Mathematical Journey with the Tao from the Circle to Square

The Mathematical Relationship of  the Tao, Jung’s Transcendent Function, and  the “Middle Way” to the Mandala and the Circle

by Curtis McKallip Jr.

The ancient Chinese wisdom embodied in the concept of the Tao is often accompanied by mandala images used for meditation. The Abstract of the Collected Works of CG Jung V. 13 (from the cgjungpage.org) explains the two as follows:

               “Tao is explained as a conscious way of uniting

               opposites, a reunion of life and consciousness that can only be

               accomplished by realization of the unconscious law of being. This union

               of opposites is neither rational nor relational, but a process of psychic

               development.”

“The middle way” ("Madhyamika Prasangika") from the work of  the 2^nd century saint Naagaarjuna called Muulamadhyamakakaarikaa, is central to Buddhist beliefs.

               “ Mandala symbols, produced by fantasies, are examined,

               and the frequent recurrence of the symbol of the circle is documented…”

The circle, of course, is a symbol of wholeness and completeness and  also symbolizes the feminine, containment, and security.

To my knowledge, the relationship of the Tao, Jung’s transcendent function, the Middle Way to the recurring mandala shape of the circle has been historically a matter of intuition by practitioners of these principles. However, a modern mathematical technique termed “Circulant Matrices”, demonstrates that this relationships may be more than intuitive. It turns out to be a mathematical fact - demonstrable by examples although perhaps not comprehensively provable.

If one makes a graph of randomly placed points and joins them by lines, one finds a diagram like the one below.  The graph very much resembles a state of psychological confusion and polarities which are not well-related.

Step 1:

Taking the midpoint of a line at A and joining it to the midpoint of the next connecting line at B and so on with midpoint C is the mathematical equivalent of the spiritual and psychological practice of “the Middle Way” and begins to simplify the diagram.

Step 2:

This is done for all the lines in the diagram:

Step 3:

The process (Steps 1 and 2)  is repeated for all lines in the new graph and the graph progressively develops more closure and a circular aspect.

End Result:

Most starting configurations develop an elliptical “egg” shape when iterated a sufficient number of times, although some may form a circle. However, the example is close enough and striking enough to suggest that there is some inherent and apparently spiritual property in the process of taking “Middle Way” that brings wholeness from chaos.

As far as I know, there is no proof that for all starting conditions an enclosed shape must be created. However, working through a number of examples does confirm that it is the most likely end result, given enough iterations, or in religious terms, “practice”.

References

Davis,P.J., Circulant Matrices

The Mathematical Experience p. 176