Self-Similarity ~ An Idea For “Knowing” and “Understanding”
September 9, 2009
In mathematics, a self-similar object is exactly or approximately similar to a part of itself i.e. the whole has the same shape as one or more of the parts. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.
Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually iterated and magnified infinitely without changing shape.
However, the Mandelbrot set is a symmetrical but scale-varient. As one zooms into it’s rough surface it changes shape in beautifully diverse ways and yet still has “memories” of the whole locked deep within itself. The M-Set is a common example of a type of affine fractal i.e. the images of the whole locked within are slightly altered via ‘scaling,’ ‘rotational’ or ‘shearing’ types of transformation.
On that note, I would like to present to you, the reader, an idea that is based on scientific observation:
Now… Keeping in mind what we have just seen… AND remembering that the above image is derived from only two fractal-like structures found in nature i.e. it could hardly be used as absolute conclusive evidence… I would like to pose a question to you, the reader. Are we using self-similar structures and processes of the “immense whole” to ponder and perceive the universe around us with? After all, the brain itself is a self referential system, that uses previous memories of experience to understand itself and the surrounding environment with…
“The question of the self has intrigued philosophers and psychologists for a long time. More recently, distinct concepts of self have also been suggested in neuroscience. However, the exact relationship between these concepts and neural processing across different brain regions remains unclear. This article reviews neuroimaging studies comparing neural correlates during processing of stimuli related to the self with those of non-self-referential stimuli. All studies revealed activation in the medial regions of our brains’ cortex during self-related stimuli. The activation in these so-called cortical midline structures (CMS) occurred across all functional domains (e.g., verbal, spatial, emotional, and facial). Cluster and factor analyses indicate functional specialization into ventral, dorsal, and posterior CMS remaining independent of domains. Taken together, our results suggest that self-referential processing is mediated by cortical midline structures. Since the CMS are densely and reciprocally connected to subcortical midline regions, we advocate an integrated cortical-subcortical midline system underlying human self. We conclude that self-referential processing in CMS constitutes the core of our self and is critical for elaborating experiential feelings of self, uniting several distinct concepts evident in current neuroscience.”
Northoff Georg; Heinzel Alexander; de Greck Moritz; Bermpohl Felix; Dobrowolny Henrik; Panksepp Jaak. “Self-referential processing in our brain–a meta-analysis of imaging studies on the self.”
So… Perhaps the idea of Grand Universal Structures being used to perceive Themselves with, is not as “outlandish” an idea as one might at first think… When one looks around in nature, we continually find self similar patterns/structures:
It has even been postulated that “many real world network systems obey a power-law scaling, just as if they were fractal shapes.”
Have you ever wonder why careful geologists always include a scale or reference when taking a picture of geologic interest? The reason is that, if they didn’t, the actual size or scale of the object pictured could not be determined. This is because most geoforms are self-similar, i.e. a fold 1 cm long looks quite the same as if it is 10 meters or 10 kilometers long. The same with most fault systems, layering, foliations, coastlines, topographic features, drainage patterns (like the one above), etc…
Self-similarity is a clue to the origin of many physical systems whose internal dynamics may be complex, in the sense that the system is at a critical state between chaos and order; a condition that has been called a self-organized critical state. A very readable account of Self-organized criticality is in the recent book by Per Bak, entitled “How Nature Works.”
For many years geologists have complained that classical math and physics are too simplistic in their representation of nature to be useful to the geologist. Now there are no more excuses for that kind of drivel. Fractal geometry, chaos theory and the science of complex systems do accurately represent many geosystems. Learning about fractals and chaos theory will considerably extend your understanding of geosystems and the workings of nature.
Even evolution is involved… Steve Jay Gould has this to say about fractals and evolution: “Finally, this pattern of long stasis, with change concentrated in rapid episodes that establish new equilibria, may be quite general at several scales of time and magnitude, forming a kind of fractal pattern in self-similarity” (Sc. American, Oct 1994).
And, as if to end this meditation on a high note… I would like to bring to the attention of the reader the following video, compiled by the late Sir Arthur C. Clarke, entitled “The Colors Of Infinity” which discusses the ideas of fractals in a novel light.
Again… I would like to bring to the reader’s attention my own intentions by writing this blog… It is not my aim to disclose a hidden meaning to life’s eternal flow. Neither is it to procure new scientific or religious standings. Nor is it my aim to put into disrepute current world views OR Religious ideals. Rather it is to ‘suggest’, using analogies recently disclosed through science, most of which have so far been reviewed within this Blog, new modes of possible understanding about ‘what’ We are and ‘why’ We came about in this Garden Of Eden that orbits around a bright furnace, one that, like all the others before it, forges all the known matter into what we are today.