02.29.08
Posted in Everything/Nothing at 5:38 am by tempo502
Unfortunately, I don’t get many opportunities to update this website. However, just to be a tease, here’s the list of all my ideas for posts and unfinished drafts:
Sleep metaphor
The Strangest Weapon
The Right to Offend
Positive-sum Dichotomy
Anonymity – Abomination
Epiphany
Cyber Death
Convenience must always tail change
Agent/Object Dichotomy
Dramatom
Warrior Caste
No Argument Here
Negativity and Nothingness
Corporate Abolitionist
Punctuated Pageantry
Project
Culture of Competence
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02.08.08
Posted in Everything/Nothing at 12:41 am by tempo502
Sometimes, I don’t recognize my own thoughts.
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02.01.08
Posted in Insightful... at 3:38 am by tempo502
Symmetry – a geometrical or other regularity that is possessed by a mathematical object and is characterized by the operations that leave the object invariant: A circle has rotational symmetry and reflection symmetry.
A face is symmetrical. A vertical line down the center divides equivalent sides. The operation rendering it invariant is a reflection. A windmill is also symmetrical — one arm rotated will produce the full turbine. The crucial element, the defining essence of symmetry, is that an ordered self-similarity occur. Some part of the figure must equal another part through simple geometric transformations.
In each of these symmetries there is some critical dimension — a plane of reflection or an axis of rotation. The necessity for such a geometric entity is not immediately clear; drawing a line down a face can show that it is symmetrical, but does the identification of this imaginary plane serve an actual use? What is the use of identifying symmetry at all?
The only obvious use is fostering simplicity. In a practical sense, half a face is easier to draw, process, or store than a full face. On a very abstract level, a symmetric figure represents less information than an irregular figure. A windmill’s turbine can be considered one blade presented three times and instantly its complexity is reduced by a factor of three. A face is only half as much information as it appears to be, barring the odd beauty mark. Incredibly complex systems can be described with a very limited amount of information: a kaleidoscope is really little more than a central element and three mirrors.
So then, the use, the purpose of symmetry can be considered to be compression of abstract information. Things may be represented by less information than they appear to require, so long as those critical dimensions (plane of reflection, etc) can be identified. Some part of a symmetrical thing can be used to define the whole.
Is a fractal symmetrical? Fractals appear to be infinitely complex. Their defining characteristic, however, is that their parts resemble their whole, that a section under magnification has the same appearance as its parent. A tree limb has a very similar structure to the trunk from which it sprouts. A branch has a very similar structure to the limb on which it grows. Trees (and many other shapes in nature) are fractals, albeit imprecise ones. A mathematical fractal is self-similar under any level of magnification.
So a fractal is a complicated structure, but its shape can be described rather simply. Knowing one part is sufficient to (mathematically) define the whole. In that sense, the information has been compressed. That precisely fits the function of symmetry. A geometric operation (magnification) has replaced information.
What, then, is the critical dimension? There is no identifiable plane of reflection, line of rotation, or point of revolution. There is no euclidean entity capable of describing the type of symmetry exhibited by fractals. Their self-similarity is that of scale, not of space. A similar question might be, “when a thing expands or contracts, on what dimension is it being translated?” If anything, the nearest conceptual analogue is the distance of the observer from the figure. A figure can be made “larger” by taking a step forward, causing it to fill more of the field of view. This is the key.
A note, first, about these critical elements required for classical symmetry. A flat image might have a plane of reflection that cuts through the page, or a circle might have an axis of rotation stabbing through the page, or a sphere might have a point of revolution that is entirely excluded from the surface. In any case, this element must be “perpendicular” to the thing it reflects: a line of symmetry can only travel in a direction that the symmetry it describes isn’t. A vertical symmetry is described with a horizontal line, a rotational symmetry is defined by a straight line through the axis.
A fractal appears the same at any level of magnification. Magnification may be considered stretching a small image into a larger one, or it may be considered to be moving the observer closer to the image. Physical magnifying glasses operate by shifting focal points so that the object appears closer to the eye than it is, so this explanation of distance seems more apt than an appeal to stretching. A fractal is then self-similar on the line of the observer, in the same way a face is self-similar on the horizontal. A face has a vertical line of symmetry, perpendicular to the horizontal symmetry.
What is the critical element of symmetry for a fractal? What is perpendicular to the observer? More precisely, what is incapable of intersecting with the line of the observer, excluded from the observer’s line by definition? Unreality? The fourth dimension, time? Unobservedness? God?
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