18
Shall I compare thee to a summer's day?
Thou art more lovely and more temperate:
Rough winds do shake the darling buds of May,
And summer's lease hath all too short a date:
Sometime too hot the eye of heaven shines,
And often is his gold complexion dimmed,
And every fair from fair sometime declines,
By chance, or nature's changing course untrimmed:
But thy eternal summer shall not fade,
Nor lose possession of that fair thou ow'st,
Nor shall death brag thou wander'st in his shade,
When in eternal lines to time thou grow'st,
So long as men can breathe, or eyes can see,
So long lives this, and this gives life to thee.
130
My mistress' eyes are nothing like the sun;
Coral is far more red, than her lips red:
If snow be white, why then her breasts are dun;
If hairs be wires, black wires grow on her head.
I have seen roses damasked, red and white,
But no such roses see I in her cheeks;
And in some perfumes is there more delight
Than in the breath that from my mistress reeks.
I love to hear her speak, yet well I know
That music hath a far more pleasing sound:
I grant I never saw a goddess go,
My mistress, when she walks, treads on the ground:
And yet by heaven, I think my love as rare,
As any she belied with false compare.
“What is she like?” “She’s a tower of ivory; a day at the beach; warm as a handshake; yet cold as a sunflower.” The answers to the question, “What is she like?” have meaning only to the extent that there are connotations in the analogies that can be applied to the character of a woman. Is there any similarity at all (Sonnet 18), or anti-similarity (Sonnet 130)? To a contemporary westerner, “tower of ivory” has little meaning, unless the Song of Songs is familiar. “A day at the beach” might mean something different to a Southern Californian and an Eskimo. And, how could a sunflower be “cold”.
Successful communication occurs when the sender and receiver speak the same language – in which the message, even if novel, can be understood. There must be a pre-existent capacity for receiver to comprehend what was sent. And, this capacity must also include the ability to learn – to receive progressive more complicated and elaborate messages. In other words, for communication to occur there must be some similarity between the vocabulary and experience of the speaker and those of the hearer.
So, what is the meaning of similarity, when we find it in Nature? We certainly understand that organisms inherit their form from their parents. And, an ecological niche can be occupied by organisms of different lineages, which, nevertheless, develop analogous structures – for example, dorsal fins on sharks (fish) and dolphins (mammals), wings on insects and birds, and “wings” on bats and flying squirrels.
Such similarities can also be found in the inorganic realm. Crystals are a manifestation of the molecular structure of particular solid elements and compounds. Such similarities in crystals of different sizes generally involve distinct symmetries: cubic, orthogonal, tetrahedral, hexagonal… Table salt crystals (sodium chloride; in solid state the mineral halite) manifest cubic symmetry at virtually all scales; break a halite crystal and it fragments into smaller crystals bounded by surfaces which terminate at right angles with adjacent surfaces.
In recent years, another kind of self-similarity has been identified, of which crystals are a subset: fractals. In Mandelbrodt’s formulation, fractal objects appear to have the same structure over a wide range of scales. In addition to a number of mathematical algorithms that can produce fractals, a number of naturally occurring examples can be enumerated: clouds, earthquake occurrences, shorelines, gas-water contacts in natural gas reservoirs. The latter examples differ from crystals in that they do not possess inherent geometric symmetries. Rather, while structures or phenomena at various scales show similar variations in some quantity, they are not necessarily congruent when rescaled.
Certain kinds of fractals have a scaling dimension that quantifies the variation similarity. For example, consider a set of earthquakes that occurs in a particular earthquake-prone region of the earth, such as Southern California, over a prolonged period of time. If the numbers of earthquakes are grouped according to their magnitudes, a simple relationship is observed. Earthquakes of small magnitudes are much more frequent than those of large magnitudes (magnitude is related to the amount of energy released by the earthquake). In fact, a plot of the logarithm of the number of earthquakes of a range of magnitudes versus the magnitude range produces a straight line 9 fig. 1). The similarity comes in with the observation that the relationship is log-linear; that is the change in number of earthquakes magnitude versus magnitude is the same at low magnitudes and at high magnitudes. The slope of the line in the logarithmic plot is a measure of the fractal dimension of the earthquake magnitude distribution.
Plot of magnitude versus logarithm of frequency of earthquakes (binned in intervals of 0.25 magnitude units), southeastern California, 1980-2005 (data courtesy of Southern California Earthquake Center, University of Southern California).
Another natural example of fractality is the dimensionality of measured coastline lengths. If the coastline is measured coarsely, the total length is less than if measured finely. A plot of the logarithm of measured coastline length versus the relative fineness of measurement produces a straight line, whose slope is the fractal dimension.
Recently, there has been shown to be a relationship between fractals and the curious concept of entropy. Fractals can be shown to maximize mathematical entropy across multiple scales, constrained by the information content of the fractal. This correspondence has interesting implications for the interpretation of self-similarity in Nature, human creativity, and Revelation(!).
Next: Understanding Entropy
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