(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, quadrature
Hence the Lunar Versor operator,
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,
And it is nearing the time
Where m is an unknown unit division in a grand cosmic cycle of
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,
(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,
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
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.
During the era following the church modes, the versor expressions of time became increasingly developed. This led to the “Age of Clocks”. Time versor operators became important parts of the music structure. The CANON is the most basic versor relation and it led to the more developed FUGE. Time versor algebra reached its greatest heights in the works of J.S. Bach, 1685 – 1750. In many ways Bach’s work is analogous to that of Steinmetz, and moreover J.S. Bach can verily be regarded the Tesla of the music world. Here is where to begin the understanding of versor algebra. A simple music keyboard, and a set of church modes is all that is required. Listening to the music of J.S. Bach replaces the reading of the equations of C.P. Steinmetz. In many ways the two are the same. As a musical reference is “Switched on Bach” performed by Carlos, in particular Bach’s “Brandenburg Concerto Number Three”.
(IV) As for the “Age of Clocks”, the versor positions of the clock face provide a most fundamental and intrinsically basic system for the study of the use of versor operators. The versor operators of music on the other hand provide a most complex but intrinsically heuristic system for the study of the theory of versor operators. For the engineer the clock represents a most important primordial advancement in the conception of versor algebra. It should be noted that the clock, the music keyboard, and the calendar, all serve as analog computers for the expression of versor position in time and/or space. The Grand Pipe Organs of Europe, some with more than 10,000 pipes, stand as the greatest analog computing systems ever to be created. Here engineering rose to heights incapable of being reached today.
The clock as an analog computer is a link to the cycles of the Earth, Moon, and Sun, in a manner of resembling that of the American Aboriginals. In the era of the clock, these analog versor systems developed into advanced analogs of the entire solar system, superclocks. These superclocks came to be known as “Planetariums”. In its fundamental expression the common clock involves the most basic cyclic functions. In its simplicity the divisions are equidistant and the hands remain unit length during progress thru the cycle. And importantly, it is divisible into four quadrants. Hence our rudimentary examination of versor operators begins with the ordinary clock face.
The basic versor operator for the clock, in hours, is given by the expression
Is the versor operator. This is to say there are 12 hours, or unit divisions, of one complete rotation on the face of the clock. All other degrees of rotation are fractional. Hence the unit division of one hour is represented by the symbol, k. This is the fundamental versor operator of the clock. The expression,
Is called “Noon”, fig (1). Hence for this versor operator, k, it is
(V) Let the versor operator
Be called Standard Time, fig (2).
Let the versor operator
Be called Daylight Time, Fig (3)
Hereby there exists two distinct time frames, let us say Pacific Standard Time, P.S.T. and Pacific Daylight Time, P.D.T. These two time frames now co-exist in the same space. In symbolic form let Standard Time be real, or h time, and let Daylight Time be imaginary, or j time. This is now in accord with A.C. theory, but with a twelfth root rather than a square root. The two versor operators, h and j, are called co-axial versors. This is to say, the h hand, and the j hand are on the same axis, fig (4). Here Daylight Time is a unit DISPLACEMENT on the face of the clock. Depending upon the time frame chosen there now exists the complication of two noons, fig 4a & 4b, but only one noon can be real. Noon by definition is that position in time when the Sun is at its highest position in the sky. This position is evidenced by the condition of a shadow being in its shortest length upon the Earth. This is the reference versor, or top dead center. In the imaginary time frame (P.D.T.) noon in that frame is one unit versor leading, or conversely real noon is one unit versor lagging. This can be expressed as,
a, real noon
b, imaginary noon
This is entirely analogous to A.C. theory. Hereby versor theory is now extended beyond its application by Steinmetz, and the versor operator j remains a transform between the real and imaginary co-ordinate systems. The time in the imaginary (P.D.T.) system must be multiplied by the versor operator j in order to transform to the real positions of noon and midnight. This in a way vindicates Steinmetz’s assertion that the most general algebraic number can always be expressed by
But this can only be true for the dimension of time.
In common language, those existing in the imaginary time frame of “Daylight Savings Time” are one over jOUT OF STEP with nature. This represents a 30 degree unit displacement on the face of the 12 hour clock. On a 24 hour cycle imaginary time is 15 degrees OUT OF STEP with the rotation of the Earth. In other words for those in imaginary time, this time is a place on Earth 15 degrees distant from the location of its natural occurrence. Your time is now in a place you are not. In A.C. Terms
(12) , percent
The Power Factor
(13) , percent
The Induction Factor
(14) unit percent
(VI) A common man conception of the versor concept is not complete without a related N.F.G. Historically, the works of J.S. Bach suffered the same fate as the works of Nikola Tesla. Shortly after the death of J.S. Bach (1750) all his important work was thrown into the garbage, this regarded as obsolete junk. If it was not for Herr Forkel, who recovered Bach’s work from the garbage and wrote the first book about Bach, all memory of Bach, and his work, would have been lost forever. Today, it is seen by some, that as with N. Tesla, the works of J.S. Bach is inimitable and profound. But it is of no value to modern society. The music of today has degenerated into the gutter sounds of “Hump N’ Slap,” and “Eubonic Barking Savages”, then the Boom….Boom Boom…….. Accordingly it is seen that the so called Science has followed a parallel path. This is known as “Quantum Mechanics”, the science of delusions. This “Science” in reality is no more than one big “Khazarian Circle Jerk”. And finally? Art! What an absolute joke. To criticize it is like slaying the slain. Today one can obtain a grant from the “Sheisenburg Foundation” to express one’s “artistic talent” by smearing human fecal matter onto a large canvas. Upon calling it “God” it will be worth millions. This is society today, COMPLETELY DEGENERATE.
So verily we are lost souls, hence the importance of reconstructing what has been lost to us. And it is to be noted that even Bach and Tesla themselves are only steps on a path, or as said by Heaviside “There is no finality.” But who is going to support this effort, or who is even interested at all? Rest assured that no support will be provided by our Babylonian society. This endeavor is more like becoming a Christian in Ancient Rome, a dangerous undertaking. See the movie “Fahrenheit 451” for an excellent portrayal of what awaits.
73 DE N6KPH