Hogenson, G.B. (2020) The geometry of wholeness. In Jung, Deleuze, and the Problematic Whole,
(Eds, Main, R., McMillan, C. & Henderson, D.) Routledge, London, pp. 125-141.
Chapter 5
The geometry of wholeness
George Hogenson
A client in my practice, whose artistic skills are highly developed, regularly carries a sketchbook with him in which he draws images from
his dreams, paintings and other renderings of his process in analysis.
Shortly prior to the conference represented in this book he brought
two sketches to our session. The first, he recounted, arose spontaneously. When he completed the sketch he ‘sat with it for a long period
of time, feeling contented and at peace’.
Several days later he attempted to reproduce the experience, but the
figure he drew in this instance left him feeling ‘disturbed and unsettled’
when he finished.
Both images are complex and reward detailed examination, but for
purposes of the present discussion I want to focus on the central geometric distinction between them. In the first image we see a centred
square that is subdivided into four triangular elements. The quadratic
shape itself balances between a space of light — illuminated by a sunlike eye — and complete darkness, while the internal quadratic subdivision demarcates the boundary between the light and dark domains
as well as marking their essential conjunction in the vertical. In the
second image the central square has been divided into two squares,
each of which is subdivided into two triangular elements. The two
squares are now balanced within a complex and quite formal structure
of incomplete circular patterns, almost mechanical in structure, with
the dark element narrowly confined between two of the incomplete
circles. The illuminating eye is still present, but like the dark element
it appears withdrawn from the central elements of the image, casting
only a representation of light, in contrast to the encompassing sense of
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Figure 5.1 Client’s first sketch
Reproduced with permission
illumination in the first image. Overall, the image has a highly planned
or intentional quality, the result of the artist’s desire to recreate an
experience that was originally spontaneous.
Jung is clear, and ample phenomenological evidence underwrites his
claim, that the most common pattern associated with the experience
of wholeness is the quadratic pattern, frequently but not always within
a circular structure. These are the classic geometric characteristics of
the mandala. Their universality is well attested, ranging from the patterns of Buddhist mandalas such as the Tibetan Kalachakra to the
pre-Columbian Aztec such as the Codex Fejérváry-Mayer cosmogram
with the fire god Xiuhtecuhtli in the centre:
In his essay on mandala symbolism Jung includes a series of
mandala figures, some of his own composition and many from his
patients, in addition to images from cultural traditions such as the preColumbian cosmogram and the Kalachakra. In all of these traditions
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Figure 5.2 Client’s second sketch
Reproduced with permission
it is important to keep in mind the correlation that inheres between the
cosmographic image and the interior states of those who engage with
the mandala. To further set the stage for the argument of this chapter
I want to add two more of these mandala-like figures, one from Jung’s
investigations of the Chinese I Ching or Book of Changes and the other
by a patient following a dream:
What is important in these two mandala images is the pattern of
quadratic movement either out from a central point or in toward a
central point, and the ambiguity of in which direction the pattern is
moving. Although Jung’s rendering of his patient’s mandala is in black
and white he reports that it was in fact coloured in red, green, yellow
and blue. He goes on to comment that:
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Figure 5.3 Kalachakra mandala
Source: WikiCommons
As to the interpretation of the picture, it must be emphasized that
the snake, arranged in angles and then in circles round the square,
signifies the circumambulation of, and way to, the centre. The
snake, as a chthonic and at the same time spiritual being, symbolizes the unconscious. The stone in the centre, presumably a cube, is
the quaternary form of the lapis philosophorum. The four colours
also point in this direction. It is evident that the stone in this case
signifies the new centre of personality, the self, which is also symbolized by a vessel.
(Jung 1969b: §651)
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Figure 5.4 Codex Fejérváry-Mayer
Source: WikiCommons
The geometry of wholeness
The question raised by these and many other examples is whether the
geometry of the symbolic patterns has a relationship to the experiences associated with them or if they are simply conventional patterns
with no deeper significance. Put another way, if we take the two images
produced by my client, is there some underlying relationship between
the patterns and his experience of peacefulness in the first instance and
disturbance in the second? Furthermore, if such a distinction exists,
does it tell us something about the experience of wholeness as a psychological phenomenon?
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Figure 5.5 ‘The River Map’, one of the legendary foundations of the I Ching
Reproduced (with permission) from C. G. Jung, ‘Concerning mandala symbolism’, Figure 2,
in Collected Works,Volume 9i
To work out a possible answer to these questions we can look at
what is now recognized as the basic geometry of the natural world,
fractal geometry (Mandelbrot 1983). The origins of fractal geometry
can be traced back at least to the seventeenth century when mathematicians began to work with recursive patterns. Recursion — the iterative repetition of a simple mathematical process in which the result
of one iteration becomes the variable for the next iteration — is central to the generation of fractal patterns. For example, one can draw
a straight line down the west coast of Norway, and derive one length
for the coastline. It would be immediately evident, however, that this
was an unusually inaccurate map. Through a process of recursive
fractioning of the original line, however, the map will become increasingly accurate. In principle this process could go on almost infinitely,
finally arriving at a map that encompassed even the grains of sand
on the shores of the Hardanger Fjord. More illustrative of the formation of fractal patterns are the correspondences that exist between
the branching patterns of trees and the vein patterns in the leaves of
the tree. Once again, one can see a process of ever-finer reiteration
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Figure 5.6 Mandala based on a woman’s dream.
Reproduced (with permission) from C. G. Jung, ‘Concerning mandala symbolism’, Figure 4,
in Collected Works,Volume 9i
of a simple recursive structure in the formation of the natural world.
Although controversial at extremely large scales, some cosmologists
have suggested that the structure of the elusive dark matter follows a
fractal pattern, and even that the hypothesized multiverse is fractal in
structure.
This last level, while disputed, is at least instructive for purposes
of examining the structure of the Kalachakra and Codex FejérváryMayer mandalas, as both of them represent a cosmological map in
the form of the domains of the deities who define their structure. The
degree to which fractal geometry can subsume phenomena ranging
from the structure of neuronal dendrites in the brain to, possibly, the
structure of the multiverse was not central to the study of this form
of geometry until Benoit Mandelbrot (1924–2010) realized the importance of scaling or the degree to which a phenomenon at one level of
scale reproduced itself at ever greater scales. This is the principle that
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accounts for the self-similarity — another concept in scaling phenomena — between the structure of the tree and the structure of the veins
in the tree’s leaf. Mandelbrot attributes his discovery of scaling to an
accidental reading of a review of a book by the American polymath
George Kingsly Zipf (1900–1950), upon the recommendation of his
uncle, a distinguished French mathematician.
Zipf was on the faculty of Harvard University, nominally with
an appointment in linguistics but with freedom to pursue a variety
of interests. He eventually referred to himself as a student of human
ecology, by which he meant the entire range of human activity. The
research he is best known for, however, involved the scaling phenomena of human institutions, beginning with the size of towns and cities
but finally grounding on the frequency of words in texts. What Zipf
found was that the relative scale of communities or the frequency of
words in a text all followed the same pattern of distribution. If, in a
given geographic area, the largest community had 5,000 inhabitants,
the next largest community would have approximately 2,500, but there
would be two such communities, and so on down to the smallest community. The same relations appear in texts and a variety of other phenomena ranging from the intensity of earthquakes to the frequency
of visits to websites on the Internet. Zipf’s law, as this phenomenon is
now known, is a power law distribution, due to the role of an exponent
in the calculation of the pattern, and along with Fibonacci numbers
and fractal geometry it seems to be among a small group of calculations that apply to situations that otherwise appear to have no relation
to one another.
In a series of papers I have suggested that Zipf’s law sheds light
on elements of Jung’s understanding of symbols and the nature of
archetypal phenomena (Hogenson 2004, 2005, 2009, 2014, 2018). The
argument in these papers has focused on what I have called symbolic
density in which some symbolic structures manifest a deep hermeneutical structure or potential for interpretation. Archetypal phenomena,
in this interpretation, are capable of much deeper interpretation than
other symbolic structures. This distinction would apply to Jung’s critique of Freud’s understanding of the symbol as basically a sign, with
relatively limited semiotic reference, in distinction to his own understanding of the symbol as referencing otherwise inaccessible dimensions of the psyche.
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As with fractals, Zipf’s law relies on an iterative process to work
out the frequency in the various patterns it describes. I now want to
argue that both fractal geometry, as found in Mandelbrot’s work and
Zipf’s theories of scaling, which played a decisive role in the development of Mandelbrot’s theorizing, shed light on the geometry of the
images associated with wholeness — the mandalas of Jung’s patients,
Buddhist meditation, and my own client.
The Mandelbrot set
The Mandelbrot set is probably the best-known example of fractal geometry, superseding the Julia set from which it derives much
of its structure. The Mandelbrot set was not, however, developed by
Mandelbrot but by the French mathematician Adrien Douady, who
named the set in honour of Mandelbrot. Much of its fame derives
from renderings of the pattern that add colour to certain outcomes of
the iterative calculation, z=z2+c. As it happens, the colours that are so
captivating are actually outcomes of the equation that fall outside the
stipulated boundaries of the set. As one will see in computer-generated
renderings of the Mandelbrot set, the patterns that fall outside the
original boundaries frequently begin, at a later point in the iterative
process, to recapitulate at ever smaller scales, the original pattern of
the set. This process of seemingly infinite recursion is one of the most
fascinating aspects of the calculation.
It is also the case, however, that the results of the recursive iteration
of the formula for the set can be graphed in a variety of forms. The
two, to which we will turn our attention here, are known as a bifurcation, or logistic, graph and a cobweb plot or graph. The bifurcation
graph typically appears in the form shown in Figure 5.8.
The bifurcations in the graph correspond to critical inflection points
on the Mandelbrot set. Essentially, what is happening is that as the formula for the Mandelbrot set evolves, the value of z – in this simplified
example – reaches critical points where the graph changes dramatically. If these values are transposed to the bifurcation graph and used
as parameter values in that graph, they correspond to critical bifurcation points in the graph, as can be seen in the illustration in Figure 5.9.
The cobweb plot, on the other hand, has the form shown in
Figure 5.10.
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Figure 5.7 The Mandelbrot set
Source: WikiCommons
What interests us here is the relationship between the Mandelbrot
set — or, for that matter, other fractal patterns that will fall close to the
same graphical outcomes — the bifurcation diagram and the cobweb
plot. Using the same parameters for the calculation in each instance,
the graphical depiction of the calculation falls out in distinct patterns,
to which we will attend.
In essence, the initial stages of the iterative calculation for the
Mandelbrot set results in outcomes that are identical or diverge in
only the slightest degree. However, the process of iteration eventually
begins to display the characteristics associated with critical dependency on the initial conditions, and divergence occurs abruptly at an
inflection point in the graphing process. The initial divergence is a single bifurcation of the plot on the bifurcation graph. This is followed,
however, by a further bifurcation, and the eventual emergence of a
chaotic regime as the divergences multiply. The point that is of greatest concern to modelling the mandala patterns illustrated above is the
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2
1.5
1
xn
0.5
0
–0.5
–1
–1.5
(i)
–2
0
(ii) (iii) (iv)
0.5
1
1.5
(v)
2
a
Figure 5.8 Bifurcation graph
Source: WikiCommons
first bifurcation. To illustrate this point we can look at a combination
of moments in all three graphs.
Subsequent bifurcations eventually lead to the chaotic regime.1
Discussion
Among the themes that are central to Jung’s system of psychology,
the ‘unity of opposites’ looms large. Drawing principally from the
fifteenth-century Cardinal-philosopher Nicholas of Cusa, Jung’s focus
is on a variety of unities, such as consciousness and the unconscious or
anima and animus. It is in these unities that Jung sees the possibility of
an experience of wholeness that he associates with the quadratic form
of the mandala:
Although ‘wholeness’ seems at first sight to be nothing but an
abstract idea (like anima and animus), it is nevertheless empirical
in so far as it is anticipated by the psyche in the form of spontaneous or autonomous symbols. These are the quaternity or mandala
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Figure 5.9 Bifurcation and Mandelbrot set
Source: WikiCommons
Figure 5.10 Cobweb plot
Source: ‘Fractal Geometry’, Yale University, Michael Frame, Benoit Mandelbrot (1924–
2010), and Nial Neger. Open source.
Figure 5.11 The initial stage
Source: Courtesy of Michael Hogg
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George Hogenson
Figure 5.12 The final point of unity
Source: Courtesy of Michael Hogg
Figure 5.13 Chaos
Source: Courtesy of Michael Hogg
symbols, which occur not only in the dreams of modern people
who have never heard of them, but are widely disseminated in the
historical records of many peoples and many epochs. Their significance as symbols of unity and totality is amply confirmed by
history as well as by empirical psychology.
(Jung 1969a: §59)
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The other aspect of this view of wholeness, which he derives in part
from his study of Gnosticism, is that the quadratic form not only gives
shape to psychic wholeness but is imbedded in the material world
as well:
The primordial image of the quaternity coalesces, for the Gnostics,
with the figure of the demiurge or Anthropos. He is, as it were,
the victim of his own creative act, for, when he descended into
Physis, he was caught in her embrace. The image of the anima
mundi or Original Man latent in the dark of matter expresses the
presence of a transconscious centre which, because of its quaternary character and its roundness, must be regarded as a symbol
of wholeness. We may assume, with due caution, that some kind
of psychic wholeness is meant (for instance, conscious + unconscious), though the history of the symbol shows that it was always
used as a God-image.
(Jung 1969a: §308)
We can now begin to see the possible importance of the relationship
between fractal geometry, in our example presented by the Mandelbrot
set, and the images of wholeness identified by Jung. The original work
done on fractal geometry was confined to the purely mathematical world of Georg Cantor and others, but with Mandelbrot’s work
and the implementation of fractal geometry on computers it became
evident that fractals, in large measure, defined the geometry of the
actually existing physical world. If, at the same time, fractals define
important elements of the symbolic world of psychic wholeness as
conceptualized by Jung it becomes possible to hypothesize a level of
psychophysical unity as proposed by Jung, particularly in his collaboration with Wolfgang Pauli.
What we find in the various diagrams presented above is the formation of a quaternity at precisely that point where the first bifurcation
occurs, as unity separates into duality. Further progress leads inexorably to a chaotic regime of dispersal, both in the physical world and in
the psyche.
Returning then to my client and the two drawings he brought to
our session, we can now see that the spontaneously drawn figure that
brought with it a sense of calm appears, as a simple quaternity, to
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occupy that point in the formation of the cobweb plot where unity is
on the cusp of bifurcation. The second drawing, intended to recreate
the spontaneous experience but failing to do so, occupies the space
defined by the bifurcation of unity into duality. A sense of wholeness
infuses the first drawing, while the dissonance of duality is reflected in
the second. In the same manner we can now see in the cosmic mandalas such as the Kalachakra or the Aztec Codex Fejérváry-Mayer a
recognition of the encompassing nature of this inflection point in the
geometry of nature and the psyche. Jung saw this correspondence as
central to his thinking about synchronicity, remarking on the importance of mathematics as a transcendent aspect of reality:
Psyche and matter exist in one and the same world, and each
partakes of the other, otherwise any reciprocal action would be
impossible. If research could only advance far enough, therefore,
we should arrive at an ultimate agreement between physical and
psychological concepts. Our present attempts may be bold, but
I believe they are on the right lines. Mathematics, for instance, has
more than once proved that its purely logical constructions which
transcend all experience subsequently coincided with the behaviour of things. This, like the events I call synchronistic, points to a
profound harmony between all forms of existence.
(Jung 1969a: §413)
The particular mathematics of Jung’s own reflections on wholeness,
of course, are forms of geometry, and we can now begin to see why
the symbolism of wholeness is in fact the form that unifies psyche and
physis.
Note
1 The process by which this pattern unfolds can be viewed in its entirety at
https://vimeo.com/13566850
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