"To see the world in a grain of sand"
All cultures prize proverbs, pithy sayings; aphorisms, parables; koans, sutras; slogans, slokas. . . . East and west, north and south, succinctness is savored.
Optics can shed light on the underlying mechanisms IMHO.
Sine/cosine "harmonic" waves in trigonometry have no beginning or end; they go on forever.?Waves in the real world on the other hand, are finite in duration.?As Fourier showed, however, finite waves can be synthesized by summing (like making a cocktail; a bit of this, a lot of that) several harmonic waves of different frequencies.?Not just finite waves; any form (a step function for example) can be expanded in Fourier series of harmonic waves.?The sharper the transition in the form to be fitted, the more terms in the series; or a broader variety of harmonic waves.
Fourier series are for fitting finite waves or functions that have some periodicity; that is, the function repeats, if not nicely like a sine wave, but all mixed up, like heartbeats.?Fourier transforms (with the summation of Fourier series replaced by an integral) extend the procedure to even more mixed up forms of no readily discernible repeats, like the binary bit stream that makes up this post, flitting through optical fibers to go from my desktop to your smartphone.?Again, if the shape (in the time domain) is discontinuous, its transform (in the frequency domain) will encompass a broad range of frequencies.?
The ultimate example is a narrow pulse or impulse (mathematically, a "Dirac Delta Function") -- with sharp transitions in both directions.?Its Fourier series or transform will comprise?an infinity of harmonic waves (as pictured in the schematic above). The two go together, as "conjugates." Either -- the pulse or the wave collage -- can be the starting point; "forward transform" and "inverse transform," or vice versa.
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Physicists use this as a takeoff point to fly high with their "particle wave duality."?Mix up a lot of waves, you get particle-like behavior.?OK, whatever.
A pithy proverb or aphorism packs in waves and waves of wisdom.?
That is, sages, prophets, inspired writers or orators concentrate a world of wisdom into memorable pulses. Conversely, it takes the disciples, fans, and critics pages and pages of commentary and gloss to unpack the pulses back into usable guidelines.
Note: This article was prompted by a rhapsody (by David Shulman; Israeli scholar of South Indian languages) on a new English translation of a classic Tamil book of wisdom in 7-word verses. https://bit.ly/3SS794t
Consulting Architect/CTO - Leadership in Enterprise Architecture and Software Engineering Innovation (US Army Veteran)
2 年The idea of waves has been on my mind this past week. As in, the waves of fads that wash over world of IT - some returning again, often from a slightly different angle - as if it were a reflected wave.
Senior Software Engineer
2 年It is an ideal, the proverbs. I used to be so annoyed by people around me using them too much, over a decade ago, that I added to my email signature the following "meta proverb" : "The set of all proverbs is closed under negation." (I can only guess how it was perceived ??). A few years later, I saw this: https://wadler.blogspot.com/2011/03/law-of-proverbs.html