HyperFlight Multi-Pointed Stars In the Heavens		"..but only some numbers can play together"
Five pointed star revelations and revelries from the solar neighborhood Two pointed star of Neptune-Pluto joins the five point star — there is Catseye looking at you, kid Eight pointed star chimes in — the eight sided octagon is musical, you know. The eight point star also has a diagonal and that may be about something else. Little discussed meaning of the diagonal gets into the Tao of the East (paisley) symbol, the most wonderful Hanub-Ku of Mesoamerica, and Venus from the land of Sumer
There are many stars and you can draw them with the number of points of your choosing. The first star has five points and it certainly is in the heavens if heavens is what you see above when you look up at night. The shape of this five point star is crafted together by Venus and Earth.
		Mayan stargazer. For better or for worse, Mayan Venus horoscope dealt with propitious times to wage war
Mayan astronomers first observed, measured, and applied the periodic relationship between the planet Venus and Earth. It turns out these two bodies have interlocked orbits in a ratio and that means that the planetary orbits of Venus and Earth are not independent of each other.
The ratio is 8/5 -- for every eight Venus orbits (cycles), our Earth completes five orbits (years). While this is not a source of orbit problems, the orbit interlock allows Venus and Earth to become effectively a single planet in one orbit. The resulting merged planet is moving close to the midpoint between Venus and Earth because both of these planets are of about the same size (have about the same mass). From the Sun's point of view the positions of both Venus and Earth periodically occupy the same two spots on the solar plane and the merged planet is then also in a spot it occupied before. Every time Venus catches up with Earth the merged planet reaches its aphelion, the farthest distance from the center of the Sun. For such merged planet the time between aphelions happens every 1.6 earth years. The interlock of the two orbits merges both planets yielding one highly elliptical orbit that rotates and repeats every eight years -- that is, the merged planet passes the same starting point every eight years while its path is curved. During these eight years the merged planet reaches aphelion five times, tracing a star with five points and taking 1.6 years between points. Historically, the aphelion is defined as the farthest distance from the Sun but you may have noticed that the aphelion for the merged Venus-Earth planet is now defined as the farthest distance from the center of the Sun. The merged planet's orbit may possibly dip into the Sun. You can imagine the "crisis" in the horoscope circles as Galileo's and Newton telescopes begin to uncover new planets. There were seven planetary influences and seven is a sacred number. Horoscope casting is a serious business for some, so what is to be done? Presently, several mainstream science publications want to make Pluto a planetary outcast but nobody is arguing whether Venus or Earth is a planet or not. We could never think of Earth as not being a bona fide planet -- yet the Sun has made the orbital decision for us. Sun sees Venus and Earth as one body that orbits in a five pointed star topology with a tight curlicue approaching the Sun between points. Perhaps there is a way to resolve all of this because Pluto's orbit is interlocked with Neptune as well. So now, planetary pairs merge computationally and each pair forms a unique orbit. While it may be difficult to prove that seven is the maximum number of independent orbits in our solar system, what this also means is that there could be more planets than there are independent orbits.		Planet is a planet because it contributes to the solar system's angular momentum through its own orbital (angular) velocity. Angular mo is the rotation energy (of a spinning bullet, say) that contributes to the solar system's stability. Because the angular momentum is proportional to the radius squared, the angular mo of Pluto, for example, exceeds those of the terrestrial planets, Earth included. For a treatment of the angular momentum on both the cosmic and atomic (electron) scales, visit ConceptThree. If you want to construct and draw the perfect five pointed stars of the pentagon, pentacle, and pentagram -- and learn of other stars from the perfect star family -- make a visit to the pentagon page where you will do it with the queen of math: geometry. Also construct the three and five sided stars directly and exactly. There are direct and indirect families of stars and some stars cannot be exact New book you will thoroughly enjoy QUANTUM PYTHAGOREANS More ..
Why not give each planet its own independent orbit? There may come a point where this is not possible. Any two-body system is tractable but a three-body system can become chaotic and intractable. An independent orbit calls for planetary separation that is great enough for each planet to be treated as a two-body sun-planet system. As big as Sun's mass is, there are but a limited number of orbits available. Orbit interlock allows merging of Venus and Earth such that the two bodies become one and the resulting body is always tractable with respect to Sun. A potential Sun-Venus-Earth chaotic trio becomes a computable duo. Giving Venus and Earth their own independent orbit could technically result in two two-body systems but would likely require greater planetary separation and greater distance from the Sun. Also, the orbital interlock of Venus-Earth, say, could results in a smaller disturbance or wobble onto the Sun: Through the orbit interlock both bodies can be physically closer to the Sun and to each other. Now, take the Venus and Earth planets as orbiting in the clockwise direction and start both planets at the top. The merged planet makes its first aphelion also at the top, which is the first aphelion (point). As planets orbit, the merged planet skips every two points of a five pointed star because the the next time Venus and Earth are closest together is at the fourth point. The trace of aphelions moves from first (top) to fourth to second to fifth to third and back to the first point. The picture of the merged planet trace is at the bottom of this page. [You may find a bit more in the picture than just a trace -- give it a slight tilt.] Being in the solar plane the star’s orientation does not have an appreciable social aspect, at least as far as the solar perspective is concerned. The five pointed star of the pentagram is intimately related to Venus and Earth.		Tractability is a technical yet exceptionally important property. When a system is tractable, it means that a mathematical solution exists that happens in real-time. Scientists routinely ignore tractability and offer "solutions" that reach the goal in, say, million years. None such solutions match reality. Scientific theories of evolution have two Achilles' heels: (R) Ignorance regarding the existence of the virtual energy. Scientists think that light is real but light is virtual and, therefore, there exists no physical pressure when light reflects from a mirror; and (L) Claiming that intractable solutions are representative of nature. The use of intractable descriptions (even intractable proofs) are not relevant and do not happen in nature. Addressing the 'traveling salesman' problem with intractable methods does not and will not work because a dramatic computer speedup does not help -- in fact, that is how intractability is defined. Getting to another solar system by traversing every inch of distance is intractable as well. [Scientists cannot go there and remain earth-bound.] By advocating and by allowing intractability in the explanation of reality, the present science cannot deliver tractable -- that is workable, solutions. Overall, the present and taxpayer supported science is truly a pseudoscience in its very definition of considering but a subset of facts and also by espousing irrelevant -- that is, intractable and useless, approaches.
Five pointed star: Pentacle with two rings -- that is, orbits		Draw the star in the air, for that is how the Sun feels the dynamics of Venus and Earth. Chart the star in your heart and make a friend.
Symbolically and fairly close to reality, all points of the star are at midpoint between two circles that are the orbits of Earth and Venus. Dot in the center is the Sun. It is said Pythagoreans first used a circle around the pentagram (making a Pentacle) but you want to use two. The double circle is paramount because the star owns its existence to two interlocked orbits. To add mystique, you can include alchemical symbols for the sun and the two planets. Also add a curlicue that may look like a lower case scripted 'e' for gals (Eves) or the Greek alpha for guys (Adams). But of course you can call it pentalpha or pentagram or make it into a pentacle. If you are into numbers then 5 and 8 and their ratio 8/5 (1.6) are relevant. When the ratio is slightly outside of 1.6, the star slowly rotates in the solar plane.		3rd-4th century BC. Circled five point star coin or a talisman.
Radius, radii. A question may come up as to the actual radii associated with rings that represent the interlocked pentacle-producing orbits. We know the orbital period of Venus and Earth, but what is the corresponding orbit radius? If Johannes Kepler were alive today he would likely have 'Dear Johan' column in the daily paper, much the same way Kepler worked in his time when he was the editor of an almanac making horoscopes and forecasts (socio-political forecasts at that). As it happens, Kepler is the author of the very equation we need for our calculation: T2=kR3. T is the orbit period (Time) and R is the radius. We do not need to know k (constant that deals with mass) because we are interested in the ratio of the outer ring (orbit) radius to the inner ring radius. The ring multiplier comes out of calculations and after selecting the outer ring's radius we obtain the inner ring radius by multiplying with 0.731. If the outer ring's radius (or diameter) is 11 feet then the inner ring's radius (diameter) is 8 feet. If you select the inner ring first then divide by 0.731 to get the outer ring's measure. If the inner ring is 9 feet then the outer ring is 12' 4". No matter what the mass ratio of the two planets may be, the pentagram will always have its points between the rings (orbits). The math in this geometry also means that these two rings are valid for planetary bodies of any size.		For perfect pentagram solution: 52 = kR3 (inner faster orbit) 82 = kR3 (outer slower orbit) (5/8)2/3 = Ratio of radii = 0.731 This result is in effect the arithmetic solution to 'doubling of the cube' of ancient Greeks, but it is easier to call it the '3D to 2D dimension step up' on the Pythagorean Tetractys. You might see this an excellent example of the benefit of rationing. By rationing the orbits we can draw orbits in accurate proportion and need not be concerned about bodies' mass. Mathematically, orbit rationing yields results that are mass-normalized. An explanation of proportioning is in the article on the golden proportion.
Pentagram's root properties. Pentagram is generally associated with protection, friendship and health. Pentagram's root is about a two-body orbit interlock (friendship) where the protection and health aspect is the ability to withstand chaotic environment. Similar associations lead to the acts of bonding and marriage. Pentagram's protection and interlocking qualities would also make a great seal. Pentagram is in the solar plane and each point is the closest Venus-Earth approach: Pentagram or Pentacle orientation cannot corrupt its symbolism. The two-dimensional aspect of the star also makes the pentagram well suited as a shield, which is generally any flat area or partition such as a curtain, door, cover, sticker, seal, table, plate, coin, or a window shutter. Perhaps the most misunderstood aspect is that the pentagram or pentacle may appear as "just an identification symbol." Pentagram, however, is a working tool and produces a strong reaction. In a somewhat crude analogy, a blade is sheathed as it is a serious tool. There are no easy or simple explanations for the pentacle, for its workings are applicable to both the cosmic and the atomic topologies. More you understand it, on the scholarly or the practitioner level, more respect you will have for its properties. If you flaunt it, you likely do not understand it and nothing will come of it.		The protection aspect of the pentacle and pentagon cannot be easily dismissed. Atomic stability issues from waves that can close around the core exactly. The waves, then, exist in exact integer multiples and give atoms their stability -- and this includes the number five. If you are familiar with Raymond Rife work, the atomic (and hence molecular) stability may just be what is needed to keep viruses from morphing and becoming cancer-causing. We give Raymond credit when discussing the squaring of a circle. Pythagorean pentacle has a single ring but then the pentacle is made into a (second?) ring of a piece of jewelry. For some stars, such as the 7 point star, the exact division inside a circle is not possible. Presently, Kepler is mostly remembered for his discovery of planetary orbits that he was able to describe mathematically as ellipses. Yet Kepler's equations are also used for any and all satellites being launched into orbit -- and 'satellite' is Kepler's word, too. We have a book review, Kepler's Witch in our archive. The rotation and revolution are closely related but it seems only in the last hundred years or so would the five pointed star become associated with revolutionary ideas. Without elections the revolutionaries then work hard for no more revolutions.
Which way is up? There appears to be some tugging and pulling regarding the symbolism of the pentagram, particularly when considering, "which way is the right way?" or "what is the meaning of the upside down star?" From the solar perspective this has no meaning, for the sun is in the center of the pentagram (and pentagon as well) and there is no up or down. The pentacle, pentagram and the five pointed star are powerful symbols. Many groups wish to claim them as their own. Up, down, or slightly tilted (Boston specialty). With circles, rays, flame, letters. Yet, with the array of attributes, the star cannot be owned by some and cannot be owned by all.		What does it mean when the star is drawn point up or down? Upside down? Isn't two points up better than one point up? Are both of your ears pointing down? The complication here is that the 'up' and 'down' designation arises strictly from the actual and physical movement. There is no net up or down momentum in the universe as a whole and there is no net spin. Once an object starts to move, however, the up, down, clockwise, or counterclockwise directions begin to differentiate on a local level. Because the momentum is conserved, a motion can arise only in two directions simultaneously [one possibility for one-hand-clapping koan]. Visit ConceptOne. This does not mean that good and bad cannot be differentiated in absolute terms. For that you will need to get into incommensurable numbers. This topic then also includes harmony and how we (humans) are put together: what is agreeable and what gets us unglued.
Venus orbit period (time) around the sun is 224.70 earth days, while the Earth's orbit is 365.24 days. The actual Earth-Venus orbit ratio is then 1.625 -- and the actual time between pentagram's points is 593.7 earth days. The Earth-Venus interlock is not in the exact 8/5 ratio (1.60) and the pentagram rotates in the solar plane. Actually, every point slides forward by close to 9.3 days in excess of each perfect interval of 1.6 years (365.24x1.6=584.4 days vs. 365.24x1.625=593.7days). So, after 39.27 points (aphelions) the pentagram makes one complete revolution around the Zodiac. Now we take 39.27 and multiply by 1.625 for 63.814 earth years for the complete revolution. Because the pentagram has five-fold symmetry, we divide by 5 and that yields 12¾ (12.76) earth years before a point overlaps the next point. You are the best to apply this number. "Maturity" and "step" come to mind. From the Earth's perspective, when Venus is on the opposite side of Earth in the solar plane, the merged planet is on its closest approach to the Sun. There is much sophistication in the pentagram, much of it on the subconscious level, and it may take some work to bring it out. For astrology, upon Earth-Venus closest approach the point of the pentagram is made and this could be projected (vectored) onto a particular constellation or a house. Because the pentacle star's point is due to the Earth and Venus being the closest, I would not call it the 'inferior' conjunction. Also, making a ceremony with properly spaced circles could and should be akin to "drawing down Venus." Well, the Moon stands for the Goddess -- wherever she might be -- but it's okay to always know more (see comment on Mayans below). In astrology there is no pairing of planets but Earth-Venus, Earth-Mars, and Neptune-Pluto combine with unusual properties. Pairing is important because the concept of harmony is available only to pairs (number 2).		For parametric information about our solar system (Oss) and each of its planets and moons, visit University Corp. for Atmospheric Research web site. It has an excellent internal search and is great for kids, too QUANTUM PYTHAGOREANS A book of stars and pyramids Planetary orbits and geometric stars are related to each other. You'll know which stars are harmonious and will actualize. The book will move you decades ahead once you know what works and why. More.. One way of looking at the circle and the stars within is from the perspective of waves -- that is, energy. Should you make a pentacle or another star from wood or clay or metal -- that's fine but materially there is no intrinsic differentiation between one star with so many points and another star with a different number of points. Yet, as soon as you understand that waves can close in a circle only if they have certain wavelengths (read energy) and do so only in particular wavelength integers (numbers) then many stars unbecome eligible because they just cannot facilitate the closure in the circle. (Such numbers are not bad, they just do not have what it takes to support the symmetry about a point.) This [of course] gets to 'All is number' credo of the Pythagoreans. As a practitioner you want to know which stars and what modality -- clockwise or ccw -- is needed for the creation of standing waves. The 'standing wave' is a technical but quite appropriate description of waves that form and actually exist in the geometry of the circle. Once you know of other stars that are also harmonious, you will have other choices in the creation of the circle of waves.
Which way is the right way of drawing the Pentagram? That is, how to make a pentagram in the right order? This one is a bit easier to answer even though it may not agree with some books. If the opening (positive) direction is the clockwise movement of Venus and Earth, then every third point is traced (Venus catches up with Earth every 1.6 years) in the sequence shown below.
This is the aphelion sequence for clockwise moving planets:	The aphelion sequence on the left results from two orbits, which are in the 8/5 ratio -- that is, for every eight orbits of the inner (faster) planet the outer (slower) planet makes five orbits. To a scientist the pentacle is a simple drawing of a five pointed star and, because every third point is connected, the scientist will label this star as the 5/3 star. If every second point is connected then it would be the 5/2 star. A scientist is not concerned with the Nature's origin of stars -- and the geometric constructs related to their circular creation -- and views it purely mechanistically. Heptagon, then, would be the 7/1 star because the next (first) point is connected as you go around making the star. A scientist would not be happy if you were to tell him that some stars are enabling (opening, harmonious, supportive) while others are disabling (closing, disharmonious, detractive), for a computer generated star has no such associations. Yeah, they understand the cw and ccw direction and have to specify it to the computer but in the end it makes no diff to them because it produces the same result. Scientists cannot leverage what they know about stars except by intolerance stemming from their lack of knowledge. If you wish to engage a scientist in a conversation, ask about different properties of chemicals having right-handed and left-handed symmetries and why one of these has an overwhelming presence in living things.
The aphelion sequence for counterclockwise moving planets is as follows:
You now have the root to compile a set of symbols associated with penta-anything. It is very potent. The meaning of the five pointed star of the pentagram is in the root of its creation. You now know who is speaking from indoctrination, from knowledge, from experience, or from a soap box. For example, if you were to create a flaming star as a way of heating up your relationship, you could indicate the rotation of the star when rendering the flame. However, it is also possible that the flaming star is some people's way of talking about a UFO, in which case the five pointed star has nothing to do with it, flaming or not. A flaming or winged sphere is probably better. As for religious applications, you are, again, the best. [My take on the "new star in the heaven ushering new age" deals with the human making a significant intellectual advancement such as the Renaissance, possibly the Internet, and of course the hyperflight.]		During the geometric pentagram construction, one encounters an abundance of the golden proportions that are inherent in the pentagram. Because the golden proportion is also found in many living structures -- both plants and animals -- the pentagram also becomes associated with life support (health) and possibly with the root of life creation. The creation part can be somewhat complex but those familiar with geometry of the Great Pyramid may be a step ahead. (The guts of the Great Pyramid are 100% math, 0% symbolism. Symbolism comes from math and your brain uses various forms of math, including shapes and geometries, to obtain superior solutions, interpretations, and analogies. Working the geometry of the Great Pyramid, you can establish the arithmetic mean, the geometric mean and the harmonic mean -- and take it from there.) The construction/creation of irrational numbers is done through the Pythagorean theorem. (Irrationals are straight-line incommensurables.) Another class of incommensurables, the transcendentals, resists such implementation and, of course, the science math guys proved that the squaring of a circle is impossible. But the scientists wish to work with 2D of a paper and do not want to think about 3D -- even as the Great Pyramid is staring in their face. [Scientists like to prove what cannot be done and slip in constraints to make it so.]
The photo below is that of the crop circle discovered on Silbury Hill (Beckhampton) in England in 2002. Speculatively, the 13 "scales" or "fans" or "sweeps" (orbits, i.e., years) between points are interesting because, according to our calculations, the Venus-Earth pentagram rotates in the solar plane and overlaps the next point in 12¾ earth years. Still, beading around the periphery in the pentacle drawing below invokes a better representation of the pentacle's rotation. 64 beads would nicely show the number of years for the star to rotate around the Zodiac. Without the actual reference to the earth year, the number of beads around the periphery could also be 39.27 if you want to show ("point out") the rotating aphelions as they happen.		Scales are not exclusive to fish because dragons have scales as well. We are not explaining the meaning of mythology here but the idea is that the scales and sweeps are related through the Pi that is the circle (alchemical dragon) and a sphere (Chinese dragon) and -- viola -- we are back to geometry. The dragon of the Mayans is slightly different. The mountain is still there but it has the oval (or clam shell or saucer) shape of their number zero. Then there is the Trout Engine of Schauberger .. .. and the geometry of the scales themselves.
Two-Pointed Star pattern is made by Neptune and Pluto If you take Pluto away, you cannot play musical orbits. The 3/2 ratio is a well known musical ratio -- but not well known to scientists. If a two pointed star is the north-south star, what happened to the east and the west?
		Orbit period is the reciprocal of frequency. Music is about frequency. Scientists are the living proof that some people use less than 10% of their brain. Physically nothing would change if Pluto were to be declassified as a planet but if it keeps scientists in the news to their own detriment so much the better. The real issue is their incompetence when it comes to global warming. This sticks out when we look at the scientist's ignorance in the area of light-gas interactions.
Neptune-Pluto orbits are also interlocked but in the 3/2 ratio -- for every turn of the outside (slower) planet that is Pluto, the inside planet makes 1½ turns. The Neptune-Pluto duo can then be merged into a single planet with the resulting approximate orbit shown below. The combined planet makes a curlicue around the central sun before making the second point of a two-pointed star. The combined planets' orbit completes and, therefore repeats, every two Pluto orbits. This orbit, tentatively named Catseye ("the eye of the pussy") also denotes stability similar to that of the pentacle. It can, of course, be applied if the outer planets are to be preferentially referenced.
The making of the two-pointed star is along the same mechanism as that of any multi-pointed star. Kepler's equation is again appropriate for obtaining the ratio of Catseye-creating radii. You may want to experiment combining three orbits into a triple ring -- but consider that one orbit (one planet such as the Earth) would become the reference for the other two. All merged orbits are continuous in mathematical terms -- that is, the points may be tight and sharp looking but all points are rounded. Having its ends curved may seem a bit of trivia -- yet some difference exists when comparing the actual combined (merged) trajectory that has rounded points and the compass drawing of two intersecting circles with discontinuous transitions claiming to have "divine" geometry. In actuality, all stars have rounded points and the geometric construction is a stylized representation of the actual (see below). Geometric ways and means, however, is the computing platform of the cosmos. You may enjoy the fact that the two-pointed star is a bit of an enigma because the general up-down direction may seem to be "not really a star." The three-pointed star could be claimed to be the "lowest point star." Not so. With the extra curlicue and some width, there certainly is the two-pointed star because the star exists in 2D.		For perfect Catseye solution 22 = kR3 (inner faster orbit) 32 = kR3 (outer slower orbit) (2/3)2/3 = Ratio of radii = 0.763
The Eight Pointed Star is made with the Red Planet The octagon and thus the number eight is sourced from the Earth-Mars orbit relation. However, there are several ways of making the eight pointed star and that introduces the geometry of the diagonal. The number eight also continues its presence from the Earth-Venus orbit ratio (8 Venus' orbits for every 5 of Earth). We are, then, dealing with several topics when working with the eight pointed star.
		Diagonal is about transformation Searching for the meaning of the eight pointed star is also a search for the meaning of the diagonal. People can make nifty eight pointed stars with pronounced semicardinal directions (Rome's St. Peter's Square an excellent example) but that does not capture the idea of the diagonal. Similarly, the second rotated square making a diagonal of the "Muslim star" does not appreciate the qualitatively different role of the diagonal. A better understanding of the diagonal is in the Greek/Russian Orthodoxy where the diagonal is directly associated with the transformation of the soul, although -- like most religions -- they want you to "transform" to their side. Surprisingly, a scientist may have easier time with this because the diagonal of the Hermit (square) matrix has properties that hint strongly at the transformative aspect of the diagonal [think conjugation]. In the book Quantum Pythagoreans, another transformative aspect is invoked at the diagonal of the hyperbole, which is treated in the context of transitioning (and the necessary transforming) of data between the left and the right brain. In the East, the transformative aspect is in the virtual ("empty" or 'negative space') S shape that is in the middle of the Tao symbol. Alternately, it is composed of two separated paisley shapes instead of a diagonal, which conveys the idea of 'not horizontal and not vertical.' The gap exists, for there must be a separation (and joining) between the real and the virtual domain. (Technically, a wave couples across the gap.) Whatever gets from one domain to another is subject to transformation. As a Tai Chi practitioner you might be ready to answer the question: "Why should I keep my foot at a diagonal?" At times the Tao symbol is shown with dots of the opposing color in the middle of paisley fields. This is an interesting addition even Buddhists fail to interpret in their literature. (Martial arts literature tends to discard the dots altogether.) It deals with alternate means of bridging the gap and you will need to get familiar with this site to understand it. In the Mesoamerican tradition the diagonal is approached via the Hanub-Ku, which combines a straight diagonal with rotation while resembling the bar/spiral galaxy. The double-line axes drawing method is really about the empty (that is virtual) axis of symmetry [very impressive]. Note the absence of the number three. Hanub-Ku is possibly the most complex symbol of all. Unlike Tetractys or Tao or Ankh or even AUM, it is likely not meant to be evolved but carefully decomposed. It has the real-virtual duality on the atomic level and their transitions via the diagonals. It has subtle even-odd symmetry interplay, too. If you get serious on this subject you want to be as authentic as possible. Many Mayan concepts such as the tribes (I call them faces) were redrawn to look cute. Love to hear your interpretation(s). One input is from David Adair's recollection of symbols he saw on the alien engine. (I call it the energy-charge overlay, even though David calls all the symbols he found on the engine just "serial numbers for parts.") It's a good start-up symbol and now the rest of it comes from both the real in the virtual domains while differentiating irrationals and transcendentals. Oh, there is a big difference between numbers and numerals. (Ten is a number, Tetractys a numeral.) The Mesoamerican Venus symbol appears to be based on the number eight but it makes quite a distinction between cardinal and semi-cardinal directions (also see below).
Octagon star Earth has an orbit interlock with Mars as well. This orbit ratio is 15/8 and it is the ratio of two notes of the musical octave. You will need to find two notes of the octave that, once put in a ratio, will yield the musical frequency ratio of 15/8. It is a ratio of two notes and that means they need to be played together. Why do these two numbers make a harmonic and pleasant sounding ratio? There are some mysteries left to be pursued. The fact that you and your friend know that this ratio sounds harmonious is perhaps another mystery. The harmonious and disharmonious nature is absolute because we are able to agree on which is which. So far, the ratio of two notes that results in 15/8 (and that is musically harmonious) is in agreement with the ratio of the Earth-Mars orbits. Just as we did in the Earth-and-Venus case, Earth-and-Mars merged planet has aphelions from time to time and this happens when both planes are closest to each other. Connecting the aphelions results in the octagon. Because of the orbit ratio, the points of the octagon happen in the ccw fashion while both the Earth and Mars are taken as moving clockwise. The eight pointed star of the octagon then also represents a harmonious relation even though points happen counter-clockwise. The trace of the Earth-and-Mars merged planet happens clockwise but the aphelion points happen (peak) in the counterclockwise sequence without skipping points. An eight pointed star that skips points, however, may or may not be harmonious. [This is more than a homework for the Muslims, for they do not seem to understand this.]
There are several eight pointed stars. The octagon is traced in sequence and different orbit ratios account for its clockwise or counter-clockwise path. Skipping every two points draws every third point and results in an eight point star that could be called the octagram or octacle. Skipping every four points results in the identical octacle except that the order of drawing the points is different. No even-pointed star can be drawn by skipping a single point because only half of the points would be traced. Arithmetically, this deals with division such as when the attempt of drawing the 6/4 ratio actually produces the same star as the 3/2 ratio does. This could be a nice instance of applying the operation of division in nature. If you want to get into multi-pointed stars in more depth, the gateway question is: "How come 15/8 is harmonious (or musical) but 9/8 is not?" This will open up to other areas such as the absence of the six and seven pointed stars in orbit topologies. The true and harmonious eight pointed star comes from the 15/8 Mars-Earth orbit ratio that is in the shape of the octagon. The eight pointed star is usually associated with Sun or Venus but you will need a better rationale besides just 'someone says' or 'someone wrote.' From the ancient carvings, the semicardinal directions are strongly differentiated from the cardinal directions. The eight-fold symmetry does not have an equal weight nor meaning when it comes to the diagonal. The point is that the "eight-pointed" star is not necessarily about eight points. The cardinal and semicardinal separation takes precedence to "counting to eight," because the symbolism of the diagonal deals with the important aspect of transformation. It is likely that the "wavy" nature in the depiction of some of the diagonals is what leads some to think of "sun rays" and hence the Sun. Not so. We have depictions of other Venus stars -- their diagonals have different graphics and do not support the symbolism of the Sun.  In the Mayan mythology Venus is called "the great fake" in connection with early Mayans confusing Venus with sun. Confusing Venus with the Sun is by itself confusing but the story goes on in that a great spirit or god "lifted the clouds" and allowed sun to be seen in addition to Venus (or allowed Mayans to differentiate Venus and sun). Esoterically, then, you may want to think of Venus as having intelligence that is much broader than that of the Sun. If you are a scientist you might think of intelligence as nonlocal (and organized) energy. This is also confusing but it's the best bridge for you.		Venus in the Mesopotamian tradition -- the land of Sumer. The "eight pointed" star of Venus strongly differentiates cardinal and semi-cardinal directions while hinting at pulsing, vibration, and counter-rotation. [Venus on the right does not look happy camper and could have high emotional content -- commenting on her pretty leg might not be returned in kind.] This symbol, while associated with Venus, is about the diagonal that is transformation. The idea, then, is to look to Venus for her transformative qualities. In general, a transformation deals with the exchange of real and virtual energies. As such, transformations can be personal or can deal with free energy extraction, martial arts, space travel, or new computational modalities. In the ancient Egyptian context, the god of wisdom and magic Thoth is depicted full of diagonals. We have a picture of him on a page dealing with the squaring of a circle. As Australians look for a star that would enhance their identity, this site contributed a particular star that does not forget the diagonal.
		Occult, by definition, means hidden. You do not want to play with someone hiding cards up his or her sleeve, yet we all like to keep our privacy. In our case, it is the relationship between two patterns that is not obvious: How the flat (2D), five pointed star relates to two rings and vice versa. You may appreciate the occult if you think the subconsciousness rules (Carl Jung & Co.). A particular eight pointed star is created arithmetically from two orbits/rings but because the orbit periods could be mutually disharmonious, the star will not happen for real orbits -- will not actualize. The idea is that if you know which stars are harmonious and which are not, you may think you would be able to affect the outcome of a relationship using such stars. Similarly, a star that is culturally embedded may give the wearer certain characteristics even the wearer does not recognize.
Occult? Well, you cannot see the multi-pointed orbits but you can compute them. The shape of the shared orbits can never be observed but can be visualized or heard. Sun indeed feels the shared obits gravitationally because the planets are computationally combined. The metaphysical question here is really the reverse: If you draw a pentagram, are you in fact computationally and subconsciously creating the two specific interlocked orbits? If you think geometry rules (and it does), wearing a steel helmet just might not work. Thinking that geometry and numbers are illusions just might not work. One way of drawing a pentacle is harmonious -- think 8/5; Another way of drawing a pentacle is disharmonious (closing) -- think 7/5. One way of drawing a pentagon is harmonious -- think 6/5; Another way of drawing a pentagon is disharmonious -- think 9/5.
		One mathematical component in merged planets and the subsequent star formation is modulo math but it harkens back to Pythagoras as well because we deal with the basic Pythagoras' and Greek work on rationing -- albeit in the orbital geometry. Modulo math is funny in a sense that applications are not readily apparent. (Perhaps the best known application is the summing of digits of large numbers to test for divisibility.) It is said Gauss enjoyed modulo math and came up with a number of mathematical relationships in this area. Only recently, the biggest and very practical modulo math application surfaced as public key cryptography.
NASA, as usual, cannot get there. When all planets are computationally combined, there exists one overall wobble onto our Sun. Unfortunately, the wobble (periodic deviation) of other solar system's suns cannot be correlated to the number, size and orbits of planets other solar system may have. That is, knowing the overall wobble cannot be decomposed into specific planets and orbits, even if we know that the orbits are interlocked in musical orbit ratios. In fact, this is the primary mechanism behind public key cryptography. Don't bother telling NASA, however, for they are busy telling us about the size and distance of a single "planet discovery" based on the overall wobble. If NASA can see a wobble of another solar system that is grossly out of kilter, they then also spend some more money misinterpreting it.
Summary		Q: We understand vibration and tones for making music. But you say that the gravitational force working the planetary (macro) bodies needs to subscribe to music as well. Is not the force of gravitation just a mindless force -- this times this divided by that? A: For some, perhaps. For others, gravitation also organizes things into tractable topologies, particularly since this allows more than two bodies in the universe.
Music is vibrations. Orbits are periods that define vibrations. The Catseye orbit results from Neptune-Pluto interlocked ratio of 3/2 or 1.5. Interlocked orbits can make a number of star configurations but the planetary interlock usually does not exceed the ratio of 2.0. (Quantum Pythagoreans book tabulates the ratios for planets and moons.) Because the musical octave spans orbital ratios from 1.0 to 2.0 then the Catseye ratio of 1.5 corresponds to the Sol note, which is a "fifth" between C and G. (Music guys make it more complicated than it is -- and so we will stay with ratios.) Both notes need to be played to obtain the 1.5 ratio and can be played at any particular octave. These tones then reinforce the symbolism of the Catseye. You can now construct the associative symbols of the Catseye that will acquire a symbolic meaning: numbers 3 and 2, their ratios, and notes of the (Western) octave C and G, and the eye shape itself. The eye shape can be rotated 90 degrees to resemble the cat's eye. [The shape then also begins to look like a winged sphere but I would not go that far.] The topic of Pythagorean celestial tones can get mystical, particularly if you get into chords and harmonics, which then reinforce or possibly create other orbits -- some claiming to be able to hear the music. The correspondence of the pentagram's 1.6 ratio to a particular tone makes for a good exercise. Another ratio is 5/3 and this corresponds to notes C and A (ratio 12/3 or 1.66). If such orbital ratio were to exist (corresponding to musical ratio of notes C and A) then the merged planet would make a three-pointed star orbit. Three-pointed merged planet orbit does not exist in Oss [yet? or perhaps it exists at another solar system].
Orbits make shapes of multi-pointed stars. Rational numbers in general and the ratios of the musical scale in particular play an important role. Orbit interlock also exists between Earth and Mars that corresponds to two notes of the octave in the 15/8 ratio. The Earth-Mars merged planet makes an eight-pointed star of the octagon -- the next point being formed in a simple sequence (no points are skipped). This is a fertile mathematical area with potential new applications. For example, the orbit interlock in the 6/5 ratio also produces a 5-pointed star but it is not the pentagram, as each point is made in incrementing sequence – and yes, 6/5 is a ratio of two notes of the octave – and no, there are no planets of Oss that would play this orbit ratio. It is a clockwise pentagon. A ratio of 9/5 produces another pentagon that traces points counterclockwise .
Additional things to consider arise when considering the 13/8 ratio. This ratio is not on the musical scale but, arithmetically, it is closer to the actual measurement of the Venus-Earth orbit ratio, which means that the rotation of this star is slower than that of the pentagram. Both 8/5 and 13/8 are ratios of two consecutive terms of the Fibonacci sequence of numbers. It makes for a good exercise to figure out the star that issues from the 13/8 ratio and what advantage, if any, this star brings to the table. Fibonacci sequence grew from but a series of numbers into the golden ratio a/b when Kepler discovered that the ratio of two consecutive Fibonacci numbers converge toward (1+SQRT(5))/2. (Reduced, the golden ratio is 1.6180.. to 1.) Because the actual Venus-Earth orbit ratio is very close to the golden ratio (1.625 and 1.618, respectively), any ratio of two consecutive terms of the Fibonacci sequence will make a ratio that is a very close fit: 8/5 (1.6), 13/8 (1.625), 21/13 (1.615), 34/21 (1.619), 55/34 (1.618), etc. If somebody proclaims that the eight or the thirteen or the twenty-one pointed star represents Venus -- you will know where he or she is coming from.		Venus symbol in the Mesoamerican tradition appears to be based on the number 8, possibly as (1) The eight Venus cycles that synchronize with Earth's five years, or (2) The eight Earth's years that synchronize with Venus' thirteen cycles, or (3) The relative importance of cardinal and semi-cardinal directions as issuing from planetary orbit observations and counting. Diagonal -- that is semicardinal -- direction is about transformation (evening & morning?) [my vote here] Numbers 0, 1, 5, 13, and 20 are prominent in Mayan numerical and calendar systems. For better or for worse, Mayans (used zero but) did not use fractions [running out of pegs on a calendar wheel because of that? Or is it because they did not get to irrationals?]
Here is the actual pentacle picture in the making -- the trace of the merged Earth-Venus planet in the 8/5 orbit ratio. The straight (shorthand, simplified) 1-2-3-4-5-1 sequence appears to be counterclockwise but the actual curved trace is always clockwise. Yep, there are five curlicues there in all.		QUANTUM PYTHAGOREANS Book by Mike Ivsin Musical tones have equivalent representation in geometric stars. Planetary orbits also have representation in geometric stars. There are, then, harmonious and disharmonious geometric stars. Quantum Pythagoreans will show you which stars are which and why some happen while others do not. Relation reversibility is about transformations between right-brain and left-brain contexts. Very important for self-organization, Quantum Pythagoreans explains the conditions for entities to move between the two domains. You will understand the virtual domain, the transformative mechanism, and the harmonious composition for the true road to reality. More ..
Go or select another topic from the gold post HyperFlight home Portal in new window  Added 8-point star and closing paragraphs summer 2005 Presently enlarged with Neptune-Pluto and Earth-Mars interlocked orbits, the original article is available in Press Releases as pdf file. © 2002 -- 2008 Backbone Consultants Inc. Copyrights Information Last update April 11, 2008