(I) What of this thing called versors, and why is it that we should be interested in them? In the most basic conception a versor operator is a means to “move about” in a given dimension or dimensional relation. In an extended conception a versor operator is a means of moving from one dimension into another dimension. In the most general conception versor algebra is the algebra of position.
In nature there exists the four polar positions of our moon. This gives four unit versor positions,
, new, conjunction
, leading, quadrature
, full, opposition
, lagging, quadratureHence the Lunar Versor operator,
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These unit versor positions occur when at a given geographic location the following exact relations exist;
Full, moonrise at sunset
Lead, moonrise at noon
New, moonrise at sunrise
Lag, moonrise at midnight
This leads to a versor algebraic expression. Hence a natural set of versor relations in quadrapolar form. It should be noted that Nikola Tesla’s original archetypal vision of alternate electric waves was his poetic versor rotation of the Sun-Earth relation. Needless to say versor algebra finds extensive application in alternating current theory. Read first part “Symmetrical Components” by Wagner and “Power and Double Frequency Quantities” by Steinmetz, (from his A.C. book), for advanced concepts.
A most important application of versor algebra is the study of complex electric waves in space. There has been no progress beyond the cumbersome and under-developed quaternions of J.C. Maxwell, nor the transverse electromagnetic vector algebra of Oliver Heaviside. This is an important task. Such an electrical condition is found in the propagation of electric waves within transformer windings. The networks of Nikola Tesla and Ernst Alexanderson follow from an analysis of complex electric waves in the common transformer. However this knowledge evades us. It is that the present state of versor algebra has not developed out of its infant form, the square root of negative one. It’s engineering application was founded by C.P. Steinmetz (1898 A.D.) and he also suggested higher order versors as given in his A.C. book, “Roots of The Unit”, and etc. Steinmetz however never applied these to engineering even though their possible application was evident. He later on became rather “stuck up” on his own established work, this a side effect of G.E. using Steinmetz to promote their own image. A more generalized conception of versor algebra was attempted by Alexanderson McFarlane but this also never developed into any engineering applications. This is where versor algebra sits today.
(II) The concept of a versor algebra is not new, nor is it limited to electro-dynamics. One of the oldest versor systems is the symbolic representations of the Aboriginals (Indians) of the American Continent. These systems found advanced development in the cultures of Central America. Noteworthy here is the “Mayan Calendar”. In versor symbols this calendar states,
For
unit,And it is nearing the time
unit,Where m is an unknown unit division in a grand cosmic cycle of
years.This in common language, equation one states “Happy Grand New Year”. A versor axis, here given as
, raised to a zero power, is like midnight, or the new moon, it is the start of a “new day”.In mechanics there exists the condition called “top dead center”. In electrics there exists the condition called “unity power factor”. In both cases it is a vertical reference, zero degrees, pointing upward. This is expressed by “high noon” on the face of the clock. This hereby establishes the position of a reference versor,
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unit(III) Another versor system of antiquity is that developed by Pythagoras of Ancient Greece (570 - 495 B.C.) This system is called “Music”. It is based upon the HARMONIC SUBDIVSIONS of an oscillating string, two to one, two to three, four to five, and etc. Each subdivision represents a versor position. This developed into a base seven versor,
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It is however than n is a fractional and sometimes complex quantity. Therefore music represents a most complex versor system. This versor algebra, or music, found slow development until the era leading to Martin Luther, 1400 A.D. – 1500 A.D. The foremost development of this era is what are known as the “Church Modes”
It is of interest to note that one Church Mode in particular has a hysteresis loop. The rising scale takes a different path of notes than that path of notes for a descending scale. Here exists a musical analog of a hysteresis cycle. M. Luther (1500 A.D.) published his archetypal musical series using the Church Modes. Of these, his “Ein Fest Burg”, is well known in church music today, it is the “standard”.
After 1600 A.D. music found considerable advancement in Italy, mostly thru the efforts of the Italian Priest, Antonio Vivaldi. (1678 – 1741 A.D.) From here forward the use of the church modes was confined to only two out of the complete group. Today these are known as the MAJOR scale, and as the MINOR scale. Transformation between frames of reference known as “keys” (the frequency bands) was not possible until the system of “Equal Temperament” by J.S. Bach (1700 A.D.). Bach’s versor system was based upon the expression
percent.This gave one unified versor system of divisions called the chromatic scale. This twelfth root relation is the basis for the diatonic scale of today. It is however that Equal Temperament is at odds with the original percentages, or ratios, of the Pythagorean System. This remains as an unsolved condition in the versor algebra called music.
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