HyperFlight I. Exact circle division by five – draw the perfect star II. What Natural (whole) numbers divide a circle exactly and equally? Meet the perfect all-star families III. Construct the real and the virtual shapes – and take them to 3D "Give me a ring and I will make it sing" Geometric construction of the pentagram, pentacle, pentagon, and the five pointed star How to draw a pentagram or pentagon using compass and straightedge construct a pentacle directly on your own circle make a five point star indirectly via pentagon tiling – and vice versa create fancy five-pointed stars plus a ten-pointed hyperstar (in a new window) Perfect three-pointed star can be had on a circle or semicircle – and it's doubled for a hexagon already Perfect square is a four-pointed star How many perfect stars can you make with geometry? With arithmetic? Geometry vs. Arithmetic is a very old topic, very unresolved, and very relevant; and Numbers are not just symbols – in a circle they make 1) stars on paper and 2) atoms in space A comment on a seven-pointed star (heptagon). Hit a snag in a circle, but .. Pentagon Pyramid. Fusion of numbers in three dimensions is good but it is not about averaging
The four sided Great Pyramid construction is via the golden proportion and has its own page (in a new window)
Intro Historically, geometric drawing of the pentagram star was considered a secret. One can appreciate keeping the "formula" and directions secret because a construction of the five point star is not obvious even if you worked in geometry for some time. Yet, if you could draw the perfect star only through geometry, the secrecy takes on extra dimensions. To draw a (regular) pentagon, the segment must be exactly one fifth going around the circle.
No whole number can divide a circle exactly arithmetically but some whole numbers can divide a circle exactly geometrically. This is the first differentiator between geometry and arithmetic. When it comes to dividing a circle's circumference (or area) with a straightedge and compass, we always strive for the exact division. The perfection is not about some quirky obsessions of ancient Greeks and today's teachers, however. If we finish our assignment in a finite number of steps and achieve an exact division, we would then claim executability and, therefore, creation of such structures could be implemented in nature. Computer's arithmetic gives us precise – but not exact – answers when working with incommensurable (irrational and transcendental) numbers, but the geometric way can be exact and have much utility just because of that. We will apply the division of a circle in the atomic construction below. Yes, the exact construction is about the (exact) conservation of energy. If the length of a circle's straight segment (cord) is exact, then the segment's length is unambiguous and can be expressed as a finite number. A finite number is also a rational number. If the cord's length were an irrational number then such number's sub-unity portion (mantissa) is infinite and we could not agree on its exact length – but we could agree on the exact distance between the two end points because the two end points can be constructed exactly geometrically. Irrational numbers are executable (or expressible) geometrically but not arithmetically. This is the second differentiator between geometry and arithmetic. You might skip on the diff between length and distance right now – later you may avail to the explanation and construction of the (in)commensurable numbers. Instructions on dividing a circle into five equal and exact parts – in five steps: Construction (continued): If you want to construct the penta~ with the circle radius of your choice, point A is one-half of the radius away from origin O. When you divide a circle into exact fifths along the periphery you now have the template and: Make the pentacle by connecting every other (second) point. If you do it counter-clockwise, you will be doing it in the 8:5 ratio, which is harmonious, and you should not have a problem with that once you understand that the underlying pattern mechanics are clockwise – it's on the Venus page. Pentacle is the easiest and the simplest to make because it does not require erasing. Does it mean it is the most fundamental? Make the pentagram from pentacle by erasing the circle Make the five pointed star from pentagram by erasing the inside (smaller, upside down) pentagon. Also see pentagon tiling, further on Make the pentagon by connecting every neighboring point and erase the circle. Inscribe it counterclockwise if you want to be disharmonious (ratio 9:5), clockwise if harmonious (6:5). (Cw and ccw makes a big difference – think modulo math.) The ~gon identifies the stars that do not skip points – polygon in general. When you say regular polygon you are emphasizing that all segments span the same distance. (Distance is more general than length.) All points of all perfect stars are on a circle and are equally and exactly spaced going around the circle. The best way of seeing it is that they have the exact fraction of a circle between them. As to the actual angles between points -- see below, for these angles could be irrational numbers. (For now, ask yourself if a protractor can give you any irrational angle.) Why should the circle division be made of equal (evenly distanced) segments? There is nothing wrong with unequal lengths if that's your fancy. However, an electron's wavelength is proportional to the electron's energy and if several wavelengths of one electron were to fit (were to close) around the nucleus then they have to do so in whole multiples of the same distance. It's about the numbers. There are other geometric ways of constructing pentagon or pentagram patterns and symbols. The construction presented here has the length of the side of the pentagon c the incommensurable (irrational) distance – that is, the measure of the side's distance is composed of an infinite number of digits (that do not repeat individually or as a group). Other constructions make the side c a rational distance, which is better suited for the Great Pyramid's purposes. In the pyramid, one half of the side of the base is (must be) a rational unit of measure, for such measure is executable and can become. (For transcendentals you may have to put a kink in that.) The connection between the five sided pentagram and the four sided Great Pyramid is through the golden proportion, a subject that lets you understand how to draw and construct the Great Pyramid and include the pinch on its side. Determination of the pentagram's angles is easy. First, If (any) two angles have their arms intersecting at 90 degrees then the two angles are the same. The central (Egyptian) star has 360/5=72 degrees between its arms. Because the arms (rays) of this star are at right angles to other angles then such angles are also 72 degrees. You will note two special triangles on the pentagram above. One has the internal angles of 72, 72, and 36 degrees while the second one's angles are 36, 36, and 108 degrees. Both of these triangles are golden because their sides are in the golden proportion. There are many other golden proportions on the pentagram but these two kinds of triangles are the most important in the micro (atomic) domain. One Other Penta~ Construction This one comes from Yosifusa Hirano of 19th Century Japan. It is elegant and also constructs the pentagon or pentacle on radius 2. There is a dual approach to the pentagon construction. You can either specify the radius of a circle or the length of a pentagon's side. In either case, and starting with the shortest unit distance of 1, the radius or the side's length ends up being 2. Pentagon construction with side length priority is on the golden proportion page. You might have noticed that in the five fold division of a circle the three points made by the compass are at the corners of a right angle triangle with sides 1 and 2. (A pin of a compass centers the radial symmetry [masculine].) It is no coincidence that the Great Pyramid's Grand Gallery has the vertical height (rise) of 1 and the horizontal length of 2 while the Trough is the hypotenuse spanning the distance of SQRT(5). (This also establishes the unit length 1 of this pyramid.) Is it a coincidence that to define Pi we need distances 1 and 2? Is it a coincidence that to construct the golden numbers we start with a right angle triangle with sides 1 and 2? Not done yet. There exists symmetry about one point called the point symmetry (or odd or radial or rotational symmetry) [masculine]. There also exists symmetry about two points called the even symmetry (or axial or line mirror symmetry) [feminine]. These two kinds of symmetries are all-pervasive in atomic construction. Yes, everything is coming up numbers. Now, how would you marry these two symmetries? There is more to 5 It is very easy to get excited about the number 5 and begin to associate each of the five points with other things. This is a Pythagorean site and we love the number five but Pythagoreans also know that using numbers for counting is introductory to the power of numbers while correspondences are about the elemental – that is differentiating, power of numbers. The five discrete elements in Wicca and Tao are fine, yet the visual-geometric imagery based on the Penta~ is about the infinity (irrationality) of the golden proportions stemming from the square root of five. With the golden numbers you construct many shapes that extend the Penta~ beyond counting. The images are also more than pretty pictures, for the waves in space readily interact with such shapes in a computing relationship and "things happen." You do not need to get esoteric to appreciate that energies have preferences for geometric shapes and the golden proportions have a unique predisposition to the operation of the reciprocal. (Yes, energies make things grow but also think about the nature's way of capturing the waves' energies.) So it is nice to recognize the golden shapes around you even though neither the star or the number five jump out at you. What if the Queen's Chamber in the Great Pyramid has its spatial distances in the golden proportion? What if the spiral on the Eye of Horus were made through the golden rectangle? How can you visualize the golden ratio in financial terms?		There are three sources – and therefore more than one meaning – of the five pointed star. One meaning has its origin in the exact division of a circle and is discussed on this page. It is a fairly complex though rewarding topic that leads to the symbolism of a star issuing from a single circle. Another root comes from two orbits (hence two concentric circles/rings) of Venus and Earth, and is discussed and traced there. (Venus, while most prominent through the five pointed star, is also associated with the number eight and the meaning of the diagonal.) The third source of the five pointed star calls on non-concentric rings {Dec 19, 2009}. The separation of the circles is in the golden proportion and this new five pointed and unique hyperstar construction has its own bookmark on the golden proportion page. This brings us to the diff between the pentagram sign/drawing, aka the Pythagorean Pentalpha, and the pentacle . The pentagram is a five pointed star drawn with five straight lines. The pentacle has a circle(s) around the star. Yet, these are but technical differences. There are three separate origins associated with the pentagram and you may want to show the pentacle to point out the root. In other words, the pentagram always issues from orbits/orbitals (circles) and there are three separate ways to do so, as follows:  1) A single circle around the star makes it the classical pentacle that comes from atomic construction, and a single circle shows the standing wave around the nucleus. Pagan-wise the classical pentacle stands for earth and if you think of it as 'materia' it's a close match.  2) Two concentric circles around the star make it the cosmic pentacle and the two circles are the orbits of Venus and Earth (can be computed via modulo math from the clockwise 8:5 orbit ratio). Points are between the two orbits and the pentagram rotates (why that is so is on the Venus page). For Pagans the double circle around the star is about 'drawing down the Goddess' and the particular circle diameters are calculated on the Venus page as well.  3) Separated but interlocked circles (rings) with centers at the "hips" of the pentagram shows two separate atoms joined in a molecule, which I call the hyperstar pentacle. This is new and I don't presently know of anybody applying the hyperstar pentacle. My feel is associating it with friendship, marriage and angels. I like using it in Tai Chi and here is an example. Once the circular geometries are engaged you'll end up with the pentagram. It is then okay to draw just the pentagram but you want to draw the various pentacles if you want to show where the pentagram is coming from. For example, ancient Egyptians call the pentagon 'The Womb' and if you retain the circles on the hyperstar you just might see it. Pythagorean pentagram/pentacle symbolism is a bit more sophisticated. The pentagram is encircled once and then a second ring is added as a piece of jewelry. The second ring is then at the right angle to the first. Regarding the Satanic or evil side of the pentagram or pentacle, consider it a feeble attempt at corruption by the self-proclaimed sign-of-the-beast creator and ex-priest Levi. You want to know where corruption comes from and then you are in position to overcome it. In this case the upside down pentagram could issue from natural rotation of the cosmic pentacle in the solar plane and then the satanic notion loses its meaning once you appreciate that the cosmic pentacle's rotation does not stop and there is no 'up' and 'down' of the solar plane to begin with (but cw and ccw can be differentiated -- think angular momentum 3D). When it comes to the hyperstar pentacle, the up-and-down points form a ten-pointed hyperstar (not ten-sided) yielding a North-South axis with very unusual properties. The axis can become absolute under spin and symbolically has a touch of Tartaros (or Tartarus) -- but, as a Pythagorean you know what the post is about. In the case of the classical pentacle the upside down notion also has no meaning, for the five-fold atomic orbital is symmetrical about the atomic core and is free to rotate without appreciable symbolism. (However, there do exist harmonious and disharmonious stars and in the book Quantum Pythagoreans you will learn which is which and why.) On the political side, particularly in the association with Communist power, you want to be cognizant that the five pointed star issues from orbits and the star is always a 2D entity. Making the star into a 3D star (the likes of the Kremlin) points to the lack of understanding on the root of the star's creation. The book you will thoroughly enjoy QUANTUM PYTHAGOREANS It is about the perfection of geometric stars and the waves that go with it. More .. The radius measure of 2 (diameter of 4) in our pentagonal construction on the left is the outcome of using the shortest applied distance as the unit 1. This is not because you couldn't divide by two (you can -- and work with ½ as the distance OA, for example), but if you construct other structures such as the the Great Pyramid with the shortest distance as the unit 1, you will always be in sync with your numbers from one structure to the next. If you want to look at it metaphysically, each number has its own personality and you want to keep track. So you think you know your numbers and you might think it's okay to reduce them to your liking. But if you construct the Great Pyramid with the golden numbers and use the shortest distance as the unit 1, you will arrive at the pyramid's base as having the length of 4. The base of the Great Pyramid is then 4 times of some unit of measure. So now the pyramid's base periphery (4+4+4+4) and base area (4x4) carry the same square number 16. You see, if you reduce the numbers and think of the pyramid's base as having the unit length of, say, two, the base periphery would have eight units of length but the base area would be but four (square) units. If you do not reduce the numbers you can think of the number 16 in the context of acceleration (unit of measure per time squared) and derive a unit of length that is most appropriate for this planet [yeah, it's a foot]. Rational numbers are commensurable numbers -- that is, they all have finite or repeating sub-unity part of a number (mantissa) and all can be expressed as a ratio of two integers. Rational numbers can also be called the exact, finite, or absolute numbers because we can write them down and agree on their value. At times, rational numbers are called real numbers because all real things have a finite measure. Rational numbers happen when we ratio two integers. All mainstream mathematicians define the rational number as the ratio of any two integers. So, a mainstream math guy would say, "Of course the rational number is a ratio of two integers -- it is defined that way." Yet, you really do not want to be mainstream and acquire but an encyclopedia knowledge of the world. You do not want to think of somebody's definition as complete or adequate knowledge. You know that a rational number is a finite number (has finite or repeating mantissa) and once it is finite it can be expressed as a fraction of two integers. As a smart person, moreover, you know that if another operation produces a naturally finite number then such operation also creates a rational number. The circumference of a circle is a transcendental number. Many of circle's round segments (arches) are transcendental numbers and their straight cord could be an irrational number -- and both of these numbers have an infinite mantissa (infinite precision). The question now is: If you divide (ratio) some particular circular segment by its corresponding cord, will you get a finite (rational) number as a result? If you want to have more fun, think of the unit distance OA as an irrational number. Even Euclid did not think of the number 1 as just a counting number. Some Pythagoreans view the number two as a problem number because it "divides the unity." Pythagoreans discourage division of the unit 1 because it could create problems elsewhere -- but the number 2 is certainly not the culprit. (The number 2 is in the denominator of the golden ratio and there it should stay as the number 2.) As you get familiar with this site the sub-unity will become applicable to atomic orbitals and hence the number 1 is the Great Divide between the macro-cosmic and micro-atomic. [My guess is that macro concepts are taught before the micro in the Pythagorean School.] Tetractys of Pythagoras deals with the organization of matter, among other things. A circle has many positive connotations. What would be the idea of dividing it? Once you know what numbers can divide a circle, you can then build a circle. Not [yet] from real things such as wood or metal but from waves. It turns out that the waves must have a particular wavelength count (a particular multiple of particular energies) before these waves are able to close in a circle -- and thus be symmetrical about a point. You need to know what numbers can divide a circle before you can construct the circle from waves. You might think this is something witches do, and you would be right, but an electron is a wave that wraps around the nucleus, too. There is (always) a bit more to this. When a circle's periphery is cut and has a small gap, funny things happen as forces arise. One could call this a circle corruption and in a way it is. Yet the forces that arise are not corrupting, for they attempt to close the circle and.. The making of a circle is also about taking a step from 1D to 2D. There, you will find the friendly transcendental number Pi. To round it off, you may want to learn more about the squaring of a circle, for it is about the straight and curving geometries. We did not forget the ancient Egyptians and use the example of the five pointed star as one of the steps in working the circle and the square. Ah, geometry Not everybody likes geometry. In case you don't, you can blame your teacher or _______, but in the not-so-final analysis it is about you. Geometry is about movement and placement in space, from an atom in your body to your ship as a whole. Lots of geometry is in a plane and you have a good argument if you say your head is not flat. So let me cut to the chase. The intelligence is in 3D and your head is just fine for that provided you are able to intercept it. Lots of free energy is in 2D and it can be harnessed there once you figure out how to relate 3D to 2D. The linear movement is in 1D while the wheel and gravitation need 0D for spin. Your challenge, desire, need, or necessity is to understand and work the tetra(ctys) of 0D through 3D because that is how the universe is built and you want to continue to be a nifty and smart participant in it. Every time you double something – think octave. Every time you halve something – think node (or fit) for standing waves. Every time you rotate by 45 degrees – think transformation. Every time you rotate by a right angle – think.. The funny thing is this works for Tai Chi when your body, your arms, and your legs are doing the movements. Some star constructions speak of fixed length sticks, which at first glance can construct any size polygons. Here is where the executability of angles comes up. In space, the irrational angle is constructible only approximately and only some angles will be actualized – think snowflake formation. Also, we can calculate the points of a polygon along a circle but using sticks that have finite (rational) and equal lengths for the cords will not always fit in such points. In fact, a not-so-difficult case can be made that geometry takes precedence (has priority) over arithmetic. [If you are a scientist, you may think of Emmy Noether who ignored the nature's beauty of snowflakes and made simplifying assumptions about space that proved the 'ignorance is bliss' postulate – for in her world everything is reduced and snowflakes and crystals don't exist.] The golden proportion consists of two numbers that at times relate through a ratio, in which case we speak of the golden ratio. The two golden numbers consist of one irrational (1 + SQRT(5)) and one rational number (2) and, because they may relate to each other through multiplication or division or addition or subtraction or.., they should not be reduced into a single number. Reduction into a single number severely limits the application of the golden proportion and that is one reason scientists like to reduce it as the Phi [scientists have reductionist tendencies – perhaps not a disease but it could be a handicap]. Reduction into one number hides other relationships the two golden numbers might have. You can calculate the area of any polygon by taking the area of the triangle and multiplying by the number of sides. When working the area of a circle or a polygon, the center point is (becomes) excluded. (If you are metaphysically inclined, think Isis looking for all parts.) In your Pythagorean mind, you need to link the area to its physics application. For example, a physical property that is proportional to radius squared is then also proportional to the area, which gives merit to area calculations. This is bigger than it seems. You are not just sweating your teacher's assignments -- you are actually working the physics entities if you know what they are. Analytically attacking all three major pyramids at Giza as one layout can earn you a label or two, but on this site Jiri starts with a square and then looks for the golden proportion – and gets very, very close to the actual measurements. Ready to bury the Pharaohs someplace else? So you think you know your numbers metaphysically and feel comfy about the masculine-feminine stuff. You might be dividing by 2 and think it feminine. Not so. Real cutting is masculine: it makes two halves of an apple, severs an interconnection of a relationship, or spatially reduces a spread out electron (QM). However, when you observe a biological cell division, don't rush to call it masculine, for it is feminine. You'll have to get into symmetries to understand this. Meanwhile, don't make the silly mistake of equating masculine with a man and feminine with a woman -- unless you want to give up on one half of your brain.
Some basic geometry. From a square angle to a square
	Here is a simple yet powerful construction that 1) Divides any distance exactly in half; 2) Erects the perfect right and square angle; and 3) Makes a true square using any circle centered at O (at the intercept of horizontal and vertical axes). A square is also a four pointed star Only straightedge and compass are needed. (Straightedge is an unmarked ruler.) Both arches have the same radius. Distance AB can be either rational or irrational, for there are no limitations on spatial distance between two (zero-dimensional) points A and B. Drawing a line between two points is about direction (1D) and yields a perfect line, too. If you want to know the minimum separation between points before the line could become a real line, take a look at Absolute Minimum Length (it's about the infinitesimal). If distance AB is irrational, should it be dashed? If so, why?

Perfect star families It is easy to draw stars using geometry's tools, a straightedge and compass. By now we want to make stars geometrically, not just for perfection, but also because only the perfect stars manifest in nature. A circle can be divided exactly into 2, 3, 5, 15, and 17 equal segments, technically called regular polygons. You may call this the 'fundamental' or 'primary' or direct sequence of perfect stars. Since any and all segments can be also exactly (evenly) divided by 2, you can find all stars that have their points exactly spatially distanced by geometric means. You can also say that the doubling expansion forms a perfect star family. For example, you can make an eight point star or a 64 point star from a two point star through simple halving of distances. From a three point star (below) you can make the exact hexagon and from there the twelve point star of the Zodiac or do a layout for a twenty four point Feng Shui star. Starting with a 2 point star – the only direct even star – you can construct 4, 8, 16, etc. stars that you could also label the 'evenly even' sequence of stars. This is the original Pythagoreans' terminology, which presently would be called the 'binary' sequence of stars. From the 3 point star you can continue to halve each side to make 6, 12, 24, etc. stars. From the five point star you can make 10, 20, or 40 point stars. Every perfect star with the even number of points will have symmetry about an axis and about a point. If you think there is no such thing as a two point star, it is on the Venus page and it is formed by the combined Neptune-Pluto 3:2 orbit. The stars that are left out from direct and doubling constructions cannot be constructed exactly. For example, you cannot make a nine point star directly – or indirectly from a three point star. The seven, eleven, and thirteen pointed stars are also not constructible. Numbers that divide a circle exactly could have a name of their own. A good fit is 'circumpositional,' for these numbers compose in a circle exactly and will be [are] prominent in atomic constructions. {Mar 21, 2006}. The most interesting (and important) aspect is that even if a star is constructible exactly via geometry, the very same star cannot be constructed exactly via arithmetic. Carl Gauss "recently" added the 17 sided polygon as the perfect star. The 15 sided polygon is in Euclid's Elements, Book 4, Proposition 16.	Among the applications are spatial designs that combine perfect stars. In watch design, for example, the circle being divided by twelve looks fine but, in addition, it is harmonious to overlay the 12 point layout with triangular, square, or pentagonal designs. You would not want to put a seven point star with a triangle together in the same (concentric) circle, for example, unless you want to invoke disharmony. Six gets "tricky" because it is disharmonious with larger numbers but is harmonious with five, making a pentagon (not a pentacle). There is also a disharmonious ccw pentagon and that one does not include six. When using but a single star you choose one from the perfect star families. When combining stars, however, you also must deal with harmony. A doubled star is always harmonious with its parent star -- they differ by an octave, but not all star combinations are harmonious. QUANTUM PYTHAGOREANS If a musical tone x is harmonious with tone y and y is harmonious with z, is z harmonious with x? Not always. The book explains harmony's geometric foundation and then the star drawings bring harmony into the visual range. Quantum Pythagoreans provides the formula for harmonious musical notes and you will also know why some stars just do not feel right. In the beginning was the number -- and the power of numbers begins .. Continue ..	One interesting property of the perfect star families is that they do not intersect. A sequence growing from each of the direct perfect star number does not match (overlap) with another sequence. That is, from 2 we get 4, 8, 16, 32, 64, 128, 256, 512, etc from 3 we get 6, 12, 24, 48, 96, 192, 384, 768, .. from 5 we get 10, 20, 40, 80, 160, 320, 640, .. from 15 we get 30, 60, 120, 240, 480, 960, .. from 17 we get 34, 68, 136, 272, 544, .. Each member of the perfect all-star family has but one origin. The perfect star families of numbers introduce some changes to our perception of universe building and how everyday reality happens to come about. Mathematicians can make all kinds of star constructions, in 2D and 3D – but only the perfect star families can begin to bridge the straight line energies, such as photonic energy, with circular orbits and orbital energies. Because the vast majority of the real energy in the universe is in the form of spinning or orbital energy – that is, energy having angular momentum, the perfect star families of numbers take the front seat. Scientists can draw all kinds of curves but these are usually fancies. Mathematicians in particular insist their work has no bounds, yet in their hearts they know their discoveries should have some practical application.
Three pointed star Construct the perfect triangle on a circle and another triangle on a semicircle -- in three steps	Instructions: Draw horizontal and vertical lines. The intersect is the origin O Draw a semicircle of radius r around O. This makes point V Draw a circle around V of radius r The larger triangle divides the circle into three exact cords of length c for a perfect three pointed star. The smaller (red) triangle divides the circle into six exact cords of length r resulting in a perfect hexagon or hexagram. You can verify (using the Pythagorean Theorem) that the relation between the cord c and cord (radius) r is: c2 = 3·r2 What physical entity is proportional to r2? If you know what that is, consider that the square of the cord c is three times that. Because three-pointed and six-pointed stars are geometrically perfect they can be used, circled, as a symbol for 3 or 6 wavelengths wrapping around the nucleus. However, a hexagon and hexagram reduce into a triangle under modulo math for harmonious ratios and do not manifest in orbits (macro) -- that is, showing six-pointed stars with two circles does not reflect nature. A six-pointed star could be of some interest regarding energy accumulation and we included it in the numerology section on the Pythagorean page. Without a circle, a triangle symbolizes 3-state systemic (complete) systems, each state being in one corner. Such triangle, then, has no metric as it is a logical, say clockwise, process. (Some systemic systems call for quaternaries -- think ancient Egyptians.) For Pythagoreans a triangle provides bounds for the ten dots of Tetractys (a triangular numeral 10), which also becomes one facet of a tetrahedron (projection from the apex [or from your eye]). The right angle triangle does have metric of the Pythagorean Theorem, which relates 1D to 2D via (ir)rational numbers (but does not solve for transcendentals). When you see a triangle with some symbol in the center (a dot, an eye, dragon), take such symbol into 3D of the apex of a tetrahedron to see if it means something to you. The puzzle of a bad souffle: Given a square, construct a new square that is exactly one third of the original square. You can be fairly certain that the person will try to partition the square in some way but the solution is to erect a triangle on the square's side and then obtain the radius for the circle that covers the triangle. Radius r is the side of the new square. [In our case the bad souffle does not cave in but runs over the rim.] It is not possible to reverse engineer a souffle without stepping back and understanding the relationships between the ingredients and their proportions – as well as the irreversible nature of the baking process. Can you apply the construction of the geometric mean in the solution of this puzzle? Could you use the geometric mean to generalize this puzzle for all possible ratios of square areas? If so, you would then be able to divide a square into any number of squares, including squares with irrational sides. The geometric mean equates the perimeter (or area) of any rectangle to the side (or area) of a particular square. Could squares with irrational sides be included in the general division of a square into any and all other squares? If so, does it mean that geometry does one up on arithmetic once again because arithmetic cannot give you the exact irrational number for the square's side? Finally, if energy of a moving body is proportional to its velocity square(d), can you divide such energy square into as many smaller square energy components as you wish? (Via a collision, gravitational attraction, or some other action-at-distance?)
Tiling of pentagons and stars Tiling does not involve direct construction but only translation and/or rotation in two dimensions. Translations are linear (straight) motions and are always symmetrical about a line (edge) [feminine] while rotations are always symmetrical about a point [masculine]. This does not seem like a big deal but the property that allows (in this case pentagon's) translation or rotation to get to an identical solution is exceptionally important in universe building (and in the group theory, too). Tiling of five pentagons to make a cool five pointed star was (first?) published by Kepler in Astronomia Nova		When the "ancients" instructed us to use the straightedge and compass, they were not really talking about constraints because they were talking about geometry. Rotation about a point is about the use of the compass. Straight movement (translation) is about a symmetry about a line and perhaps you could see now that the line of symmetry is a virtual line – that is, the line of symmetry is an empty slit. (Would you go as far as to have Justice brandishing her sword with a slit down the middle of the blade?). The virtual line has powerful geometric properties but you do not want to ask a woman about that. Not that you couldn't, it's just that the explanation is nonverbal. The pentagon template for the illustration on the left was obtained with MS PowerPoint by selecting AutoShapes .. Basic Shapes. Pick the pentagon object.
If you tile five pentagons you get the five point star in the center. Now, if you take five five-pointed stars and arrange them around with their points touching, do you get a pentagon in the center? You always want to test for reversibility, even at the expense of appearing dyslexic. Relations are reversible only under certain conditions and you want to know what they are. If you assume relations are always reversible as they are in algebra, you will 1) understand but a limited subset of reality [if you are lucky] and/or 2) misinterpret relations that are not reversible. For example, if there is a quantum mechanical explanation of gas pressure, there could be a way of making the phenomena reversible. Now, how would you reverse the rotation of a light mill? (Give it a thought and get the answer.) We readily apply force to get things moving. So, how would you reverse 'something' and have the force arise?		Testing for reversibility is crucial in the understanding of relationships. Dyslexia is a condition that is constantly reversing relationships in all modalities: verbal, tonal, geometric, written -- to see if the reversal acquires another meaning, valid or not. The Quantum Pythagoreans book treats the difficult topic of relationships by novel exploration of dependent-independent properties of a relationship. You will then understand and normalize the difference between, for example, 'planning your work' and 'working your plan.'
The tiling construction – that is, movement about a point and/or translation along a line, of some objects may result in the appearance of another object. This is at times referred to as "negative space." While it is true that the original object is real and in some respects positive, the 'negative space' label is but an introductory way of looking at it (and the left-brain way at that). A good way is to see this as the act of creation of the virtual object. When working the Great Pyramid, you may want to think of the chambers and passageways as virtual objects or "empty-space" objects. It really helps.		Self test:-) Straighten up two adjacent fingers. Do you see a difference if you think of these fingers as two closely spaced pencils – or as an empty slit or space that is between the pencils? Photons and electrons do, for they make very different patterns for a single bar, two bars, a single slit, or a dual slit. For homework: 1) How is it possible, and 2) What is the utility of the result that one pattern ends up in the left side of the brain while the other in the right side?
Geometry versus Arithmetic Ancient Greek-speaking scholars debated geometry and arithmetic, and understood the complexities even without a PC. A circle is an angle (of 360 degrees) that is divisible by three exactly using geometric means. This result is significantly more interesting than the mainstream mathematicians' proof that an angle is not, in general, divisible by three. If you think of a circle with the orbit (cosmic) and/or orbital (atomic) applications in mind, you will see there is lots of fun in figuring out what works[, rather than beefing up your resume with things that don't]. So, the ancient riddle about dividing the angle into thirds has more than one answer and no answer is the wrong answer. It is, however, a parting, or the "Tau" riddle that to some makes all the difference. A circle cannot be divided by 7 or 9 equally and exactly. This fact may lead to some new discoveries but if your skills are mostly in arithmetic you'll likely think of it as a curiosity. That is the basis of reductionism, for a reductionist first makes a claim that arithmetic is just as good as geometry (brain "grouping"), and then happily ignores the advantages of geometry. Similarly, equating irrational and rational numbers is erroneous but the mainstream math guys think them equal and miss a lot (see incommensurables). Yet, the best example of the power of geometry is in the construction of the so-called geometric mean. Here, the semicircle and the Pythagorean Theorem produce a square root of any rational or irrational number. In addition, the geometric mean can multiply two irrational numbers together and produce an exact result, the infinite mantissa and all. No computer can do that. It is now time to visit the angles of a circle. Can we map the angles in such a way as to obtain correspondence between geometry and arithmetic?
What's your angle?
Arithmetic makes it strange In a calculator, the angle of 360 degrees is divisible by 9 without a remainder, but this is but an arithmetic computation. In geometry, the circumference of a circle issues from Pi, which is a transcendental number and so you cannot be arbitrary about the length of the circle or the exactness of an angle inside a circle. Consequently, a division of a circle into an arbitrary integer quantity of equal and exact segments (or angles) is not possible. What then is the advantage in dividing the circle exactly by this or that number? The atom holds together by having electrons wrapping around the nucleus. Because the electron's momentum is also a wave (de Broglie), the electron's wave must evenly, that is exactly, close upon itself to form a standing and a round wave that is symmetrical about a point. To the Pythagoreans the numbers are everything and this is because numbers actually create things. The mainstream scientists' argument that "computer's representation of an irrational number is close enough" is, unfortunately, not relevant to atomic construction. Scientists just do not know how to interpret 'precise' and 'exact' in an applications setting. The scientist can divide the circle by nine to a very large number of decimal places, but there will never be a wavelength that would fit nine times around the circle of the orbital. Incidentally, 'fit' is the original (superior?) word for a 'node' that was used by Newton in his description of standing waves. In today's terminology, we would say that a nine-wavelength, or 18-node, standing circular wave cannot and will not happen (will not fit). Numbers 7, 9, 11, 13, 19, 21, 22, 23, 25 and others cannot divide a circle exactly. Most of these numbers are incomposite (prime) numbers. Number 9, though, is a composite number but it cannot be used to divide a circle exactly. [Does this mean the Chinese Emperors could not sing? Having said that, they might have been good golf players.] Number 5 is incomposite but can be used to divide a circle exactly. What is needed, then, is a class of numbers that compose in a circle, instead of just being composite numbers (composed of products of other numbers). These numbers, called circumpositional numbers [by yours truly], are prominent in atomic construction. Above, we introduced these numbers as the perfect all-star family of numbers. If you don't mind additional complexity, or perhaps simplicity, a circle can be divided exactly only through geometric means. Another way of saying 'geometric means' is 'spatial distance means.' Yes, the circumference of a circle is a transcendental number and a division of any transcendental number by any real number remains transcendental. The computer can use only real numbers and the length of the circumference is then rounded off if it is to be stored in a computer. What this also means that a computer cannot give you a perfect star. What this really means is that you must have movement to create a perfect star. In other words, you cannot make a perfect star via placement or measurement (statically, topologically) with a computer or a computer-calibrated protractor. As a Pythagorean, you might realize that you cannot construct the perfect star without a compass – that is, you need rotation to complete exact constructions. A larger implication is that there must be movement even at the atomic core level -- and under radial symmetry the movement is also about frequency. (You might guess here is one of the gateways to gravitation and yes, the philosopher's stone opens up to the alchemist.) The necessity of a movement permeates everything. Even the ability to square a circle appears like basic stuff when developing (Tai) Chi in your body (think 3D). Arithmetically, no star creation is perfect. Worse yet, arithmetic operations with real numbers give no clue if a particular number is or is not divisible into a circle exactly. A circle's periphery divided by any rational number remains a transcendental number – that is, arithmetic keeps transcendental numbers unchanged. But what if you could geometrically divide one incommensurable (transcendental) number by another incommensurable (irrational) number and get an exact result? There may be more to it than just being nifty, for you are dividing one infinite (mantissa) by another infinite and get a rational number with finite mantissa as a result. [Does it mean we could do one up on the gods?]	Arithmetic makes it practical Pythagoreans had a category of numbers they called 'abundant.' Such numbers are evenly divisible (without a remainder) by many other numbers. Number 24 (hours in a day) is a good example as it is divisible by 2, 3, 4, 6, 8, and 12. As convenient (abundant) as the number 24 is, we presently divide the day only by 2 (am and pm) and by 3 (work part), and possibly by 12 (entertainment). Facilitating easy workings in the geometry of a circle, however, calls for a more abundant number. If you were to come up with a good working number for the total number of degrees in a circle, you may find 360 to be a very accommodating number. 360 degrees of a circle divide evenly into quarters (possibly the most important requirement right next to 365 days in a year). It divides evenly by 5, 10, and 20, too. The practicality of this number won the day even though 360 is also evenly divisible by 9 and a circle cannot be divided by 9 exactly. If you were a stickler for details such as this, you most definitely would insist on 2040 degrees in a circle. 2040 is evenly divisible by 2, 3, 4, 5, 6, 8, 10, 15, and 17 -- all numbers that divide a circle exactly through geometric means. The good old 360 is not evenly divisible by 17 and that means that exact geometrically-obtained angles do not necessarily have a whole numbers of degrees if we stick to the present 360 quantity notation. Both 360 and 2040 are not evenly divisible by 7, 11, and 13 -- as it should be. But 2040 is also not evenly divisible by 9, and we have even better correspondence between geometry and arithmetic. The year 2040 could be the most harmonious year coming up. [But don't tell IRS. They'll put this number on a form and spoil it.] Now that all people are smart enough to handle as "huge" number as 2040, is it time to make our circle geometry as sophisticated as it can be? Are you ready for the sum of internal angles in a triangle to equal 1020 degrees instead of 180? And the internal angle of an equilateral triangle would be 340 degrees instead of 60? Even if you could legislate the change – and during the French Revolution they "legislated" 100 degrees in a circle and 100 minutes in an hour -- the bottom line is that there is no perfect number for a quantity of degrees in a circle that would be a whole number or even a rational number. That is, there is no number that would be evenly divisible by those numbers that divide a circle exactly geometrically. In our example of the 2040, this number is not evenly divisible by 16. [Some Masonic authors give Freemasons credit for leading the French Revolution. If so, they would certainly be quite ignorant on what to do in the aftermath – all their Gs notwithstanding.] Whole numbers and rational numbers are called real numbers – a good name. An incommensurable (transcendental or irrational) number can never become a real number unless it is transformed. The transformation is irreversible because we cannot save an irrational number such as SQRT(2) in a computer and retrieve it as (convert it back into) the original irrational number without first taking a nip off the number – think Ouroboros and visit Circle and Pi. Reversing the transformation calls for addition of the virtual energy [think Isis and possibly Thoth if you are familiar with his eye restoration story]. If you want to get deeper into transformations of rational (real) and irrational numbers – think ancient Egyptian fractions [here, you will need to appreciate three things: 1) Ancient Egyptian fractions are quite sophisticated; 2) Present day scientists are clueless as to the ancient Egyptian fraction applications or origin; and 3) Our present civilization is not necessarily advancing.] When dividing a circle with the straightedge and compass, the goal is to make the number to become, for the number's geometric construction creates something specific to that number. There is, then, more to numbers than philosophy, and you may want to visit the original number apps guys, Pythagoras and his fellow Pythagoreans. There is a treatment there of real, virtual, and irrational numbers. Irrationals and transcendentals are in the family of incommensurables but transcendentals are not constructible through the Pythagorean Theorem (from 2D of the curve to 1D of the hypotenuse), while additional differences between irrationals is based on applications. [There are good and bad numbers and some of them have an infinite mantissa.] What does a star in a circle represent? What is the meaning and symbolism of a star inside a single circle? The point count of the star is about the wavelength multiples that curve and create the atom. Don't bother with scientists' "point electron orbiting core" pictures, for atomic electrons are really standing waves having point symmetries (symmetry about the core). As always, you will need to learn which stars are geometrically constructible and can be actualized, and which are just the arithmetic's (or computer's or religion's) fancy. Is the atomic core composed of standing waves? You bet. Scientists have way too much invested in the solid and static core hypothesis and so it is safe to talk about the pulsing and standing waves of the core. Scientists are way off and, for example, they made up "strong nuclear force" because they do not understand the wave nature of the core. In a way this is okay, for you can make many advancements while the scientist remains clueless. For example, the core's shape is not necessarily spherical. With all their equations, scientists think highly of whatever it is they describe with them. Saying that the scientist will remain clueless is no idle talk, however. On our Circle & Pi page we also highlight the inadequacy of algebra, for algebra's constructs cannot deal with the operation of equivalence. Algebra completely misses irreversibility, too. As a Pythagorean you want to figure out what entity will prevail in the interaction with a standing-wave electron and with a standing-wave proton. Why, could you then do a precision surgery on the atom? The eight pointed star has diagonal (semicardinal) directions and if the meaning of a diagonal is in some respects special (it is) then the eight pointed star could be about something else besides counting points. It is about Venus, too.
Seven Pointed Star A circle cannot be divided by seven exactly. Yes, we can leave it at that except that the number seven is the first number with such property. When things are happening inside a circle – that is, when things are spinning and evolving, the wave folding encounters the number seven and consequent inability to fit around the circle (and make a perfect star). When you examine the Mesoamerican Hunab Ku symbol, there are two seven-sided areas (heptagons) and now the challenge is to explain that. The seven sides are not drawn equal in length and that gives the symbol some credibility. The heptagons pivot into 3D and now it gets really intriguing. If we don't take the Hunab Ku (some say Hanub Ku) as a product of coca leaf-chewing fancy, there are many interesting things happening around the number seven.   Historical Note Johann Balmer was the guy who opened wide the barn doors of quantum mechanics. He came up with the relation that produces a sequence of numbers matching the wavelengths of light coming from hydrogen. These are not just any numbers – they are wavelengths corresponding to particular electron jumps and no other. Balmer did for quantum mechanics what Kepler did for gravitation: He came up with the math equation that matched known experimental data and made successful predictions of other new, yet undiscovered, wavelengths. But there is a bit more to it. Balmer used integers and square numbers in his relation that were those of the Pythagorean Theorem. Well, good ol' Pythagoras was not only right all along but the breadth of his -- some say HIS -- teaching was also the foundation of quantum mechanics. Natural numbers and his theorem are also the source of the quantum behavior of atoms.		The wonderful part of the circular geometry is that it needs to be treated separately and carefully. Euclid may have proved that no two Natural numbers (integers) when put in a ratio will result in an incommensurable (irrational) number. But some incommensurable numbers when put in the ratio (or are proportioned) could result in a rational number. You may want to reflect on what it means. As far as Euclid goes, not much. After all, Euclid talks about what does not happen. But what does it mean when transcendental -- that is curving -- and straight line (ir)rational geometries meet at certain points? Think transformations. Then there are the scales of the fish [yeah, the dumb fish]. The sweeps of the scales subtend a particular angle. Do you think a fish could fly or extract energy from the swirling water around it by using the geometry of its scales? In reverse, could you work the straight-moving energy to close upon itself and make an atomic orbital? Think pyramid geometry.
Squaring A Circle
This topic is expanded and has a page of its own. To reconcile the straight and the curving, you will be dealing with the squaring of a circle. Algebra works fine when things are straight or polynomial but, generally, when geometry picks up another dimension and lines start to be circular the equations are not enough as the transcendentals come up. The relationship of the squaring of a circle to this page's perfect division of a circle is in the possibility of linearizing the curved segments (arches) of a circle and then making a tractable exchange between curving and straight topologies. So there is a continuation to 'how to draw a star' and it deals with energy. Can we say that geometry is about energy? Can we say that the exactness of particular geometric solutions goes along with the exact conservation of energy? Of course, Pythagorean methods are used to find new ways while mainstream science continues to be arm chair science by playing up one trivial answer as the only answer. At times politicians pick up the 'square a circle' analogy and then you should know they are trying to find excuses and "explain" failures to their supposedly dumb constituents. Alchemy The complexity of our environment is oftentimes worked through alchemy. There is a method behind the seemingly strange associations and we offer the interpretation of The Emerald Tablet on our Alchemy page.
Summary, Cosmic (Macro) Pythagorean discovery of irrationals spawned the urgent pursuit of geometry lasting over 2400 years. Kepler brought arithmetic to the forefront by establishing the mathematical and arithmetic relations of heavenly orbits. In effect and in fact, Kepler introduced the parameter of time in the mathematical context, which made it possible to make planetary position forecasts – forecasting being Kepler's life long passion. Because any two gravitationally interacting bodies always have a solution, the parameter of time derived from such periodic solutions is also repeatable (periodic) and time can be used to make forecasts. Even though time is always a derived (dependent) variable, the mathematical solution establishes reversibility and allows the time parameter derived from this system to be used. (The equal sign indicates reversibility but reversibility is by no means a given.) Another way of seeing the mathematical solution and consequent time reversibility is that the spatial distance (space) and time form an overlay. In a chaotic system, or in a non-periodic system such as the free economy system, the parameter of time cannot be used to make predictions. Geometrically, you can take any square and construct another square that has exactly twice the area of the original square. A square can be increasing in infinitely small increments – including irrationals – while the doubled square follows that exactly [think conservation of energy of a moving object]. This is something your computer cannot do. If you think this is no big deal and it is something for the ancients to contemplate – that's fine. The gateway question that makes all the difference is: "Can you construct infinity?" Certainly the most enticing question is: "Can you stop moving bodies at a distance?"	Summary, Atomic (Micro) Light is understood as moving or standing linear -- that is, straight, waves following Newton's analysis of fringes (first observed as fringe rings). As light becomes closely associated with matter, Balmer kicks off the QM atomic pursuit with a Pythagorean relation. A wavefunction is understood as a probability distribution of an atomic particle -- a great step forward by Heisenberg and von Neumann. A moving particle has momentum but momentum can also be worked as a wave -- a second great step, this time by de Broglie. A moving electron now gets to become a wave as well, but this wave must curve – that is, become circular, and close upon itself in a symmetry about a point (about a core). The circular (or rounded) electron orbital and the straight path of light need to be energy-reconciled through the squaring of a circle – the first difficult hurdle. Ether is taken out of science's purview, which is the Great Reduction making the scientist that much poorer in the end. Scientists cannot make headway and talk about impossibilities. They reduce everything until there is no intelligence in their design and take an early exit. (In their last hurrah the angry mob bashes and trashes cold fusion.) Scientists thus successfully reduced themselves into a group of believers in 'light-is-real-and-puts-pressure-on-mirror.' While much of today's physics rests on it, the scientist has no guts and no brains to perform the actual experiment measuring the presumed pressure light puts on a mirror. Scientists are not able to face up to the truth that a light beam does not and cannot put pressure on a mirror and so they are stuck perpetuating, defending, and proselytizing their dogma. Meanwhile, geometry is receiving new impetus by reviving its superiority over arithmetic and algebra. The golden proportion, the infinite and instant wavefunction superposition, the understanding of irrational and transcendental numbers, linearizing particular segments of a circle, and the possibility of creating electron waves with harmonics-series energy components just might get the atomic understanding going again – perhaps in another country, perhaps by another group of professionals.	The other Five If planetary orbits interlock in 8/5 or 6/5 ratios, a five point star orbit results when the two planets are computationally combined. Venus and the pentacle get their say and, because the pentagram/pentacle also originates from the orbits, there is yet another way of drawing the pentagram/pentacle that deals with the "proper sequence" and "proper orientation" of making the points. Different ratios correspond to particular clockwise and counterclockwise stars. A good question for orbit ratios 6/5 and 8/5 is: Arithmetically, we can calculate the angles of the pentagon and derive the equation that allows the computer to draw the pentagon. You may have also constructed stars on paper, but how does Nature draw the pentagon or a pentagram on the canvas of space? The planetary pentagram and the five fold atomic orbital make the five pointed star a joining symbol for both the macro and the micro. While the atom's orbitals are symmetrical about a point (have radial symmetry), the valence orbitals in a molecule need to close around two points of symmetry (the cores of the atoms are some distance apart). Yes, the HyperStar has an answer to that on our golden proportions page. {Dec, 18, 2009}
	Pentagon Pyramid
Self-test:-) If you are not happy about the pentagon base and/or the pyramid being "split up" or "broken up" or "separated," you are not getting the picture. You may want to think about the red part as being the tangible component while the blue part is the intangible (knowledge, virtual) component. If that does not help, stay in 2D [earthbound?] where the pentagon is continuous. If you think Two rather than Four is feminine, you are very close. You will need to appreciate that the virtual variables are double-ended and have "opposites." Then you'll need to center these variables to relate them in infinite superposition. (Quantum Pythagoreans book helps in this area too.) The engagement of Three and Four is just that: It can be supportive in some contexts and in others it could be conflicting, in which case rebalancing work is needed. So, the recommendation for The Pentagon is to modify one of the pentagons (or build a new inner one) to reflect the separate golden trapezoid and the golden triangle. The new construction slants upward toward a point. It does not need to top off in a point as long as the edges are converging toward a point (and you know why).	Three and Four must remain separate even though they are also joined. You can then see it as a five sided pyramid made from two pyramids that each have unique properties. The overall 3D structure has almost all numbers in it but the point is that the numbers must play together to their best advantage (rather than just being represented). Do you see the number zero? Or is it the infinity? Could it be both? Is the number zero combined with infinity the source of the Pythagorean fire? The root of the pyra-mid? The Central fire? The hearth of Zeus? Archimedes' fulcrum of the heaven and the earth? Just a plain (free) electron? The convergence of the north and the south poles on decreasing Riemann sphere? A point of the zero dimensional (0D) geometric construct? 'The One' of alchemy? Something even smaller than the infinitesimal of Newton and Leibniz? The dot in the semicircle on the AUM symbol? A computational construct for all of the above from "all of the above?" The top dot of Tetractys? My favorite: The eye Thoth sends to look for Tefnut when she runs away to Nubia (yes, the eye finds her). If you think this pyramid is about marriage, you are on the right track. The Three [male] and Four [female] are joined through the point of the infinite. This joining is applicable to the actual marriage where the joining is through God – all there is. The alchemical marriage would have the female part becoming more abstract although the four-sided pyramid geometry continues to be needed for dealing with the infinite. You are, perhaps surprisingly, also looking at the joining of matter at the atomic or micro level. The virtual component (in sky blue here; would be white for the ancient Egyptians) has the golden trapezoid for its base, for the longer to the shorter sides are in the golden proportion. The real component has one of the golden triangles for its base. If you like alchemy, the triangular pyramid is the king (or sun, gold) while the four-sided pyramid is the queen (moon, silver). Yes, this is the 3 vs. 4. In the ancient Egyptian context, the crown of Egypt has two separate components: The white (upper, virtual) and the red (lower, real). The gap between the two is the ancient Egyptian blue crown and is invoked at war. (The gap is white in our pentagonal pyramid.) There is one symbol that uses a circle framed by two vertical lines. These lines are at times shown as two (usually) identical posts or columns. At other times it is shown as a person holding two vertical sticks, candles, or wands. This symbolism, however, is not about the 3 vs. 4. Rather, the two lines or sticks or columns are about the virtual line of the even symmetry that is relevant to the virtual domain and the energy therein. The upcoming book (summer 2010) will deal with the construction in the micro domain via the stars, the rings and the symmetries.    QUANTUM PYTHAGOREANS  Book by Mike Ivsin Pythagoreans use the knowledge of numbers to arrive at harmonious and stable systems. Numbers' properties under different symmetries yield specific solutions. Numbers create the duality while the engagement of the two components leads to organization. Quantum Pythagoreans applies the Tetractys template and that results in all observable cosmic topologies. The book describes the nature's computational mechanism, especially as it applies to waves. What it takes to transform energies. Your body is the component and it is not the only one. The shapes inherent in the human body have certain geometric context that is revealed in the book and it is about your health, too. You will like and appreciate the simplicity and the power of numbers. The Pythagorean management of numbers takes you on the road to reality and invites you to drive it as well. Continue ..
Go or select another topic from the gold post HyperFlight home Portal in new window  © 2005 -- 2010 Backbone Consultants Inc. Copyrights Information Last update July 5, 2010