I have no doubt that Jung would have been fascinated by the work of Mandelbrot. Conventional Mathematics will never be able to solve the issues deeper that fractal geometry raises. Indeed - as I suggest in this post - the current interpretation of "fractal" dimensions represents - even in mathematical terms - a basic confusion. Such dimensions are more mysterious than the mathematicians care to admit.

In 1968, the French mathematician, Benoit Mandelbrot asked an intriguing question.

"How long is the coastline of Britain?"

The surprising answer is that there is in fact no definite answer. One can indeed approximate an answer by defining one’s scale of measurement. However as these measurements are increasingly refined the answer eventually becomes so large that it is theoretically infinite.

Thus though the (two- dimensional) area is clearly finite, the curve tracing its perimeter is (theoretically) infinite.

Mandelbrot referred to these naturally rough terrains (such as coastlines) as fractals.

Numerous examples can be found in nature displaying fractal features such as leaves, trees, mountains, clouds, galaxy clusters etc.

In so many ways fractals raise the intriguing - and highly important - question of the relationship of finite and infinite notions.

I will illustrate this by just three well-defined mathematical fractals.

A classic example "one-dimensional" is the Cantor Set. Here we start with a straight-line interval. We then delete the middle third of the line. This leaves two smaller intervals. Again we proceed to delete the middle thirds of these new intervals and proceed indefinitely in this same manner.

Ultimately, the total length removed will be equal to the length of the original interval (so that it no longer remains). Yet we will be left with an infinite number of points!.

The best known "two-dimensional" fractal of this type is the Koch snowflake (named after Van Koch).

Here we start by drawing an equilateral triangle (i.e. where the three sides are of equal length.

Again we take the middle third of each side to construct three new equilateral triangles (extending outwards).

Now if say the perimeter of the original triangle was 3 units to start with, then is easy to demonstrate that the perimeter of the new figure will be 4 units.

However we can continue this procedure indefinitely (i.e. constructing new equilateral triangles on the middle third of each exposed line of our figure).

So again we generate a significant paradox that though the area of the "snowflake" tends to a finite limiting value, its actual perimeter will be of infinite length.

Other "two dimensional" examples are the Sierspinski Carpet and Sierspinski Gasket. In the first case we start with a square and cut out the middle area (1/9) of total area.

We then divide the are remaining into smaller squares (8 in total) and apply the same procedure, continuing on indefinitely.

(In the latter case we cut out equilateral triangles rather than squares)

The Sierspinki Carpet and Gasket can be extended to "three-dimensional" cubes (i.e. the Menger Sponge).

Here we start to cut away "middle" sections. Clearly as we create more and more "holes" in the cube its surface area will continually increase and will eventually tend to infinity. However the volume of the sponge will have a finite value.

Mathematical fractals, in fact seriously challenge conventional notions of dimensions.

In conventional terms we define dimensions in terms of unambiguous directional movements.

Thus we trace out a one-dimensional line by moving in a clearly defined direction (length).

We trace out a two dimensional figure (such as a rectangle) by moving unambiguously in two directions (length and height). We trace out a three dimensional cube by moving unambiguously in three directions (length, breath and height).

This is precisely how we define our conventional physical notions of dimensions. Space can be either one-dimensional (as a line), two dimensional (as a plane) or three dimensional (as a solid). The fourth dimension of time is treated as somewhat separate (though it can be treated – as in Relativity Theory as an "imaginary" space dimension).

When we proceed with this integral understanding of dimensions, all measurements appear to be finite.

However our simple mathematical fractals directly challenge this very notion, for here in each case we have a mixture of finite and infinite notions of measurement.

Again the line of zero finite length has contains an infinite set of points.

The snowflake of with a well-defined finite area has a perimeter of infinite length (the Koch Curve).

The sponge with a definite finite volume has a surface area that is infinite.

In fact, with these fractals, conventional notions of directional movement break down leading to an entirely different concept of dimension.

The key feature of the three fractals above is self-similarity (i.e. the parts are similar to the whole but on a reduced scale).

Now if we take a D-dimensional figure and divide it into N entirely similar parts, the similarity ratio, R between the entire figure and a single part is given by R = N raised to the Dth root (i.e. the reciprocal of D).

Thus if we take a straight line and split it into N equal pieces then each piece will clearly be 1/N of the length of the whole. So the similarity ratio R will be N.

This is exactly the value we get from our formula by taking D= 1. (This is what one would expect as the line is one-dimensional).

With the Koch snowflake there is a definite pattern whereby a perimeter length that starts with 3 units is increased to 4 .

So R is equal to 3 and N = 4.

The value of D which gives the dimension of the Koch Curve is obtained by the formula 3 = 4 (raised to the power of 1/D).

This can be simply solved by taking logs of both sides

So that log 3 = (1/D).(log 4).

D in fact has the surprising value of 1.2619 (correct to 4 decimal places).

Now I must take issue here with the conventional mathematical interpretation of this result.

It is generally used to prove that the dimension is fractional.

This strictly speaking is incorrect and betrays the reduced rational bias of mathematics.

Properly speaking a fraction is a rational number.

Now the number we have generated from the formula is not a rational but rather an irrational number (i.e. a finite number with an infinite random expansion of its decimal sequence). It only has a reduced rational interpretation when one approximates the value (correct to a defined number of decimal places).

From a qualitative point of view this is a highly significant point. The very word "fractal" to describe the dimension is something of a misnomer.

The dimension of the Koch Curve is not fractional but rather irrational. We thus have to explain the true meaning of irrational (and not just fractional) dimensions.

Fortunately Holistic Mathematics can throw considerable light on the qualitative significance of "fractals".

Remember a basic axiom of Holistic Mathematics is that every mathematical notion with a quantitative interpretation (in conventional mathematics) has a corresponding qualitative interpretation (in Holistic Mathematics).

Thus Holistic Mathematics can provide a deeper qualitative interpretation as to what is meant both by a fractional dimension and a true irrational dimension (as in this case of the Koch snowflake).

Let us go back now to the conventional notion of dimensions. Remember they are dependent on clearly defined one-directional interpretations of movement leading to unambiguous finite interpretations.

The psychological corollary of this is the use of a rational one-directional logic geared to merely (reduced) finite interpretations of reality.

It is very important to understand properly the key dynamics (or rather lack of dynamics) of linear rational understanding.

In experiential terms we have the continual interaction of perceptions and concepts. Our notions of objects are directly rooted in perceptions while our notions of dimensions are directly rooted in concepts.

Linear understanding – by its very nature – requires that we freeze our interpretation of this dynamic interaction. Without this, definite knowledge of reality would not be possible.

So with linear interpretations concepts are fixed and perceptions are viewed against this static dimensional background.

For example if I take an example of a number e.g. "7", this represents a particular number perception which is viewed against the general background concept of number. Thus we can have one general number concept (serving as a qualitative mental background for multiple number perceptions).

Indeed the conventional notion that time is one-dimensional (i.e. one-directional) is directly rooted in this interpretation of concept. Time thereby serves as the general qualitative background in which we locate "three-dimensional" space objects. Indeed the very ability to form definite perceptions regarding scientific objects regards freezing the dynamic interactions (involving perceptions and concepts) to a marked degree.

Though it is moving ahead of our story the true dynamic interpretation of dimensions is provided by a qualitative interpretation of the complex number system. Here space and time are truly symmetrical and bi-directional (with positive and negative polarities). "Real" space is simply "imaginary" time, and "real" time is "imaginary" space. The well-known Mandelbrot Set (which serves as the basis for the generation of beautiful computer graphics) not surprisingly is based on the (quantitative) counterpart of this dynamic number system.

So quite simply in experience by fixing the qualitative concept as general and universal, we fix time as one-dimensional (and space as three-dimensional).

This enables us to deal with the merely quantitative aspects of holons in a reduced manner.

In conventional quantitative interpretations "lower" level parts are included in "higher" level wholes. However the reverse does not apply. "Higher" level wholes are not included in "lower" level parts.

Thus for example if I cut a cake into a number of slices, in terms of conventional interpretation, each slice (as "lower" part) is included in the "higher" whole (i.e. the cake). However the "higher" whole (i.e. cake) is not included in the "lower" part (i.e. each slice).

Thus by freezing the interaction as between the qualitative and quantitative (concepts and perceptions), we are able to carry out this one-directional quantitative interpretation. This in turn confirms our notions that time is one-dimensional (i.e. moving forward in a positive manner) and that spatial perceptions are three dimensional (with positive directions).

This limited interpretation of true psychological dynamics is the very basis for our notions of rational science. Here quantitative objects as parts are divisible to the nth degree. They are literally viewed qualitatively as rational numbers (i.e. as fractions of larger objects). However the qualitative dimensions are viewed as whole and indivisible.

Science is based on the understanding of nature. It is very interesting how this natural aspect is preserved in terms of our understanding of dimensions. We literally view them in terms of the most common natural numbers.

Our scientific rational understanding only extends therefore to (quantitative) objects.

Surprisingly it does not extend to our understanding of (qualitative) dimensions.

However from a correct dynamic perspective this scientific interpretation is invalid. As physical dimensions are holons, clearly they must have a part (as well as whole) interpretation. However this part interpretation is not reflected in scientific understanding.

The true experiential relationship of concepts and perceptions is bi-directional (not one-directional). It is intuitive as well as rational; it deals with infinite as well as finite reality.

Let me illustrate this point with relation to the interpretation of a number.

Now in rational terms a number perception (e.g. "7) is viewed as invariant with respect to its number concept. In other words the dynamic interaction of the number concept with its corresponding perceptions does not alter our quantitative interpretation of numbers. In other words we have here a reduced one-directional interpretation of number.

Now in dynamic terms number perceptions dynamically interact with number concepts (and number concepts with perceptions).

Thus the number "1" as perception has a unique relationship with the number concept (to which "1" is related). However equally the number concept has a unique relationship with the number perception "1". The relationship is thus bi-directional.

Likewise this two-way relationship applies to the numbers 2, 3 4 etc. and by extension to any number.

In conventional (quantitative) terms we have a one-directional interpretation of the relationship as between whole and part.

The parts are included in the whole; however the whole is not included in the parts.

Both wholes and parts are confused with merely quantitative interpretations of reality.

Thus going back to the cake example, if I divide a cake into 5 slices, each part slice (i.e. 1) is included in the whole (i.e. 5).

However the bi-directional interpretation is far more subtle.

From one perspective the parts are transcended and included in a "higher" level whole. This involves the movement from the quantitative appreciation of perceptions to the qualitative appreciation of corresponding concepts. Thus the individual (quantitative) numbers are transcended and included in the qualitative number concept.

However from the opposite reverse perspective the whole is made immanent and included in each "lower" level part. This involves moving from the qualitative appreciation of the general number concept back to the quantitative perception of a particular number. In this manner the qualitative number concept is made immanent and included in each quantitative number perception.

Thus in this bi-directional interpretation we can see clearly that the parts are included in a "higher" whole (transcendence). Equally however the whole is included in the "lower" parts (immanence). This is the qualitative counterpart to mathematical self-replication psychological terms. Now an alternative way of expressing the same relationship is that objects are included in dimensions (and dimensions in objects).

Thus both objects and dimensions have both part and whole aspects.

Thus in this dynamic sense there is a matching qualitative concept for each quantitative perception.

Thus we never step into the same river twice! When we vary the perception the corresponding concept likewise varies; equally when we vary the concept its (corresponding) perception also varies.

Now this bi-directional interpretation is directly rooted in intuitive rather than rational understanding. Not surprisingly therefore it is entirely missed in rational one-directional interpretations.

The implication of the bi-directional interpretation is that when we now attempt reduced rational translations two equally valid interpretations are possible.

Conventional science concentrates on the merely quantitative aspect of experience (the division of perceptions into multiple parts).

However there is an equally valid qualitative interpretation (the division of concepts into multiple parts).

Thus properly understood it is not only objects that are fragmented in rational experience. The dimensions themselves are equally fragmented. However this inevitably missed in a quantitative one-directional interpretation.

Thus we can give an exciting new holistic mathematical interpretation to the conventional structures of the rational linear level. Properly understood the (horizontal) concrete operational structures involve quantitative fragmentation of perceptions (i.e. rational objects) against a fixed conceptual background (dimensions). However the (vertical) formal operational structures - properly interpreted - represent qualitative fragmentation of concepts (i.e. rational dimensions) against a fixed perceptual background) objects).

The (diagonal) vision-logic stage then involves the two-way interaction of rational objects with rational dimensions.

However invariably in science formal recognition of the role of intuition is largely missing. Thus the qualitative aspect is reduced to the quantitative and intuitive dynamics are themselves greatly impeded.

Thus vision-logic which should lead to "higher" spiritual stages often fails in this role due to lack of sufficient intuition.

Now in holistic mathematics concepts (dimensions) and perceptions (objects) are horizontal and vertical with respect to each other.

Again, the two-way interaction of rational objects with rational dimensions (i.e. the vision-logic) stage is the natural gateway to the "higher" spiritual stages. In holistic mathematical terms these are precisely defined as irrational combining both intuitive and rational interpretations of experience, (which are paradoxical in terms of each other).

The very means of moving to these "higher" more intuitive stages is by beginning to appreciate the two-way complementarity at a rational level.

Now there is an amazing parallel in quantitative mathematical terms. Any (horizontal) rational number raised to a (vertical) rational power or dimension is irrational (except for trivial cases that reduce to integral solutions).

The complementary qualitative interpretation provides a precise description of the true vision-logic stage (in holistic mathematical terms). Recognising the dynamic complementarity of (rational) perceptual and conceptual understanding is the very means of moving on directly to "higher" spiritual (i.e. irrational) understanding.

However considerable confusion exists which is exemplified by Ken Wilber as to the precise significance of the vision-logic stage.

In a previous post "Content and Context", I demonstrated the nature of this confusion. Ken portrays psychological development in transcendent holarchical terms. However his approach to research is very much an example of the reverse immanent approach (where understanding of the "higher" whole is induced through knowledge of the "lower" parts). However because Ken in effect uses a one-directional approach, he is unable to reconcile this apparent inconsistency.

So let me end by tracing the obvious parallels as between the quantitative and qualitative interpretation of "fractals".

Self-similar fractals (i.e. wholes replicated in parts and parts replicated in wholes) lead to paradoxical situations where finite and infinite notions are inseparable.

Conventional notions of integral dimensions break down and are replaced by irrational dimensions.

Rational understanding is geared to finite understanding and intuitive understanding to infinite notions.

Self-similar spiritual understanding (where wholes are included in parts and parts included in wholes) leads to the development of irrational structures. These involve the explicit interaction of both rational and intuitive understanding (which are paradoxical in terms of each other).

Quite literally therefore conventional notions of dimensions break down to be replaced by irrational dimensions.

Quite simply the deeper meaning of mathematical "fractals" cannot be grasped in terms of conventional rational understanding. It can only be understood in terms of "higher" spiritual understanding (where we can give a directly complementary qualitative interpretation).