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 2nd 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
