unitary radius circle.

which repeats itself indefinitely radially rotating clockwisee, where the beginning of a process coincides with the other's maximum.
The recursion may be displayed also by drawing in a linear way the shapes of the opposite processes:

that seems a wave, though it is not since not sinusoidal but composed by alternate semicircles.

• System border: closure

The system border is the external circle Cext of radius R, which implies as the system dynamic is of the operational closure type. The two possible directions of rotation - clockwise and counterclockwise - are traditionally, like other symbols such the swastika, associated to a further polar duality - creative if clockwise, destructive if counterclockwise -.

• System matrix

It is represented by the infinite plane external to the symbol and represents, in a necessary metaphorical way, the indescribable and unknowable Tao which the emergent symbol is the Teh:

- 25 -

There was something formless and perfect

before the universe was born.

It is serene. Empty.

Solitary. Unchanging.

Infinite. Eternally present.

It is the mother of the universe.

For lack of a better name,

I call it the Tao.

It flows through all things,

inside and outside, and returns

to the origin of all things.

The Tao is great.

The universe is great.

Earth is great.

Man is great.

These are the four great powers.

Man follows the earth.

Earth follows the universe.

The universe follows the Tao.

The Tao follows only itself.