Text: QUATERNIONS by Ben Iverson & Dale Pond Quaternions were developed in 1843 by W. R. Hamilton (1805-1865) in Dublin, Ireland. The background of these is rather hazy and is rather moot. These were a precursor of the actuality and reality of force and energy as manifested in the form of longitudinal, transverse and vortexian vibrations (Energy manifests rhythmically over Time). Quaternions are not an invention but parts of the procedure had to be invented in order for quaternions to be applied. The use of quaternions has a profound effect on science because:1) It recognises a greater scope in physics; and 2) they add recognition of the dynamism inherent in energy. However, in nature there is not the necessity of having imaginary numbers. Nature is real and hence it uses real quantities. Quaternions resemble to no small degree the well established arthimetical methods used in the study and development of music and sound (as music) dynamics. In fact one could almost say that what are known as computations of "summation and difference" tones between musical notes from the primary through the higher chords are in fact a subset of a quaternion-like math. These computations are a form of arithmetic/algebra hybridization. Putting together this 'virtual' relationship and making it workable is quite another thing. That is for putting it together into a workable mathematics for use in science. A large part of the backdrop is now in place but more explanation is needed. Everything in science and nature is quantum; i. e., discrete quantities. That is everything works and is numbered by whole numbers except (on first view) quaternions in themselves. They are the undercurrent of the whole number system or a "look behind the scenes". In addition to being in whole numbers, all values must be based within the system of prime numbers. This much is covered by the books on Quantum Arithmetic1. Essentially, a Quaternion is the square root of a prime number. They never actually appear individually in any calculation, but are combined in various combinations of them. The quaternion represents the latent unmanifested, unseen potential locked into a given triple vector system. Because this triple vector system is in reality a compound vibration this latent energy or potential may be quantified in terms of "polarization" or "degrees of polarization" (when dynamic) as opposed to a raw individualized and independent quantum of energy (when latent). John Keely referred to this undifferentiated triple vector complex as the Full Harmonic Chord of vibratory energies. The ancients referred to this as "white light" or undifferentiated (unrefracted) light from whence comes all degrees and forms of manifestation; i.e., Ra, Om, Yod, etc. The involved prime numbers are combined in various and sundry groups such that their products in each group is a whole number. This combination takes place until there are only four whole numbers, some of which are prime and some of them are composite numbers. These four whole numbers must then be able to fit into a Fibonacci type arrangement. They must also be formed of not more than seven prime numbers for all four whole numbers. This is easier to demonstrate than to describe. If one takes the combination of whole numbers to be 15, 7, 22, 29 for the Fibonacci configuration. The prime numbers here are 5, 7, 11, and 29, and the quaternions are the square roots of these six integers, plus the square root of 1, 2, and 3, to make the set of seven quaternions: Integers Square Roots Square Roots (Common) (Quantized) 1 1 1/1 2 1.41421... 14169/10019 3 1.73205... 18985/10961 5 2.23606... 7900/3533 7 2.64575... 10583/4000 11 3.31662... 13031/3929 29 5.38516... 19015/3531 Now, these seven prime numbers will always contain the square roots of 1, 2, and 3 in every set of quaternions. These are what sets the unit-of-measure for nature and creation. The other prime numbers, in their relationships of one to another, are what sets the parameters of the energy of the thing created. Going back to the set of four integers, they show the deficiency in Fibonacci's postulate. The configuration is that given by Euclid in Book VII Proposition 28, in what we call "sum and difference numbers". (But these were in use more than 2000 years before Euclid). Of the four numbers, in that configuration, the first and last are the extremes. The third is the mean, and the second is the variation as they apply to quaternions. Here we can begin to see the computation of "beat frequencies" or chords (whichever the case may be) as is done in musical tones. Having seen this musical connection to higher physics we can begin to understand the heavy emphasis John Keely placed on music throughout his entire system of vibratory physics. In the example, 15, 7, 22, 29 the 15 and 29 are the extremes. 22 is the mean, i. e., . 7 is the variation from the mean, i. e., 29-22 = 7 and 22-15 = 7. The quaternions will be expressed as: This product of square roots is then divided by 7. The value of n is the actual distance from the mean in integral units, from either extreme for the point being measured. If the extremes are designated b = 15 and a = 29; the mean is designated as d = 22, and the variation is designated e = 7, then the formula will be for y and x: y = ((ab)1/2 (d2e2-n2)1/2)/e x = nd/e In this formula, the quaternions are used but the exact value of them will be the square roots of a, b, d, & e. In essence, the formula will be the product of three square roots: a, b, & (d2e2-n2), or the product of three quaternions divided by the variation. Broken down further, each quaternion will be the square root of a prime number which is a factor of the product of the four larger numbers, a, b, d, & e. The only variable in the equation is the value of n. When n is equal to de, the result will be zero. Then n = 0, then the value of the equation will be at its maximum. The value of n will progress from de to zero and then back to de. In this case, de = 7 x 22 = 154 and d2e2 = 1542. The term d2e2-n2 as part of the value for "y" in the above formula for the scalar is what Hamilton called the quaternion. In his formula, "the square root of -1" is a reintroduction of our number which we call The One, or The Ubiquitous One.4 That "One" originiates in the higher dimensions and it is suspected that Hamilton put his math in and through the higher realms and had to then call these numbers imaginary numbers. Another major difference is Quantum Arithmetic and Sympathetic Vibratory Physics does not recognize nor use negative numbers in the conventional sense. All quantities are real amounts. In lieu of this we use a natural phenomena called polarity. In many instances this polarity may be referred to as male or female numbers or even as positive or negative. All discrete quantities have real significance but are generally phase matched or opposite in polarity just as no coin can exist without two opposite sides. Conventionally a negative number is generally considered as "less than 0" which is a misconception and can lead to tremendous illusions. In Hamilton's system the -1 caused the equation to rotate 90° and further discussions of "negative Time". In QA and SVP a change in polarity also causes a 90° rotation. So to derive a full 360° sweep there must be a positive/positive, a positive/negative, a negative/negative, and a negative/positive orientations which of course there are. This will make little sense to the experienced mathematician or scientist in contemporary terms. The background behind it will be explained in Quantum Arithmetic, Vol. 2, Bk. 2, pages 25-322 and also in Pythagoras and the Quantum World, Vol. 1-4. The numbers derived from the working of this equation will give, not a sine curve, but an equivalent elliptical curvature, which is the true formation created by the quaternions. This might be considered as the epi-sinusoidal curve. Each point so calculated will be a quantum point in which most of the values will be irrational numbers when reduced.3 However, at least two of the derived answers will be whole integers and they will be located at the latus rectum of the elliptical curve. These two equations are for calculating the coordinates of a point on a 2-dimensional surface. Three or more dimensions may be calculated individually and then superimposed on this 2-dimensional surface. What makes this new type of calculation possible is the replacement of invented calculus, along with other extraneous invented features of conventional mathematics, with an entirely natural Quantum Arithmetic. In this new method, there are no "imagnary" or "irrational" numbers and all answers are simple and entirely logical. Children love the simplicity of this arithmetic and geometry when exposed to it. The above description is a bit difficult to follow. A repeat of the description in a different frame will help to understand quaternions from a different angle. There is a change from the accepted information on quaternions, much like there is a definite change from the way Fibonacci described the parameters of Fibonacci numbers and the way in which Euclid described them. By these changes we can see that Quantum Arithmetic is evolving from older ideas into a more complete whole. In the case of quaternions, there are major changes. Quantum Arithmetic takes quaternions out of the field of the virtual and imaginary numbers, and their background is changed in many ways. But this new method is only the basic beginning of Quaternions. The approach to quaternions is through Quantum Arithmetic. The current contemporary texts describes quaternions as: "An invented associative, noncommutative algebra based on four independent units or basal elements." This is changed in two ways. In the first place it is now based on four interdependent variables. In the second place, it is changed from a non-commutative algebra to simple arithmetic. The four interdependent variables are dependent on any two of themselves, and the four will be in Fibonacci configuration. The adaptation of simple arithmetic precludes the necessity for use of calculus to obtain a resolution. This is the simple arithmetic of Quantum Arithmetic. The origin of quaternions lies in the series of elliptical equations for the Quantum Ellipse. So it is first necessary to define the difference between an empirical ellipse and a quantum ellipse. Only the quantum ellipse need be described. A Quantum Ellipse: In the Quantum Ellipse, the measurements are exact, and most of the measurements, 16 of them, are always measured in exact whole integers. For every empirical ellipse there is a comparative Quantum Ellipse. The measurements of this Quantum Ellipse will be within 0.001%5 of the comparative values for the empirical ellipse, but the unit-of-measure will be changed to quantum units in every case. Once the Quantum Ellipse is derived, the four numbers will be obtained. These four interdependent integers will be coprime6 between themselves, just as Euclid stated in Book VII, Proposition 28. These four integers will involve from five to seven prime numbers as their factors. The quaternions, per se, will be the square roots of these prime numbers. In the Quantum Ellipse there are four measurements along the major diameter. These four measurements are in the proportion of the four integers which generate their factors, and the square roots of the factors are the quaternions. The four measurements are the perigee and the apogee of the ellipse. These are the "extremes" or 15 as b and 29 as a in our example. The mean of these extremes is the semi-major diameter of the ellipse. It will be designated as d or 22. The mean will vary from the extremes by the fourth integer which is the "variation" and will be designated as e or 7 as given. These are the "basal measurements" which are termed above. To obtain the actual quantum measurements of these four segments, they will again be multiplied by d, to give the linear measurements along the major diameter. The actual measurement of the major diameter will then be: db = J = the perigee; da = K = the apogee; de = the variation; and d2 = D = the mean, or the semi-major diameter. ab = the semi-latus rectum; d2 + e2 = the vector from either focus to the opposite semi-latus rectum; d2 = D = the vector from either focus to the midpoint n of the ellipse There are two of each of these above the major diameter and two more of each below the major diameter. All sixteen measurements will be integers. This provides the backdrop of the quantum ellipse behind the quaternions. For the purpose of constructing a Quantum Ellipse on the basis of rectangular coordinates, the major diameter is divided into quantum segments. Each segment will be d/e in length, along the major diameter. There will be 2de such segments along the major diameter which is 2d2 in length, (d/e)(2de) = 2d2. These segments provide the x-axis for plotting by rectangular coordinates. This brings us to the origin of the quaternions which relate to the y-axis coordinates at each of the x-axis segments. The segments of the major diameter are numbered, (n), beginning with zero at the center and increase incrementally to the value of de at either end of the ellipse. This value is the amplitude of the wave. Then: y = ((ab)1/2(d2e2-n2)1/2)/e This gives the scalar value for the vertical vector (n). This combines the quaternions into a list which is handled like an ordinary grocery list. At the midpoint of the ellipse when n = 0 the value of the scalar is: so . This is the origin of quaternions in 2-dimensional calculations. In the case of other calculations which get into higher dimensions another calculation, using the quantum number of the next dimension is superimposed on this calculation giving the j, k values of the quaternions superimposed on the above calculations. These calculations will apply in cases of the lissajou, ellipsoid solids, etc. This is not the full story of the origin and usage of quaternions but it is a significant beginning. As stated earlier, quaternions are an invented system. The system which is given here is not invented. It is derived from natural mathematical functions and relationships, which the remainder of Quantum Arithmetic consists. In due time, others will derive the extension of this usage of quaternions, in which the whole idea of quaternions disappears, along with invented imaginary numbers and Gaussian integers. These were a step in the right direction and served their purpose for a time. One feature which may confuse many mathematicians is the necessity for conversion to quantum units-of-measure in order to utilize the equations. That conversion is a linear coefficient which is obtained in the conversion process. After quantum answers are derived this same conversion coefficient is again applied to return to conventional units-of-measure. In following Quantum Arithmetic procedures absolute answers are obtainable. 1 An interactive computer program exists for QA/SVP. For a complete list of the QA and other books request free catalog from Delta Spectrum Research, Inc. 5608 S. 107th E. Ave., Tulsa, OK 74146. 2 At the time of this writing this book is in the hands of the printer and will be available 12/93. 3 For purposes of creating harmonically balanced systems the answers of such equations must never be reduced or allowed to become irrational. All irrational numbers can be 'quantized' to a ratio of whole numbers to any desired degree of accuracy. 4 In Quantum Arithmetic and Sympathetic Vibration Physics there is a process we call "Quantizing to One". Basically this is a method whereby all values are set as relative to each other in a parametric fashion. This can be easily done by using algebraic type notation or even music symbols as representations of values and their relationships to each other. 5 Any degree of accuracy can be easily established in these equations. One of the seeming discrepancies shows up when a simple ratio is reduced to an irrational number. In these equations and most others developed within QA and SVP all ratios are used as they occur. These ratios are never reduced because a whole number is a discrete quantity (quantum) and an irrational number is not. 6 COPRIME: Co-Prime Numbers are two different numbers which are not divisible by any single prime number. Examples: 8 & 9; 15 & 22; 6 & 35. 7 Ben Iverson; Institute for Technically Applied Music; 11466 SW Royal Villa Dr., Tigard, OR 97224.

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