Thursday, December 31, 2009

Fractal Dimensions

One of the great contributions of Fractal Geometry is that it leads to consideration of the corresponding notion of fractal dimensions. For example this is beautifully illustrated with Koch's Snowflake.
See Mathworld.

So to construct this Snowflake we start with an equilateral triangle. Then marking each line into 3 equal divisions we take the middle third and erect another equilateral triangle on each side. Then we continue to proceed in the same manner (constructing a new equilateral triangle on the middle third of each exposed side).

Theoretically, we can continue in this manner an infinite number of times.

The implication of this is that the perimeter boundary of the Snowflake thereby increases without limit. In fact we can easily see that the initial construction of equilateral triangles on the middle third of the original 3 sides of the starting equilateral triangle increases the perimeter length by a factor of 4/3. Thus as we can keep repeating this procedure indefinitely (in each case increasing the perimeter by 4/3) the line can grown without limit even though the area of the entire figure clearly is of a finite magnitude.

In effect the perimeter line (through this process) encloses to a degree the overall 2-dimensional area.

This leads to a new notion of dimension for this line. To acquire it we basically find in this case what root of 4 (= the dimension) gives an answer of 3, the answer which is 1.261859507..

Strangely enough such a dimension (which necessarily falls between 1 and 2) is generally referred to as a fractional dimension.

However this conceals a very important conceptual difficulty.

We commonly think of fractions as rational numbers. However such a dimension (as in this example) will always be of an irrational nature (which thereby cannot be represented as a rational number).

Indeed the only way to get a rational dimension is to attempt to divide the line with irrational number sections!

This is all very interesting from a qualitative perspective.

In proper qualitative terms, finite notions are associated with (linear) reason and infinite notions with (circular) intuition respectively.

Now the irrational notion relates therefore to a situation where the infinite in some ways becomes embedded in the finite.

So corresponding to irrational understanding in qualitative terms is the derivation of an irrational number dimension (reflecting the paradoxical interplay of finite and infinite notions).

The clear implication of all this is that a corresponding qualitative interpretation attaches to the mathematical notion of the irrational.

In psychological terms this becomes equated with the transition from 1 to 2-dimensional understanding. Initially before two clear polar directions of phenomenal understanding crystallise, one goes through a linear phase where rational understanding increasingly interpenetrates with intuitive understanding (with finite phenomena now in some way genuinely embodying spiritual infinite notions).

This also provides an answer to the clarification of dimensions. So now we can look at irrational dimensions as basically representing transition states as between the whole number dimensions. Basically each new higher dimension (as whole number) represents a new more refined form of rational appreciation (with the corresponding number of directions). Therefore the irrational dimensions in between represent various states (whereby reason and intuition are combined in a - as yet - somewhat confused manner).

Once again in quantitative terms a "fractal dimension" between 1 and 2 represents the extent to which the 1-dimensional line can fill in a 2-dimensional area.

In corresponding qualitative terms, a "fractal dimension" (that is indeed of an irrational nature) measures the extent to which one can embody as it were 2-dimensional appreciation coming from a 1-dimensional perspective. This can never be achieved fully. However it can be approximated more closely through a progressively greater degree of intuitive understanding informing rational interpretation.

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