Similarity and dissimilarity are at the heart of human communication. Tell a story; either your listeners are hooked into another story heard (or read) before, or they are not “hearing” you. Your story is either boring, repellant, or riveting, depending on with what it connects (in the memory of the hearer), and the extent to which it then provides something new.
Oddly enough, we’re beginning to recognize that similarity/dissimilarity can characterize the processes operable in much of the rest of the physical universe, also in the context of what could be termed “communication.” That is, the existence of similarity is more than a matter of conscious recognition by sentient beings; similarity is part existence is manifested.
Consider messages sent and received; the senders must first construct the messages in such a way that they themselves understand. Then, whatever the medium of transmission, receivers must “hear” the same language (in all of its dimensions) as the sender, in order to fully comprehend the message. In a sense, then, there needs to be self-similarity in message construction and cross-similarity (or complementarity) in message comprehension.
Similarity at the same scale, or within the same domain, has been implicit in human understanding for centuries. That is, we can speak of the similarities of human beings, person to person, within the same sex and between the sexes. Where there are dissimilarities, especially between the sexes, those dissimilarities may often be seen as complementary. Most obviously, propagation of the human species requires complementary organs to conceive new life.
Mathematics makes explicit use of symbols and equations. When applied to natural observations, such symbols acquire scientific significance. Equations describe by analogy natural phenomena. The symbols within the equation represent the phenomena in terms of its dimensionality and parameterization, and relations of the parameters to one another. In some cases, a single symbol encapsulates more than one level of similarity. Thus “=”, the equality symbol, implies equivalence (similarity in every respect). “~”, the proportional symbol, indicates linear self-similarity.
Consider the previous paragraph -- and this one. They are loaded with “similarity.” Letters and spaces are conventional symbols. They define words which in turn symbolize meaning. Some words have meaning that incorporates some aspect of similarity: explicit, symbols, equations, observations, significance, analogy, dimensionality, parameterization, relations, encapsulates, level, implies, proportional, linear, words, meaning. However, each word differs in meaning, sometimes depending on context, explicating the kind and dimension of implied similarity.
Within life, speciation is partially recognized on the basis of similarity and complementarity, especially the ability of individuals of opposite sex to mate and propagate (within those orders which reproduce sexually). Further, the recognition of evolutionary lineages requires the delineation of similarities and progressive or abrupt development of dissimilarities (from ancestor to descendants and between descendants).
The presence of similarity is critical in electronics. Circuits, with their transistors, capacitors, and resistors, are designed for particular kinds of signals. They are, in effect, appropriately scaled to create, propagate, retransmit, and interpret specific kinds of signals. Such signals may include text, speech, or music, which are intrinsically self-similar, in varying degrees.
Consider music in general and baroque music in particular. The western musical scale is explicitly fractal. The frequency of each note of the scale is one-half the frequency of the equivalent note one octave higher. Thus Middle-C has a frequency of 264 Hz (cycle per second). High-C has a frequency of 532 Hz. Playing a melody centered on Middle-C, then shifting it one octave higher produces similarity across the two octaves.
Manufacturing devices are designed for the scale of the particular product generated. There are implicit similarities in manufacturing that approach near complementary identity. The mold produces the cast. Two-dimensional perforations produce three-dimensional spaghetti of the same diameter as the perforations.
Geologists engaged in study of the earth’s surface exploit similarity as they map sedimentary rock units. Layers of sandstone, shale, and limestone are correlated from outcrop to outcrop based on similarities in rock composition, texture, fabric, fossil content, and/or stratal thickness variations, as well as position in vertical sequence. Isotopic age dating and measurement of the magnetic polarity of the rocks can further facilitate the process. Oil geologists make use of similarities in electrical, sonic, and radioactive properties of rocks penetrated by the drill (supplementing rock cuttings and cores) from borehole to borehole, to correlate potential hydrocarbon reservoirs, seals, and sources. Geophysicists make use of similarities in measured acoustic properties of rock via seismic surveys, to provide nearly continuous characterization of the rocks of the subsurface, prior to the drilling of exploratory or development boreholes.
In the late 1970s Mandelbrot introduced “fractals”, mathematical objects that possess self-similarity, not only at one scale, but at many (theoretically, undefined) different scales. After exploring a number of such mathematical objects, such as the Julia set, Mandelbrot showed how a variety of natural phenomena demonstrate scale-independent self-similarity. Primary examples are coastlines, plants, clouds, and earthquakes. Even stock price variations seem to show self-similarity.
(12/27/2004)
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