

Harmony of the SpheresThe planets of our solar system (as well as the satellites of these same planets and many of the other denizens of the deep space) have several unique mathematical relationships which are often ignored in astronomy textbooks. Such textbooks invariably include a discussion of Bode’s Law  a thoroughly discredited attempt to fit the distances from the sun to the planets into a coherent scheme. Primarily known as an excellent example of the use of Fudge’s Factor and Finagler’s Theorem, Bode’s Law is a linear based rule (as opposed to cyclical) and ignores the variable distances of each of the planets as they circle the sun in elliptical patterns  their actual distances varying significantly. And yet Bode’s Law is still part of astronomy’s tradition, while the really interesting stuff is ignored. A Book of Coincidence and the below tables provide, respectively, a geometric and algebraic treatment of the more “interesting stuff”. Table 1 considers the EarthMoon system, where for purposes of clarity, it should be noted that, for example, “7!” is known as “seven factorial” and equals 7 times 6 times 5 times 4 times 3 times 2 times 1. (This table is also included in Nines.]
*The period of time (years) for a complete revolution of the precession of the Earth’s axis. The units employed in the above are based on the English system of measurements. This is noteworthy, in that the more mainstream tendency to use the metric system will result in many of the interesting features being missed! Technically, units are arbitrary. That is, one defines a unit of distance (e.g. a yard) by what appears to be a totally arbitrary reason. However, once defined, the unit then is no longer arbitrary in measuring other dimensions. For example, miles, feet, and degrees can be considered to have been arbitrarily chosen. But once “a mile” is defined as being, for example, equal to 1/7920 th of the Earth’s diameter, then the unit of a mile is fixed thereafter for other measurements. This means that the product of the first seven numbers (seven factorial) equaling 5040  which turns out to be equal to the sum of the Earth’s radius (3960 miles) and the Moon’s radius (1080 miles)  is significant! In other words, in the second case we are no longer dealing with an arbitrary definition, having already defined the “mile” in the first case. [In Etymology, a “mile” supposedly stems from the Latin mille, “one thousand”, which once referred to 1,000 paces of the Roman legion’s formal parade step, left foot and then right foot, each pace equaling 5.2 feet. This of course, yields only 5,200 feet, instead of the 5,280 feet we now use. Perhaps, there was a hop, skip, and jump added every 1,000 feet.] More substantially, a review of Table 1 suggests that the English System of Measurements may be based on something not simply arbitrary, but on what could be construed as esoteric or other profound considerations. Furthermore, Table 1 suggests the EarthMoon system may be obeying some heretofore unknown mathematical law or set of laws! This is critically important! There may be “physical restraints” on planetary dimensions and orbits (just as in quantum physics where electrons can only take certain orbital positions), and thus Sacred Geometry and/or other mathematical disciplines (such as Numerology) may be critical to a complete understanding of the “harmony of the spheres”, or of astronomy in general. With respect to Table 1, it should be pointed out that there is a minor flattening of the Earth at the poles such that the radius of the Earth actually varies between 3964 and 3950 miles. The value of 3960 miles is, however, considered sufficiently accurate  particularly in light of the fact that the Earth has mountains four and five miles high, and thereby making any greater socalled accuracy, pointless in the extreme. In comparing other planetarysatellite systems within our solar system, modern astronomy has observed that the Moon is far larger than might be expected for a planet the size of the Earth. Our Moon is sufficiently large, for example, that its apparent size in the sky (based on its distance from the Earth) is virtually identical to that of the Sun (which is why we are able to observe total solar eclipses from the Earth’s surface). [Why do you suppose that is?] The lunar disc subtends within the sky an arc which varies from 29’ 22” to 33’31” (or 29.3666’ to 33.5166’) for an average of 31.4416’  roughly equivalent to that of the Sun. Another way of looking at it, is that the ratio of the Moon’s distance from Earth to the Moon’s diameter varies from 102.53 to 116.99 (average of 109.76), while the ratio of the Sun’s distance from Earth and the solar diameter (109 times that of Earth) is 107.68. This represents an error of from 0 to 8.65%, or an average of 1.9%. It has been pointed out that the distance of the Moon from the Earth (which varies from 221,460 to 252,700 miles) very nearly equals 60 times the radius of the Earth. The actual number is 60.27, which implies that the Moon is not precisely 60 times the radius of the earth by a “discrepancy” of some 1033 miles. Curiously, this discrepancy may derive from the fact that the Moon is currently receding from the Earth at a rate of a quarter of an inch per year. By extrapolating back in time (a very risky proposition, but one to which speculations are inclined to pursue), one can calculate that the Moon will have receded 1033 miles in 261,803,520 years. Thus the exact ratio of 60 might possibly have been realized some 250 millions years ago. Probably on a Thursday. This corresponds to approximately the time of the Permian Extinction (when some 95% of the species on Earth suddenly  suddenly on a geological time scale  became extinct). This also begins the Age of Reptiles  the Triassic being initiated 250 million years ago, the Jurassic, 155 million years ago, and the Cretaceous, 130 million years ago. This is, of course, highly speculative, but one can wonder if, perhaps, there is a connection with the Moon being at a particular distance from the Earth and the advent of the dinosaurs. This speculation might also lead to some even more speculative conclusions as to how the dinosaurs managed to stand  their apparent weight being too large for their legs. One farout view is that the force of gravity was less than now, and this may be due to the location of the Moon with respect to the earth. This seems unlikely, but it is just the sort of thing that makes the universe “stranger than we can imagine”. (Keep in mind also that there may have been a time in Earth’s history When the Earth was Moonless! Or even a reason why there were once, respectable Lunatics.) Things get even stranger when we also consider the Precession of the Earth’s axis as it rotates every 25,920 years  carving out a circle in the sky and periodically changing pole stars from Polaris to Vega to Alpha Draconis. The fact that 1/12th of 25,920 years equals 2160 years, while the Moon’s diameter is 2160 miles, is nothing short of, well... amazing! Furthermore, the idea that the size of the Moon is related to the Earth’s precession of the axes is simply not obvious from any socalled “laws” of mainstream physics. Considering all of these “anomalies”, one might begin to think the Moon was customized for the Earth. This statement doesn’t necessarily imply the Moon is artificial (but neither does it imply it is not), but rather that the process whereby the Moon and Earth came together as a unit, may somehow be obeying some higher authority  a law of physics not yet well understood, divine intervention, or some other even more incredible reason. The 11/3 Earth Moon ratio of diameters (and radii) is a case in point. For example, to a three decimal place accuracy, the MoonEarth ratio equals ÖF  1 [where F is the Golden Mean.] Or we can calculate: 11/3 @ 8  7f. This curious ratio is repeated in that the maximum distance between the planets Venus and Mars (the two planets which come the closest to the Earth), divided by the minimum distance between these same two planets, is equal (to within a 0.09% error) the same 11/3 value! This intriguing circumstance of planetary orbital characteristics having precise and similar mathematical characteristics prompts us to look further into the matter. The idea of a mathematical relationship between various planet’s orbits is, of course, not new  having been considered before in everything from Kepler’s “The Harmony of the Spheres” to the “TitusBode Law”. However, the flaws in Bode’s Law are legion. First, the formula which was developed before the outermost planets were discovered, failed miserably in accounting for the orbital distances of Neptune and Pluto. Two, the formula itself is heavily dependent up Finagler’s Theorem and Fudge’s Factor. Three, the concept of assigning a single number to a planet’s distance from the Sun may be a fundamental error. All of the planets, including Earth, have elliptical orbits with their distance from the Sun constantly varying. This ranges from the nearly circular orbit of Venus (whose distance varies from 66.7 to 67.6 million miles) to Pluto (whose elliptical orbit places it anywhere between 2766 and 4566 million miles from the Sun). However, a definitive measure of a planetary orbits can be obtained by discarding the linear thinking employed in Bode’s Law, and resorting to a cyclical mode of thought. Using orbits as our measure we begin dealing with the time it takes for the various planets to orbit the Sun, the orbital period, a very consistent measure. Table 2 provides several examples whereby the orbital periods of many of the planets are related by simple addition. Table 2 Mercury Venus Earth Mars Error 0.2409 years + 0.6152 years + 1.0000 years = 1.8561 years @ 1.8808 years 1.31% Earth Mars Jupiter Saturn 2 x [ 1.0000 year + 1.8808 year + 11.862 year ] = 29.4856 year @ 29.457 year 0.10% Mercury Venus Earth Mars Saturn Chiron Uranus 0.2409 + 0.6152 + 1.0000 + 1.8808 + 29.457 + 50.682 = 83.8759 @ 84.014 years 0.16% Ceres Saturn Chiron Uranus 4.6000 + 29.457 years + 50.682 years = 84.7390 years @ 84.014 years 0.86% Saturn Chiron Uranus Neptune 29.457 years + 50.682 years + 84.014 years = 164.153 years @ 164.79 years 0.39% Uranus Neptune Pluto 84.014 years + 164.79 years = 248.804 years @ 248.2498 years 0.22% Ceres Athena Juno Vesta Jupiter Saturn 4.60 y + 4.61 y + 4.36 y+ 3.63 y + 11.86 y = 29.062 y @ 29.457 year 1.34% Chiron Earth Pluto 5 x [ 50.682 years  1.000 years] = 249.31 years @ 248.2498 years 0.06% Note that while Chiron is currently considered to be a comet by mainstream astronomers, the fact remains that Chiron is in a planetary orbit (ranging between Saturn and Uranus), and is, therefore, justifiably included in the above formulas. At the same time, if only the Mercury + Venus + Earth = Mars and the Uranus + Neptune = Pluto equations were relevant, this would still be impressive and intriguing data. Also note that Ceres is by far the largest asteroid, accounting for more than 50% of the mass of the asteroid belt. Thus its inclusion is also relevant. The numbers shown in Table 2, however, are only the tip of the iceberg. A more complete set of relationships  those between planets of our solar system  are provided in Tables 3 through 6. Developed by the author and based in part on the extraordinary geometrical drawings and discussions in John Martineau’s classic volume, A Book of Coincidence [1], these equations demonstrate that not only do the planets correlate with each other via simple mathematical relationships, but they also do so with an uncanny and profound association with various forms of The Golden Mean. The Golden Mean or Golden Ratio is one of the most intriguing number in mathematics. It is commonly denoted by the Greek letter, phi, and is given in either to two forms by the equalities: F º 1.618033989... and f º 0.618033989... (where “...” means a continuation of the numbers  See Transcendental Numbers). The Golden Mean was known to the ancients (and moderns), who considered these numbers so sacred that monuments from the Giza Pyramids and Greek Parthenon to Notre Dame Cathedral and the United Nations Building in New York City have been based on these fundamentals of Sacred Geometry. Martineau [1] first pointed out the Golden Mean relationships between the outermost planets. The additional formulas of Tables 1 thru 6 were derived, in some cases, by converting Martineau’s geometries into algebraic expressions, and in other cases, by observation. In all cases, the percentage error is less than one percent. The existence of any percentage error, however, may involve the possible nature of the Transcendental Numbers (F, p and e), and their apparent requirement for slight inequalities in making nonlinear systems perform optimally. Table 3 F^{ }= f + 1 = 1.6180339887... Error 1 Pluto (aphelion) / Neptune (aphelion) = 1.6255 0.46% [Pluto (perihelion) / Neptune (perihelion) = 0.9993] 2 Pluto (perihelion) / Uranus (perihelion) = 1.6280 0.61% Neptune (perihelion) / Uranus (perihelion) = 1.6292 0.68% F^{2} = F + 1 = f + 2 = 2.6180339887... ( @ cos 36°/sin 18°) 3 Pluto (aphelion) / Chiron (aphelion) = 2.6112 0.26% 4 Chiron (mean) / Jupiter (mean) = 2.6331 0.57% 5 ½CeresEarth½^{max }/ ½CeresEarth½^{min} = 2.6072 0.41% 6 Jupiter (mean) / Earth (mean) = 5.2032 = 2 x 2.6016 0.63% [2 cos 30°]^{3} = 5.1962] [0.14%] 7 Earth (mean) / Mercury (mean) = 2.8540 1.32% Earth (diameter) / Mercury (diameter) = 2.6141 0.15% Note: Unless otherwise indicated, aphelion is a planet’s furthermost distance from the Sun, perihelion is a planet’s closest point to the Sun, and the mean is a planet’s average distance from the Sun. Also,½xy½ is the distance between planet x and y, and may be further defined as the maximum possible distance between the planets or the minimum. In all of the above, the resulting numbers represent the ratios of the two distances. This is incredible, but it is only the beginning of the truly astounding. As Martineau [1] observed, not only does Earth have a Golden Mean connection to the planet furthermost from Earth in the direction of the Sun, but Earth also has a similar connection to Saturn, the visible planet furthermost from the Earth in the direction away from the Sun. In this case (Table 5), Saturn’s mean orbital distance and physical size are both approximately 4F +3 times that of Earth. These observations, as shown by Martineau’s 5pointed and 30pointed stars, are important, and must not be dismissed as some random occurrence. [In fact, the title of John Martineau’s book, A Book of Coincidence, should not be construed to imply that the “coincidence” is to be defined as “a remarkable concurrence of events or circumstances without apparent causal connection”, but rather that the primary definition of “occurring or being together” more aptly describes the multitude of examples in the book. These socalled “coincidences” are thus examples of coinciding  possibly, either with a purpose, or resulting from some physical reason and/or constraint. Also note that Martineau has recently published a follow up to his original effort, the new book entitled A Little Book of Coincidence, Wooden Books, Wales, 2001 (in the USA, Walker Books, New York). Inasmuch as A Book of Coincidence is outofprint, the availability of the second volume is good news.] There are many curious mathematical relationships demonstrated in these tables, but for the less mathematically inclined, it might be wise to skip to the conclusions below, do a quick trip to Satellites of Jupiter and/or Hyperdimensional Physics, or go directly to A Book of Coincidence, for a more graphic, visual description. zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz In Table 3, it is important to note that equations 1 and 2 have in common, ten (5 x 2) circle geometries, while equations 3 and 5 result from nested 5sided pentagons, 4 and 7 result from a 5pointed star, and equation 6 results from two, nested 5pointed stars. The 5pointed stars and pentagons also inevitably involve the ratio of cos 36°/sin 18°, which is equivalent to F^{2} (within a percentage error of 0.0000014%). We might also observe that equation 7’s ratio of the Earth/Mercury orbits (i.e. 2.5840), when squared, equals 6.6769. This is equivalent to (within a 0.47% error): 3 (F + f) = 3 Ö5. Finally, we must not forget the intimate Golden Mean connection between F and/or f and Ö5. Finally, in Table 3, there is the relationship  illustrated in equation 6’s Jupiter/Earth ratio  between the nested 5pointed stars and the 6pointed, Star of David configuration. Using the latter, we derived the relationship of: [ 2 x cos 30° ]^{3 }(shown in Table 3). This effectively connects the 5 and 6 geometries by the equation: 2 F^{2} @ [ 2 x cos 30° ]^{3}. The latter is faintly reminiscent of Kepler’s Law of Periods, where the planet’s orbital period, T, is proportional to the planet’s mean distance to the sun, A, i.e. T^{2} = k A^{3}, where k is the constant of proportionality. Not shown in these Tables is the dodecahedron which relates Mercury and Earth  as well as Venus and Mars  and the icosahedron relationship between Earth and Mars. These three dimensional relationships are noteworthy because of the twelve, 5sided pentagonal faces of the dodecahedron, and the fact that the icosahedron, with its faces of equilateral triangles, is derived by taking lines from the adjacent centers of the dodecahedron faces, and is thus considered the dual of the dodecahedron. In Table 4, equations 8, 9, 10, and 11 are based on two nested pentagons, and equations 12 and 13 are based on four nested pentagons (as distinct from equations 3 and 5, which were based on five and three  i.e. an odd number of  nested pentagons, respectively),. Note also that in the VenusMars connections (equations 13 and 14), which includes both 3 and 4 nested pentagons, the difference is nothing more than a factor of cos 36°. Table 4 4 x f^{2} = 4 x (F  1)^{2} = 4 x (1  f) = 4 x (2  F) = 1.527864... Error 8 Mercury (aphelion) / Mercury (perihelion) = 1.5185 0.62% 9 Venus (mean) / Mercury (aphelion) = 1.5409 0.85% 10 Mars (mean) / Earth (mean) = 1.5241 0.25% 11 Ceres (perihelion) / Mars (aphelion) = 1.5306 0.18% 12 ½MercuryMars½^{max} / ½MercuryMars½^{min} = (1.5262)^{2} 0.21% 13 Mars (aphelion) / Venus (perihelion) = (1.5233)^{2} 0.61% 8 (F + f  2) = 8 (Ö5  2) = 1.8885... ( = 1.5278 / cos 36°) 14 Mars (perihelion) / Venus (aphelion) = 1.8954 0.36% 15 ½MarsEarth½^{min} / ½VenusEarth½^{min} = 1.8917 0.17% 16 Ceres (mean) / 2 x Earth (mean) = 1.8850 0.18% 17 Jupiter (mean) / Ceres (mean) = 1.8783 0.54% 18 2 x Uranus (mean) / ½UranusPluto½^{min} = 1.8943 0.31% 19 Pluto (mean) / Chiron (mean)  1 = 1.8788 0.51% In Table 5, the VenusEarth connection has multiple attributes (equations 20, 25, and 26). The first uses an enclosed 4sided square between the two orbits, while the second uses a 5pointed star/circleinscribing pentagon combination and the third a series of 8 circles. This has the effect of relating the Golden Mean 5 to the 4 and 8 geometries. Meanwhile, in equation 27, we have again encountered the 30pointed star which connects both Saturn and Earth’s distance from the sun and their physical sizes. Table 5 F^{4} = (1 + F)^{2} = 3 F + 2 = 3 f + 5 = 6.85451... Error 20 ½VenusEarth½^{max} / ½VenusEarth½^{min} = 6.8684 0.21% 21 ½JupiterEarth½^{mean} / ½MercuryEarth½^{mean } = 6.8515 0.04% 22 2 x Jupiter (mean) / Mars (mean) = 6.8281 0.38% F^{5} = 5 F + 3 = 5 f + 8 = 11.09017... 23 ½SaturnMars½^{max} / ½SunEarth½^{mean} = 11.06 0.27% 3  F = 2  f = 1 + f^{2} = 1.3820... 24 ½MarsSaturn½^{max} / ½MarsSaturn½^{min} = 1.3799 0.15% 25 Earth (mean) / Venus (mean) = 1.3826 0.05% 26 Earth (mean) / Venus (mean) = 1 + sin 22.5° = 1.3827 0.05% 4 F + 3 = 4 f + 7 = 9.472136... 27 Saturn (mean) / Earth (mean) = 9.539 0.070% [From a 30pointed star: 1/sin 6° = 9.5668 0.029%] Saturn (radius) / Earth (radius) = 37,449 mi/3,963 mi 0.024% Slowly but surely it should be more and more apparent that these “coincidences” can not be thought of as random events. Clearly, all of the planets (and an occasional comet) are profoundly connected via the Golden Mean, and in a sufficiently strong fashion that one must assume that physical forces are requiring some form of quantum limits to stable orbits. This latter point is extremely significant, and can not be overemphasized. A curious aspect of Tables 4 and 5, is that all of the planets are well accounted for except Saturn  which appears only in the more complicated relationships of Table 6. The need for the more unusual forms of F may call for some additional consideration  particularly inasmuch as Saturn dominates Table 6, where p and trigonometry are used. Equations 28, and 29 in Table 6, for example, have the added connotation of relating the circumference of a planet to another planet’s orbital radius or diameter. For example, the radius of Saturn’s orbit equals the circumference of Mars’ orbit, while the diameter of Neptune’s orbit equals the circumference of Saturn’s orbit. Another Saturnian oddity? Table 6 p = 3.14159265358979323846... Error 28 Saturn (mean) / Mars (mean) = 6.2592 = 2 x (3.1296) 0.38% 29 Neptune (mean) / Saturn (mean) = 3.1513 0.31% 10 ( p  3 ) = 1.4159... (similar to a Square: 2 sin 45° = 1.4142... or just Ö2) 30 ½JupiterSaturn½^{max} / 2 x Jupiter (mean) = 1.4159 0.00134% ! 31 ½JupiterNeptune½ / ½JupiterNeptune½ = 1.4183 0.17% 32 Earth (aphelion) / Venus (mean) = 1.4162 0.02% 3sided Triangle: 1 + cos 30° = 1.8660 33 Venus (mean) / Mercury (mean) = 1.8687 0.14% 7sided Polygon: 1 + 2 x sin 360°/14 = 1.8678... 34 Venus (mean) / Mercury (mean) = 1.8687 0.05% 9pointed Star: 1 / sin 10° = 5.7588... 35 Neptune (mean) / Jupiter (mean) = 5.7774 0.32% The unusual nature of Saturn’s relationships to the other planets is also observed in the Saturnian satellites and their respective distances from the mother planet  especially when compared to the Satellites of Jupiter. Finally, another 4F^{2} relationship which was not included in Table 5, concerns the tilt of the Earth’s axis! This wholly unrelated  or what has thus far passed scientific muster as wholly unrelated  physical characteristic of the Earth is extraordinary to say the least. The basic concept is that by projecting from the viewpoint of the pole, the Tropic of Cancer (or Capricorn) onto a plane located at the equator, one is then able to measure the radius, r, of the projection and compare it to the radius, R, of the equator. The ratio of the two is: R = r x 1.5278... = 4F^{2} r. The identical relationship holds for the larger projected circle of radius R = 4F^{2} R (of the equator). The Tropics of Cancer and Capricorn are, of course, the result of the tilt of the Earth’s axis of rotation with respect to the plane formed by the Earth’s orbiting the Sun. (The angle of this tilt is 23.45229°, and is the major factor in the Earth having seasons.) A Book of Coincidence shows this fact in a much more graphical fashion. zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz Conclusion (Skipping Stop Point) Clearly, the ratios of orbits of the planets and moons of our solar system depart slightly from precise equalities. This, in fact, may be a requirement or constraint of planetary quantum geometries, and directly related to Heisenberg’s Uncertainty Principle  which, in principle, limits the degree of accuracy for which an electron’s position and momentum can be simultaneously determined. In effect, without a recognition of the need for slight inequalities or minute departures from precise symmetries in the design of mechanical and/or electromechanical new energy systems, the devices may simply not work in the optimal condition envisioned. It’s as if without slight imperfections, there can be no interaction with other entities. [I.e., if one has a rotating perfectly smooth, perfectly spherical sphere, then it cannot mechanically interact with anything else. It’s like trying to change directions on a perfectly smooth surface, such as an icy surface. Skates work only because they mar the smooth surface with a nick, which is used to push off on.] It appears that the planets of our solar system, the moons of Jupiter, and other orbiting bodies have strong preferences for discrete distance and inclination windows  similar to the quantum physics requirement of electrons existing only in discrete energetic orbital levels within atoms. Accordingly, it might then well behoove the National Aeronautics and Space Administration, i.e. NASA (an acronym also for “Never A Straight Answer”) to consider that orbiting artificial satellites might undergo significantly less “degradation of their orbits”, if they are situated in orbital windows which are more conducive in allowing the satellites to stay in their assigned orbits for much longer periods of time. In other words, harmonize with the Harmony of the Spheres! Duh! For Quantum physicists, applying these principles of Sacred Geometry might elicit some additional understanding of the geometrical restrictions on everything from electrons in orbit around atoms to nucleons within a nucleus to internal spin characteristics of any and all elementary particles to Superdeformation of heavy nuclei. The latter may prove to be verrrrrrrrrrrry interesting Sacred Geometry Cycles Vibrations Music Forward to: Kepler A Book of Coincidence Satellites of Jupiter ___________________________ References: [1] Martineau, John, A Book of Coincidence, Wooden Books, Powys, Wales, 1995


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