HyperFlight Circle division by five – draw the perfect star Divide a circle by .. .. – design the perfect family of all-stars Construct the real and the virtual objects -- and take them to 3D Geometric construction of the pentagon, pentagram, and the five pointed star How to draw the pentagram using compass and straightedge; How to make the pentacle directly on a given circle; and How to create the five pointed star indirectly via pentagon tiling Perfect three-pointed star can be had on a circle or semicircle		"Give me a ring and I will make it sing"
How many perfect stars and star shapes 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 Pentagon Pyramid. Fusion of numbers in three dimensions is good but it is not about averaging
Take this page out of frames Historically, the geometric creation of the pentagram star was considered a secret. One can appreciate keeping the "formula" and directions secret because the five point star construction 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 pentagon, the segment must be exactly one fifth going around the circle.
Not all numbers can divide a circle into exact and equal segments, and this is the first differentiator between geometry and arithmetic. When it comes to dividing the 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 finite number of steps and achieve exact division, we would then claim executability and, therefore, the 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 exact division of a circle in the atomic construction below. If the length of a 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 segment's length were an irrational number then its 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. Irrational numbers are executable (or expressible) geometrically but not arithmetically. 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 by five -- in five steps: 1. Draw a horizontal line and erect a vertical line. The intersect becomes the origin point O 2. Make point A on the horizontal line at any unit distance 1 from O 3. Draw a circle of radius 2 around O. This makes point V 4. Draw an arch around point A through V and make point B 5. Distance VB is the length of cord c that makes sides of the pentagon		There are two sources -- and therefore more than one meaning -- of the five pointed pentacle star: One meaning has its origin in the exact division of a circle and is discussed here. It is a fairly complex though rewarding topic that leads to the symbolism of a star inside a single circle. The other root comes from orbits (hence two circles/rings) of Venus and Earth, and is discussed there. Venus, while most prominent through the five pointed star, is also associated with the number eight and with the meaning of the diagonal. Rational numbers are commensurable numbers -- that is, they all have finite or repeating 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 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 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? New book you will thoroughly enjoy QUANTUM PYTHAGOREANS More ..
		The radius measure of 2 (diameter of 4) in our pentagonal construction 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. 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 unit 1 is the Great Divide between macro-cosmic and the micro-atomic. [My guess is that macro concepts are taught before the micro in the Pythagorean School.] For the four sided pyramid, particularly the Great Pyramid, you want to start with the irrational and rational golden numbers and their proportions. The pyramid template results from the new way of the golden numbers construction that also reveals geometries of the inside of the pyramid. The meaning and benefits of proportion(ing) and rationing are also there. Again, we would not pass up an opportunity to talk about the one-up of geometry over arithmetic when it comes to irrationals. When you do 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 [and this can get cosmic]. Pyramid's base periphery and base area then also carry the same square number 16. The number 2 (half of the base) is the denominator of the unreduced golden ratio. If you like Japanese calligraphy, you might appreciate directions starting with the horizontal and then moving to vertical. Then left, right. The lower closure happens last. The horizon is also important to the ancient Egyptians. If you want to have more fun, consider that the unit distance OA could be an irrational number. Even Euclid did not think of the number 1 as just a counting number. 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. 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 the circle, for it is about working the straight and curving geometries. You will like the connection between the exact construction and the (exact) conservation of energy. We did not forget the ancient Egyptians either.
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. [Every time you double something -- think octave. Every time you halve something -- think node (or fit) for standing waves. Every time you rotate by a right angle -- think..] 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 mechanics are clockwise. [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. [Connect it counterclockwise if you want to be disharmonious (ratio 9/5), clockwise if harmonious (6/5).] The ~gon identifies the stars that do not skip points -- polygon in general. When you say regular polygon you are emphasizing that all segments have the same distance. (Distance is more general than length.) All points of all perfect stars are on the 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 computer can give you an irrational number.)	There are other geometric ways of constructing a pentagon or pentagram. 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. Other constructions make the side c a rational distance, which is better suited for the Great Pyramid's purposes. (The Quantum Pythagoreans book presents the pentagon construction with a rational segment length.) In the pyramid, one half of the side of the base is (must be) a rational unit of measure, for such measure is executable. (Transcendentals may 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. Some star constructions speak of fixed length sticks, which, technically, can construct any size polygons. Here is where the executability of angles comes in. 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 case can be made that geometry takes precedence (has priority) over arithmetic -- see below. [If you are a scientist, you may think of Emmy Noether who ignored 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 physical entities if you know what they are.
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 semicircles 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. If distance AB is irrational, should it be dashed? If so, why?

Perfect star family
Using geometry's tools, a straightedge and compass, 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. 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 will have some symmetry about a point and about an axis. 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 star is 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 start is constructible exactly via geometry, the very same star cannot be constructed exactly via arithmetic. Carl Gauss "recently" added the 17 sided polygon. 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 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 the disharmony. Six gets "tricky" because it is disharmonious with larger numbers but is harmonious with five, making a pentagon. When using but a single star you choose one from the perfect star family. 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.. QUANTUM PYTHAGOREANS If a is harmonious with b, and b is harmonious with c, is c harmonious with a? Not always. The book explains harmony through geometry and star construction. While much talk is about harmony, Quantum Pythagoreans provides the formula for notes that actually are harmonious. In the beginning was the number .. and the power of numbers begins .. Ready to construct the atom? You will need knowledge and not just raw power. More..	One interesting property of the perfect star family is that they do not intersect. A sequence growing from each of the direct perfect star number does not cross (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 family of numbers introduces some changes to our perception of universe building and how the everyday reality happens to come about. Mathematicians can make all kinds of star constructions, in 2D and 3D -- but only the perfect star family can begin to bridge the straight line energies such as photonic energy with circular orbits and orbitals. Because the vast majority of energy in the universe is in the form of a spinning or an orbital energy -- that is, energy that has angular momentum, the perfect star family of numbers takes 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.
Construct the perfect triangle on a circle and another triangle on a semicircle	Instructions: Draw horizontal and vertical lines. The intersect is the origin O Draw semicircle of radius r around O. This makes point V Draw circle around V of radius r Larger triangle divides the circle into three exact cords of length c. Using the Pythagorean theorem, verify that the relation between cord c and radius r is: c2 = 3·r2 or (1/3)·c2 = r2 What physical entity is proportional to r2? If you know what that is, consider that the square of cord c is three times that. 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. Could squares with irrational sides be included in the general division of a square into many other squares? If so, does it mean that geometry does one up on arithmetic once again because arithmetic cannot give you an exact irrational number? The six-pointed star could be of some interest regarding energy accumulation and we included it in the numerology section on the Pythagorean page. While the circle is divisible by three exactly, we need six nodes for three wavelengths to wrap around the center (or around the nucleus in the case of an atom). We are then talking about the hexagon.
Pentagon 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).	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.
If you take five, five-pointed stars and arrange them around with their points touching, do you get a pentagon in the center? (yes) 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.) The pentagon template for the illustration was obtained with MS PowerPoint by selecting AutoShapes .. Basic Shapes. Pick the pentagon object. Tiling of five pentagons to make a five pointed star was (first?) published by Kepler in Astronomia Nova
The tiling construction (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. (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. Now, 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?) When working the Great Pyramid, you may want to think of the chambers and passageways as virtual objects or "spatial" objects. It really helps.
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 the 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. 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 the circle depends on 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, 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 that 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. So, if you use a protractor to construct the angle of 72 degrees, say, this angle (and the corresponding mark) cannot be obtained exactly if this protractor was created or calibrated by a computer. 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. [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 is the meaning of a circled star? The 'star inside a single circle' is about the atomic creation symbolism. 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. 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 the 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. While many books talk about Georg Riemann accomplishments, they are written mostly by mathematicians. It is likely Riemann did not even try to find applications for his math and the science writers don't try very hard either. Mainstream physicists bring up Riemann applications in a superficial way of "curved space is possible." So far, nobody applied his work to atomic closed topology orbitals and to atomic computability of curved pathways. So far, nobody linked his discovery of incomposite (prime) number availability to the "forbidden" electron orbitals. This task would be simpler if the incomposite numbers of the Pythagorean origin were not renamed 'prime,' for incomposite numbers cannot compose while prime has no relevant applications association. The label prime makes its descriptive value so weak nobody even thinks of it as a hole in the wall. We have three book reviews on Riemann.
This brings us to wider applications of rational and irrational (incommensurable) numbers. These numbers are important if you want to appreciate the interplay between geometry and incommensurable numbers. A nice definition and some apps are on their own page, with the following few lines serving as summary: For commensurable (rational) numbers the applications are clean. Any whole number divided by another whole number could also become a fraction, yet such fraction always remains a rational number because any fraction (such as the decimal fraction) sooner or later starts to repeat, and such fraction can be converted back into a ratio of the two whole numbers from which the fraction can again be made. The ratio of any two whole numbers or its computed fraction can be reversibly converted into each other without loss in fidelity. If you have difficulty visualizing why a repeating fraction is really finite, you will need to get into the ancient Egyptian way of representing fractions [yes, this is loaded]. The computer, then, can store rational numbers just fine (at least in principle), while the present computer does not, and never will, manage the infinite mantissa of the irrationals. Numbers with infinite non-repeating mantissa are incommensurable numbers. Applications of incommensurables are rooted in their infinite mantissa. In the space of geometry, moreover, distances can actually be irrational. Here is also where the 'number line' reaches the limit of its usefulness, for the number line is about the magnitude while the incommensurables are not. (Advanced -- need to differentiate between 'distance' and 'length/magnitude,' and the reason it is advanced is not because it is complicated but because Euclid did not do it. Euclid did not get primed very well in the Pythagorean tradition even thought he understands that 1 is not just a counting number. Distance vs magnitude benefits are treated in Quantum Pythagoreans book and applied to energy.) 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) do 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. Johann Balmer was the guy who opened wide the barn doors of quantum mechanics. He came up with the equation that produced a sequence of numbers matching wavelengths of light coming from hydrogen. These were not just any wavelengths -- they were particular jumps of particular wavelengths and no other. Balmer did for quantum mechanics what Kepler did for gravitation: He came up with the 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. Whole numbers and HIS theorem are also the source of the quantum behavior of atoms.
Squaring The Circle This topic is expanded and now 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 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 the 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.
Summary, Cosmic 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 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 time parameter 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?" [Think those pesky irrationals and glorious transcendentals.]	Summary, Atomic 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?
	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.	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 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? Something even smaller than the infinitesimal of Newton and Leibniz? A computational construct for all of the above from "all of the above?" 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 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). 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.) QUANTUM PYTHAGOREANS Book by Mike Ivsin Pythagoreans use the knowledge of numbers with straight and curving geometries to arrive at stable -- that is harmonious -- systems. Numbers' properties under different symmetries impose specific solutions. Numbers create the duality, and engagement of the two components of the duality are responsible for organization. The Quantum Pythagoreans book also treats the reversibility of interactions and transformations as components of stability and creation. In Quantum Pythagoreans the force of gravitation is one of the variables that applies the Tetractys template and results in all observable cosmic topologies. Yes, the book proposes the gravitational mechanism as well as the computational mechanism. 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. You will appreciate the simplicity and the power of numbers. Pythagorean management of numbers and operators then takes you on the road to reality and invites you to drive it as well. More ..
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