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This article is dedicated to the memory of Marie-Louise von Franz
Abstract
According to the archetypal hypothesis (AH) of Jung and Pauli, physis
and psyche may be seen as complementary aspects of the same reality. At
the basis of all physical and mental phenomena, there exist primal ordering
archetypes which operate as fundamental dynamical patterns of behaviour.
Jung expressed an interest in the archetypes associated with the small
natural numbers, and research into number archetypes has been carried out
in depth by M.-L. von Franz. Pauli felt that not only the infinite series
of integers but also the geometrical continuum has an archetypal nature;
however this idea has not been further elaborated as yet. The present work
is a first attempt to extend the concept of number archetype beyond the
small integers to the Golden Number that is derived from the Golden Section
or Mean. This number archetype in various representations is present in
certain physical and mental phenomena as an optimal structuring pattern
with an inherent dynamism. It also has an affective charge and numinosity.
Consequently the Golden Number bears all the essential features of a fundamental
archetype, thus adding a qualitatively different type to the range of number
archetypes.
Key words: number archetype, golden number, analytical psychology
An extraordinary collaboration between
Carl Gustav Jung, one of the originators of twentieth century psychology
and founder of analytical psychology, and Wolfgang Pauli, a brilliant Nobel
Prize-winning quantum physicist, led to the formulation of the so-called
archetypal hypothesis (AH). Jung and Pauli were ultimately brought to the
archetypal hypothesis as the result of perceiving parallel developments
in depth psychology and quantum physics. Jung noticed that research into
the behaviour of the psyche led to an encounter with certain "irrepresentables"
, the archetypes, while quantum physics similarly had led to "irrepresentables",
namely the elementary particles which constitute matter but for which no
complete space-time description is possible. He concluded that since physis,
the realm of matter, and psyche, the realm of the mind, "...are contained
in one and the same world and moreover are in continuous contact with one
another and ultimately rest on irrepresentable, transcendent factors, it
is not only possible but fairly probable, even, that psyche and matter
are two different aspects of one and the same thing,"the unus mundus".
(Card 1996) The idea of unus mundus is founded "...on the assumption that
the multiplicity of the empirical world rests on an underlying unity, and
that not two or more fundamentally different worlds exist side by side
or are mingled with one another." (Jung, 1958). Pauli postulated the existence
of "...a cosmic order independent of our choice and distinct from the world
of phenomena." ( Jung and Pauli, 1954) The propositions articulated by
Jung and Pauli that eventually were to constitute the archetypal hypothesis
may be summarised as follows (Card 1996):
1) Physis and psyche represent complementary aspects of the same transcendental
unitary reality, the unus mundus.
2) Archetypes act as fundamental dynamical patterns whose various representations
characterise all processes whether mental or physical.
3) Archetypes acting simultaneously in both the realms of matter and
mind account for synchronistic phenomena.
The unus mundus contains all of the
preconditions which determine the form of empirical phenomena, both mental
and physical. These preconditions are archetypal in nature and are, therefore,
completely non-perceptual, pregeometrical, and prelogical. When they reach
psychic perception, they take on specific representations in the form of
images of geometric or numerical structures. Such preconditions, i.e. archetypes,
are the mediating factors of the unus mundus: when they operate in the
realm of psyche, they are the dynamical organizers of images and ideas;
when operating in the realm of physis they are the patterning principles
of matter and energy (Card 1996).
Jung and Pauli formulated their archetypal
hypothesis in the late years of their lives, so their heritage mainly consists
of broad ideas and leading principles. Jung realised that associated with
the set of discrete small natural numbers were some of the most primitive
archetypes, i.e. the number archetypes. According to Jung: "...I always
come upon the enigma of the natural number. I have a distinct feeling that
number is a key to the mystery, since it is just as much discovered as
it is invented. It is quantity as well as meaning" ( von Franz, 1974).
He held that: "...[number] may well be the most primitive element of order
in the human mind...thus we define number psychologically as an archetype
of order which has become conscious" (von Franz, 1974).
Pauli was interested in developing a
more general concept of archetype. This "...should be understood in such
a way that included within it is the 'mathematical primal intuition' which
expresses itself, among other ways, in arithmetic, in the idea of the infinite
series of integers, and in geometry, in the idea of the continuum...I think
it would be of interest to work out more precisely the specific qualities
of the 'archetypal ideas' which form the basis of the mathematics in comparison
with more general archetypal concepts." (Pauli, 1964)
Research into the archetypal nature of numbers was significantly advanced
by Marie-Louise von Franz (1974), who had worked closely with both Jung
and Pauli throughout their collaboration. She came to conclude that, "natural
numbers appear to represent the typical universally recurring, common motion
patterns of both psychic and physical energy." As a result, "...the human
mind can, on the whole, grasp the phenomena of the outer world....The existence
of such numerical nature constants in the outer world, on the one hand,
and in the preconscious psyche, on the other...is probably what makes all
conscious knowledge of nature possible." ( von Franz 1974; Card 1996).
Following her investigations, Card (1996) has restated the original
hypothesis as a general archetypal hypothesis:
1) All mental and physical phenomena are complementary aspects of the
same unitary, transcendental reality.
2) At the basis of all physical and mental phenomena there exist certain
fundamental dynamical forms or patterns of behaviour called number archetypes.
3) Any specific process, physical or mental, is a particular representation
of certain of these archetypes. In particular the number archetypes provide
the basis for all possible symbolic expression.
4) It is possible that a neutral language formulated from abstract
symbolic representations of the number archetypes may provide highly unified,
although not unique, descriptions of all mental and physical phenomena.
In Number and Time, von Franz was primarily
concerned with describing the characteristics of a quaternio of archetypes
that are associated with the first four integers. Up to the present, no
substantial arguments have been made which further extend the number archetype
concept, particularly in the directions suggested by Pauli. In the following
sections it will be argued that the Golden Number, which derives from the
Golden Section or Mean, must be considered to be a number archetype as
well. This represents a limited task in terms of Pauli's more comprehensive
agenda, yet it is qualitatively distinct from the quaternio of number archetypes
explored by von Franz, because the Golden Number, as an irrational number,
represents something quite different archetypally from the small integers.
When a segment of straight line is divided
into two unequal portions such that the ratio of the lengths of the larger
portion to the smaller is the same as the ratio of the original segment
to the larger portion, then the Golden Section (or Golden Mean, Golden
Ratio, or Divine Proportion) has been obtained. The numerical value of
the ratio of the larger portion to the smaller portion can be shown to
be an irrational number (1.61803...) and is often denoted by the Greek
letter, Phi, and referred to as the Golden Number. The Golden Section was
known in antiquity, and its mathematical properties have been subjected
to an extraordinarily broad scrutiny from the time of Pythagoras up to
the present day. Because of its ubiquitous presence within geometry, art,
architecture, and the morphology of certain natural objects, plants, and
animals, the Golden Number has a long and celebrated history of unrivalled
status as an enigmatic mathematical curiosity.
Previous authors have sensed the fundamental
significance of the Golden Number for nature and man and have referred
to it as an archetype ( Huntley, 1970; Ghyka, 1927, 1931, 1977; Reading,
1997). However, no real analysis has been performed in terms of a detailed
assessment of the properties of the Golden Number to determine if it truly
possesses the essential characteristics of a number archetype. To accomplish
this, a set of criteria, consistent with the work of Jung and von Franz
and encompassing the essential elements of the archetype concept, has been
established:
a) The number archetype functions as a primal ordering archetype whose
representations should appear in both the realms of psyche and of matter.
In the psyche, the representations should appear both in manifestations
of the unconscious and as the objects of conscious deliberation.
b) No single representation of a number archetype expresses the full
nature of the archetype. There is a necessary distinction between the archetype-as-such
and any particular representation of the archetype. Consequently, representations
of the archetype may present themselves in novel and surprising ways.
c) The number archetype has an inherent dynamism.
d) The number archetype has an affective charge and numinosity.
We shall proceed with a systematic analysis
of the Golden Number according to the above criteria.
Some of the most remarkable and universal
evidence demonstrating the role of the Golden Number or Golden Section
as an unconscious ordering operator in the human psyche comes from the
analysis of archaeological or ethnographic artefacts which display proportions
of the Golden Section or proportions related to this Section. Several recent
studies of pottery and other clay artefacts from the Neolithic period or
from antiquity reveal the ubiquitous presence of the Golden Section in
their proportions (Morariu et al. 1990; Morariu, 1996a; Morariu 1996b;
also various books on the history of arts, for example Tatarkiewicz, 1978;
or literature cited by Neuveux,1995 ). For example the harmonic analysis
of a clay cup of the Neolithic Age ( Millennium 5 BC) found near Cluj-Napoca,
in the middle of Transylvania, Romania (in the collections of the National
Museum of Transylvania, Cluj-Napoca) reveals that the ratio of the height/width
is the Golden Section. Also the harmonic analysis reveals the main details
of the cup as well as the positioning of the painting are clearly related
to the Golden Section. More examples of the presence of Golden Section
as the main proportion or into other details of the shape or decorative
motifs of archaeological or ethnographic artefacts may refer to: Neolithic
pottery ( 5000 BC), Iclod culture (province of Transylvania, Romania);
Cucuteni culture (province of Moldavia, Romania); Neolithic clay idols
(Cucuteni culture) ; Neolithic dwellings (mud huts) and mud sanctuary (province
of, south-western Romania); pottery of Roman Empire; (province of Transylvania,
Romania); clay rushlights, Roman Empire (Transylvania ); bronze surgical
instruments
Roman Empire (Transylvania); wooden
carved gate, 19 th century, Romania (Transylvania); wooden carved spoons,
19 th century, Romania (Transylvania). All these examples are collected
from the following references: Morariu et al. 1990; Morariu 1996a, 1996b,
1998 and Morariu, unpublished work.
These examples show that widely different
artefacts have in common the Golden Number proportion. At the same time
the suggestion that these findings might imply the conscious use of an
art canon involving the Golden Number may be easily dismissed. The subtle
geometry of the vases and anthropomorphic figures from the Neolithic period
is amazing, even more because it is beyond doubt that such forms have been
made without the use not only of advanced knowledge of mathematics and
design but of even the most modest level of conceptualisation; the artefacts
have been simply created as a result of an "inner force" or inspiration.
The same surprising geometry can be seen, as well, in the artefacts such
as carved wooden spoons, gates, etc., of the traditional rural communities
of the Romanian peasants of the 19th century of Transylvania (Morariu,
1998).
On the other hand it is doubtless that
the artefact makers transmitted from one generation to another some concepts,
motives, stiles, and technical rules. However to the best of our knowledge
it cannot be demonstrated that either the main proportions or the details
of an artefact, above mentioned, have been deliberately made by obeying
quantitative rules, particularly proportions represented by irrational
numbers. On the other hand the proportions represented by rational numbers
render themselves easily to measurement, drafting, and planning at a simple
level of knowledge of elementary mathematics.
While the examples above come from several studies limited to a small
region of Europe over about seven thousand years of prehistory and history,
many similar examples may be expected to exist around the world regardless
of place and time. However it should be clearly stated that the present
work does not intend to make a exhaustive report on the presence of the
Golden Number in artefacts all over the world and during the whole mankind
history.
At some time in the antiquity, the Greeks
realised the aesthetic value of the Golden Section and deliberately transformed
this knowledge into a canon, i.e. a basic rule to be consciously applied
to the creation of the art and architecture. The Parthenon, for example,
was constructed using the proportions of the Golden Section. At a later
stage even the minor arts such as pottery may have followed this rule,
thus explaining the Golden Proportion found in the masterpieces of antique
pottery manufactured in the advanced workshops of the time (see for example,
Tatarkiewicz, 1978). Even to the present day, this rule is well known to
contemporary artists and is a part of their basic instruction.
No one can say exactly how, when and
where the canonisation of the Golden Section occurred. One can only guess
that through long empirical experience, the Greeks came to realise that
such a proportion is an expression of beauty and harmony and therefore
made of it a precise rule which was later taken over by the Romans and
passed along to the present. However, only with the development of mathematics
in the Middle Ages did the special mathematical properties of this proportion
come to light. The coming of the Golden Section into human consciousness
is a good illustration of how the archetypal representations evolve and
develop from an initial stage of unconscious representation ( the pure
action of the "inner force"), through the stage of conscious representation
as an empirical rule ( based on the intuitive feeling of its numinous value),
to the final stage of conscious representation based upon a profound mathematical
and intellectual understanding of the concept.
The Golden Section is however
not the absolute rule for harmony and beauty. The human preference is in
fact a very complex phenomenon and besides the Golden Section we may find
other key proportions such as (PHI)1/2 or proportions based
on irrational numbers: (PHI)1/2 , 21/2 , 31/2
, 51/2, 51/2/2 (Ghyka, 1977), or even more
complicated proportions based on powers of ? (Morariu et al, 1990). More
recently we can find an elegant study concerning the preference for chaotic
figures which are quantified in terms of fractal dimension and Lyapunov
coefficients (Aks and Sprott, 1996). Why such various preferences do exist
in no terms of Golden Number, is a question which may only show the complexity
of the problem. Therefore by no means the value of the Golden Number should
be absolutized as the only one valuable paradigm.
Although we have presented here evidence
in support of the Golden Number at both unconscious and conscious levels
in separate sections, it is of no intention to support the romantic oposition
between “le bon sauvage” and the helpless, intellectual man. In fact all
along history since the scientific knowledge has been developed both the
unconscious and conscious use of the Golden Number was ever present in
the mankind’s work and this represents an important evidence in support
of our claim.
In the natural world the role of the
Golden Number as a characteristic proportion for living creatures regardless
of their position on the evolutionary scale is so well known that the very
presence of the Golden Number as a proportion is regarded as a fundamental
characteristic of the living world and its spatial and temporal characteristics.
The basic morphological structures of many plants, insects, animals and
man have many general and detailed features which correspond to the Golden
Proportion (for examples, see: Ghyka, 1927, 1931, 1977; Ciofu, 1994; Reading,
1997).
Beauty and harmony are usually associated
with the Golden Proportion. For example, despite the great diversity of
human faces, the average "portrait" of a human face that results from the
superposition of many individual photos is an ideal face which obeys the
Golden rectangle rule--i.e., it can be inscribed in a Golden Rectangle
(the ratio height/width = Phi). Furthermore, many of the details of the
face can be easily found to be located at strict geometrical sections or
cross points resulting from the harmonic analysis of the Golden Rectangle,
and the human eye can readily detect minute departures from such a harmonic
structure (Ghyka, 1977). This is a relevant example of how living nature
works archetypally. In fact, it reveals the archetype's statistical mode
of behaviour--it is an "average" rule, not an "absolute " rule. It permits
diversity and a certain degree of randomness, while it keeps such variances
within certain limits.
As well as the presence of the Golden
Number in the proportions of morphological characteristics of living creatures,
similar proportions can be found at the cellular or biochemical levels.
Consider the following examples (Ciofu,1992):
a) the percentage of the globulin fractions (alpha 1, alpha 2, beta
and gamma ) is an increasing series progressing with a factor of (PHI)1/2;
b) the ratio of the minimal to maximal concentration in urine of urea
and glucose respectively is (PHI)1/2;
c) the ratio of monocites, neutrophiles and erythrocytes respectively
in the human blood represent terms of a progression series with a factor
of (PHI)1/2. The same type of series also represents:
d) the ratios of the oxygen partial pressure in the venous blood, in
the arterial blood and in the atmosphere respectively;
e) the partial pressure of the carbon dioxide in the following series:
in atmosphere, in the expired air, in the arterial blood and in the venous
blood respectively.
These kind of examples may appear as
unusual or just simple coincidences compared to those found widely exposed
to the eye i.e. whole organisms or part of the organisms. While the presence
of such a proportion is common in the anatomical structure even into some
details, it is not unreasonable to seek it at cellular level or in physiological
properties. However it remains to be established what are the advantages
of these spatial or temporal ordering at this level of the living matter
or how could be described the principle lying behind such a rule.
The presence of the Golden Number has
also been discovered in other areas of biology, including physiology, neurophysiology,
epidemiology, and also in the biological rhythms (Ciofu, 1994). In psychology
and sociology, occurrences of Golden Number proportions appear in studies
of reaction times, memory and learning, IQ, behavioural simulation, subjective,
projective and interpretative evaluations, etc. ( Ciofu, 1994).
Less known is the presence of the Golden
Number as an ordering principle in the physical universe, in both the micro
and macrocosms. This circumstance is not surprising because the aesthetic
value of the Golden Section was the first to be realised, and this stage
remained for about two thousand years. Only with the later development
of science from the time of Kepler onwards has the ubiquity of the Golden
Number in the physical world become evident. At present, examples of the
Golden Number may be found in the physics of elementary particles, ionising
radiation, atomic phenomena, chemical processes and structures, the periodic
system of elements, and the temporal and spatial structure of the universe
( Ciofu, 1994). More specific examples refer to: the ratio of the distances
to the Sun of two neighbouring planets; the sideral period of revolution
of the planets; the ratio of the average speed on the orbit of the Moon
and Earth; the equatorial radius of the six satellites of Saturn; the ratio
of the shortest path to the longest path of alpha particles emitted by
Uranium and Actinium series respectively; the periodic table: the place
of the elements in groups and periods; the boiling points of helium, hydrogen,
nitrogen and oxygen; the ratio of the carbon dioxide, oxygen and nitrogen
percentages in atmosphere; the burning of a candle: various structural
and caloric details. All these examples are discussed into some details
in Ciofu’s book ( 1994). However these examples seem to be less spectacular
than in the living world and the more frequent proportion refer to (PHI)1/2
and quite rarely the phi itself. A more careful investigation in the area
of physics and chemistry is needed. At present this area of evidence seem
the most in need of new and well documented examples.
Just as in the case of the evidence
coming from physiology or cellular biology, it is necessary to seek a basic
principle behind these observations. At a very general level the Golden
Number is seen as an expression of the principle of minimum action (Ghyka,
1977; Ciofu, 1994). On the other hand we know that physics text books tell
nothing about Golden Number or its significance for its basic laws. Some
isolated findings mention the presence of Golden Number in non-linear phenomena,
chaos and aperiodic crystals ( Shibayama, 1984; Reading, 1997)
When the lengths of the two portions
of the Golden Section are compared, they are found to be incommensurable.
When the Golden Number is evaluated, it is found to be an irrational number-a
never-ending, never-repeating decimal. The Golden Number appears in many
guises that seem to bear no relationship to each other, such as the geometry
of the pentangle, the logarithmic spiral patterns of the chambered nautilus
or the seeds of the sunflower, structure of the cosmos, the trajectory
of a moth approaching a candle flame, or the shape of a human face.
It emerges again and again from the
pattern of numbers that Fibonacci first discovered when considering the
growth of a population of rabbits but which also appear in the arrangement
of leaves on a branch or petals in a flower, in the arrangement of "bulbs"
on the Mandelbrot set, in the multiple reflections of light from glass
plates, in the patterns of transition of electrons in atoms, and a multitude
of other examples. Thus, no single representation of the Golden Number
can be said to completely characterise its essence, and the consequence
of this is that after more than two millennia it continues to be discovered
in new and surprising circumstances. Most recently it has been discovered
in various aspects of chaos theory where it, "...characterises something
like the last barrier of order before chaos sets in." (Peitgen, Juergens
and Saupe, 1992)
The Golden Number has an intrinsic dynamical
character--it acts as a condition for harmonious balance between part and
whole. Fibonacci's series, for example, describes many types of growth
in populations of plants and animals in which the Golden Number appears
as a characteristic constant of this growth.
If one looks at the whole range of representations
o the Golden Number, then one discovers that the Golden Number is related
to what "happens", "flows", "evolves", "develops", or to what has a " relative
beginning and an end". This is in contradistinction to timeless structures
such as the periodic crystals which do not contain the Golden Number.
Aestheticians have conceptualised this
difference through the distinction of static proportions from dynamic proportions.
According to Jay Hambidge (cited by Ghyka, 1927, 1970), static proportions
refer to rational numbers such as 4/3; 4/1; 3/2; 3/1 ( it should be noticed
that there is a fundamental distinction between a periodic yet rational
number, versus an irrational number). The dynamical proportions are represented
by irrational numbers resulting from ratios like: 21/2/1; 31/2/1;
51/2/1; 51/2/2; (51/2 + 1)/2 = 1.618...
A particular case is the ratio 1/1 or 2/1 which is has both static and
dynamical properties. Hambidge's classification may be regarded as an empirical,
broader basis of what static and dynamic qualities may signify within an
archetypal theory. This might represent the seeds to a further widening
of the archetypal significance of continuum in respect to our present understanding.
The Golden Number has long been associated
with the aesthetic judgement of harmonious proportion. This aspect of the
Golden Number has, in fact, been the subject of several psychological experiments,
and the outcome in each case confirmed its unique aesthetic character.
In 1876, the German psychologist Gustav Fechner conducted experiments in
which variously shaped rectangles were presented to a number of individuals
who were asked to give their preference for the most aesthetically pleasing
rectangle.
Fechner found that as the relative proportions
of the sides of the rectangle approached those of the Golden Proportion,
the frequency of selection of the rectangle increased. Conversely, when
the subjects were asked to select their least favoured rectangular shape,
the Golden Rectangle shape was selected least frequently. Fechner's experiments
were later repeated by Witmar, Lalo, and Thorndike, and in each case similar
results to Fechner's were obtained ( Huntley, 1970).
These results help to provide a psychological
basis for understanding why there has been a long history of reverence
for the Golden Number by artists, architects, mathematicians, and others,
evoking from them typical responses of pleasure, surprise, and awe. For
some it has even become the inspiration for a mystical attitude, in which
the Golden Number in its various representations takes on a sacred character.
These responses are all testimony to the numinosity of the Golden Number,
an aspect which helps to confirm its archetypal nature. There are even
occasions when the numinosity of the Golden Number has led to a pathological
fascination (as may occur with all deeply affecting archetypes) whereby
an individual has become fixated upon it at the cost of reasoned judgement.
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Notes:
1. First communicated at the Symposium Physics and Mind, 19 Oct.1997, "Babes-Bolyai" University, Cluj-Napoca, Romania and in revised form at the Assisi Conference, The Confluence of Matter and Spirit, Woodstock, VT, U.S.A., 26 April 1998. BACK