All is number
I think therefore I am
I build therefore I become

HyperStates, the working symbol

Pythagoreans and their methods for the organization of matter

There are many elements in nature that compete with each other. Physical elements, rather than being created in conflict, are formed in balance in one of the hyperstates shown here on left.

By the numbers. Heavy duty
Macro and Micro. What issues from the numbers may work for you
What is behind the numbers. Putting numbers on Tetractys
One application: The Barbury Castle Tetrahedron. Twenty years and one book later
Living with numbers. Pythagoreans do it with numbers and operators and degrees of independence
Numerology with shapes. Intro -- also has masculine-feminine differentiation. For Pythagorean marriage numerical aspects and alchemical marriage also look at the pentagonal pyramid
Harmony. This topic is not presently understood but receives much attention in the new book: the formula, equivalent stars and all
The Pythagorean Y
The Theorem. Yes, it's HIS
The Pythagorean School -- it's out there
You, a Pythagorean. It does happen
Mother Goose of Tetractys -- For personal Pythagorean discoveries and for fun, too

By The Numbers {2/22/2002}

In the Pythagorean tradition the universe is running on Natural (whole) numbers and on rationing and on squaring. Numerology, then, was the way of looking at just about everything. We are not staying with or departing from tradition. Yet, the picture of the HyperStates is mostly about integers and near-integers. One hyperstate is a ratio. One other hyperstate, although it is computable and potentially real, does not seem to actually form. As of now, there are ten realizable and observed topologies in the cosmos

There is much similarity between the colorful dots of hyperstates and the Tetractys of Pythagoras. Verbal interpretations of the Tetractys from the Pythagorean tradition such as the "universal order" in literal terms and "nature's spring" in the figurative sense are indeed close to hyperstates. A new platform, if true, is not the end but the beginning of the next building phase.


 The Tetractys of Pythagoras -- The symbol HE gave to his fellow Pythagoreans

HyperStates start with Tetractys of Pythagoras. Tetractys is a numeral 10 rather than just a count of 10.


 Enumerated Tetractys of the Pythagoreans -- the Hyperstates

 One (blue) state is added and all eleven states are placed on one facet of the tetrahedron. We arrive at the triangular pyramid geometry.

I named the collection of dots the HyperStates. Each dot is a particular tractable solution (a construct) but each dot can also become a component of another solution with a change in scale and this can (and does) go on without bound. The multiple solutions overlay is what prompted me to call this symbol the HyperStates. There is yet another overlay in that a particular cosmic body may belong to several solutions. Moving throughout the cosmos, one is under several HyperStates at any one place. Very nice. But of course, it's in the book.

If you came here looking for some good applications of the Pythagorean Theorem, we use it in the unique construction of the golden proportion. Another app is the geometric mean, which makes a square out of any rectangle and also gives you a square root of any length (including infinite irrationals).

If you have questions regarding the story that Pythagoreans feared irrational numbers, read up on it in this footnote on free energy. It may be based on the mysterious applications of irrationals that "they" do not understand and their interference comes from upholding the status quo.

The pentagram/pentacle is the Pythagorean not-so-secret flagship symbol. We've got three pages on Penta~
 1) Venus page (has the Pythagorean ring, too),
 2) Pentagon construction page (also has the diff between pentagram and pentacle); and
 3) Golden proportion page (with a new way of the golden numbers construction)

Operators also issue from numbers at the commencement of the realization of the second dimension because a movement or a measurement is required. Say hello to the friendly transcendental number Pi.

As soon as you construct a circle [the meaning of a circle takes a while], you may get an itch to divide a circle into exact and even segments. This brings the power of geometry home. Oh, and the stars, too.

Once you know which numbers can divide a circle, you may want to tackle the squaring of a circle. Transcendentals and irrationals can be tamed but this is a new thing where magic becomes a component of geometry. Then again, atoms do it every day.

We also have a summary of Pythagoras' accomplishments on our credits page.



Fundamentally, numbers come first. The Tetractys with 1, 2, 3, and 4 dots also refers to 0, 1, 2, and 3 degrees of independence because 1 is associated with a dot (geometric point), 2 with a line, 3 with an area (plane), and 4 with volume (some say solid). Degrees of independence are, again, numbers. Degrees of independence form the fundamental constructs for geometry. Yes, each degree of independence has its own geometric rules and, for example, you need to understand the squaring of a circle if you want to move between 1D and 2D. Overall, you need to move from 0D to 3D and back -- that is, you'll need to know what it takes and how to go about it. Topology is about building things as it is a static subset of geometry, and could be seen as the 'placement' in the ancient Greek tradition or the 'measurer' of the Mayan Hunab Ku it's about the real numbers in general. In addition to placement there is also movement (and the second book Stars and Rings will get into that ~ mid 2013).

Dots of the Tetractys plus one (blue) dot form hyperstates.

Hyperstates are about the creation of the real environment that is tractable (non-chaotic) and that has its origin in the virtual domain, which is oftentimes called hyperspace or ether or "fire." Hyperspace is intangible (invisible) and is not explicitly shown in the HyperStates picture because HyperStates is about the one single facet of the triangular pyramid where all real (visible, hard) topologies manifest. We can represent the hyperspace as an eye and put it on top, which results in a triangular pyramid of tetrahedron with the eye at the apex. Presently, the HyperStates picture includes three axes that project the real plane from the hyperspace or from one's eye. The projection from one's eye reveals the individual and the collective way of building the real world on the real plane because it is our cumulative knowledge that continues to compute and create the ever growing and organized universe.

Although the solutions for the axial parameters converge toward integers, the integer value is not fully reached but their sum adds up to three exactly*. The sum being the integer three stems from the fundamental maxim of the real domain, which deals with tractability limits. (Tractability is a subset of computability.)

If we define the triangle as the "summing triangle," then the outer hyperstates would be on the triangle's edge. The triangle can also be defined axially as the "bounding triangle," in which case the outer hyperstates would be just inside it. In either case, all hyperstates are in the plane formed by the triangle and that is the reason for calling the triangle the template of the real plane. Technically, each and every point on the real plane is a solution but because the plane is a logical plane, there is no measurement metric associated with it. The creation of realities, then, has no prescribed size. If we apply integers to the real plane, the summing of 3 out of 4 numbers (0, 1, 2, and 3) in a way that results in integer three yields ten integer-summing sequences, which correspond to 10 hyperstates.

The presentation (orientation, rotation, view) has no overwhelming preference. The spherical galaxy hyperstate is at the top corner as magenta. It may be said that spherical galaxy's organization is closest to the "harmony of spheres." However, bodies have no independent movement in hyperstate 3,0,0 because they all move in synchrony. Orbits also are not purely spherical but have a large degree of symmetry. Hyperstate 0,3,0 could also claim the top and the sun is indeed an awesome sight — yet the sun can also go nova. Hyperstate 0,0,3 is a simple one and, while common and uneventful, there is opportunity there for new growth because the growth is mostly localized. In the alchemical tradition a discourse on orientation would fill many pages. [If your ship is in hyperspace, you do not want to materialize in 0,3,0. But if you are building a centralized organization, that is the place to be..] Hyperstate 3,0,0 has periodicity while 0,3,0 does not. Hyperstate 0,0,3 has translational (linear) repeatability that can be called periodic as well since the velocity there is constant. In summary, Hyperstates' triangular plane has neither the "top" state nor a direction in which it is pointing (up or down or sideways).

Separating the micro and macro

Macro: Pythagoras' Tetractys guides and even directs the creation topology on the macro scale. Here, the real and irrational numbers are prominent while the virtual numbers deal with the periodic reduction of the gravitational wavefunction.

Micro: Organized topologies also hold on the atomic scale and it is likely we will find additional hyperstates on the Tetractys plane in addition or subtraction to those shown here. Dynamic and event-driven hyperstates define the micro of the atomic scale. On the atomic scale the hyperstate realization hops around because the plurality of electrons can instantly transition into another tractable state on the real plane. (The real and temporary states are at times called eigenstates. In the macro the hyperstate realization stays put -- see below.) Squaring of a circle is unique to micro and the transcendental-irrational number interplay is the most prominent. Much promise is in understanding the tractability of the core, for completely new elements could be created. A [easy] case can be made that the core's tractable solution is periodic and that its period is constant.

Tetractys is best applied at the macro. (I'd use Hunab-Ku for the micro.) Returning to the cosmic (macro) scale, the manifestation of a particular hyperstate is the solution – that is, a particular hyperstate realization is the last step in itself. Hyperstates do not change or evolve because they are the solutions ("final condensate") resulting from unbounded and concurrent computations of the whole. One can argue that a gradual change from a dual-sun system to a sun-planet system can be called evolution but in fact it is a two-body system that is the one and the same hyperstate. New hyperstates, though, are created every day through hyperspace and that is the norm (Pythagorean spring, source) of universe creation and expansion. When a sun parts into a dual sun system a new hyperstate is formed. When a new sun forms from ether (Atum's mound, scarab-spin, Mayan Hunab Ku spider-spin), another new hyperstate is formed.

Because the macro hyperstate is the final solution, there are no migrations or evolutions. There may be logical similarities such as a discus, ring, or one-body orbit (moon or planet) topology states but there are no in-betweens and this is because the solutions coagulate around integers. We will not find two or more planets in an identical orbit where there would be a gradual migration to a ring of planets. Similarly, we will not find a planetary ring that would gradually become a moon because, again, the ring is a particular hyperstate that is a final solution yielding a ring topology. Finally, we will not find a discus that would reduce into a body. When the hyperstate is not composed of (near) integers, as is the blue dot, we see greater topological similarity as in the bar and spiral galaxies.

A good way of looking at hyperstates is that it is a framework for real solutions. A particular hyperstate is a tractable topology and such topology (such solution) is independent of scale. We can find a particular hyperstate manifestation in a planetary system, not in a solar system(s), but then it forms again in a galactic system. Scale independence results from nonlocal computational aspect of quantum mechanical gravitation having, as one of its characteristicts, an instantaneous reduction of its wavefunction. 'Independent of scale' is not the best phrase here. An entire solar system shrinks to a point with distance and despite the scale shift such solar system continues to serve, this time as a point, on the Tetractys template. (In the Quantum Pythagoreans book the independence-dependence of relationships is developed in depth, including the reversible and irreversible property of a relatioship.)

A nice way to conclude the micro-macro section is in discussing just one aspect of the power of numbes. It is a logical process and it centers on the division of the Unit 1 {May 18, 2010}.

    In the micro the Unit 1 is the orbital circumference (circle/ring) and the consequent geometry that divides a circle exactly. This brings in the circumpositional numbers. The virtual numbers give you the wavefunction for the nonlocal electrons in their orbitals. Somewhat similar situation happens for the core. The squaring of a circle calls for adding irrationals into the mix to reconcile the curving energy of orbitals and 1D energy of photons that affords a measure of stability. All irrationals are via the Pythagorean Theorem.

    In the macro we divide the Unit 1 as follows: For Unit distance we work the 1D aspect (length, distance) by dividing Unit 1 with any real and everyday number and we also bring in the irrational distances (yes, all irrationals are in 1D). For energy we divide the Unit square (2D) via the geometric mean into as many squares as we want. Okay, you figured out you also need to to divide the Unit cube into at least two cubes and this will make a nice brainwork for some (think Archytas). Yes, you have to do it geometrically because, well, how else would you solve for planetary orbits? (Think Kepler but stay with geometry.)

    To top off the Tetractys, the 0D (a point) cannot be divided but it makes The One, the most important and universal geometric construct for the creation of the radial symmetry, which is applicable to both the macro and the micro. There is much more to The One and the knowledge about The One serves as the determinant the universal credentials if you will for anybody trying to tell us something about the workings of the universe.


The book you will thoroughly enjoy


 To Publisher...

Of all things Pythagorean

Applying the Tetractys, Quantum Pythagoreans book describes and explains all observable cosmic topologies. For example, the hyperstate 1,1,1 is the bar galaxy. The spiral and bar galaxies differ in but one parameter. All topologies are particular stable solutions and they do not "evolve" from one to another. Because all topologies are covered in the book, Newton's two body and Kepler's orbit laws are included as well.


Pythagoras, son of Muesarchos, a Samian, who was the first to call this matter by the name of philosophy, assumed as first principles the numbers and the symmetries existing in them, which he calls harmonies, and the elements compounded of both, that are called geometrical. And again he includes the monad and the undefined [nonlocal] dyad among the first principles; and for him one of the first principles tends toward the creative and form-giving cause, which is intelligence, that is god, and the other tends toward the passive and material cause, which is the visible universe. And he says that the starting-point of number is the decad; for all Greeks and all barbarians count as far as ten, and when they get as far as this they return to the monad. And again, he says, the power of the ten is in the four and the tetrad. And the reason is this: if any one returning from the monad adds the numbers in a series as far as the four, he will fill out the number ten; but if he goes beyond the number of the tetrad, he will exceed the ten. Just as if one should add one and two and should add to these three and four, he will fill out the number ten; so that according to the monad number is in the ten, but potentially in the four. Wherefore the Pythagoreans were wont to speak as though the greatest oath were the tetrad: 'By him that transmitted to our soul the Tetractys, which has the spring and root of ever-flowing nature.' And our soul, he says, is composed of the tetrad; for it is intelligence, understanding, opinion, sense, from which things come every art and science, and we ourselves become reasoning beings.

Aetius, Placita, 1st Century CE

Tetractys is not a component of the cabala. Any method could be used in an attempt to explain Tetractys, yet the original oath of the Pythagoreans that speaks of the Tetractys has the most authentic root of its meaning.

The interpretation of Tetractys with cabalists' tetragrammation methods, for example, does not advance Tetractys symbolism and does not give Tetractys a measure of application. Some cabalists attempt to modify or corrupt Tetractys by claiming that the corner states are not real. The corner states (3,0,0; 0,3,0; and 0,0,3) are very real. At times cabalists substitute Tetractys dots with Hebrew letters without rhyme or reason -- and end up with graffiti. Putting IHVH or YHWH or whatever in place of dots -- which some may think are geometric 0D points -- could also be seen as an attempt at obfuscation, derailment, blocking a path to the truth, or just plain corruption. Cabalists do not understand Tetractys and their frustration is reflected by their attempt at corruption.

Cabala contains no geometry, which may have been the intent. (Geometry issues from numbers and is the first step toward creation.) Cabala partially addresses modulo math through summation and recycling of numerical values, which, incidentally, is very Pythagorean and precedes cabala by hundreds if not a thousand years (Aetius, Placita, 1st C. CE).

Quantum Pythagoreans book explains the purpose and the need for the transformative agency, which is altogether missing from the stick-to-orthodoxy Old Testament.

Mystical Pythagoras
Mystic or mythical label stems from two aspects of the Pythagorean knowledge

1) Keep the knowledge foundation broad for the widest possible applicability.

    Tetractys is but the triangular numeral ten. Take it from here.. [What? This is worse than one-hand-clapping koan!]

    Even though the hyperstates issue from Tetractys, there is no attempt to replace Tetractys with the HyperStates symbol. Tetractys serves as the platform for geometry in general (0, 1, 2, and 3D) and, also, the ratios of the triangular progressions 4:3, 3:2, and 2:1 form the basic harmonious cords of the Western musical scale. Forced substitution of Tetractys in effect cuts off other applications and can be seen as corruptive. Yet, in its naked (virgin) form the ten dots of Tetractys seem impenetrable and, to some, mystical.

2) The richness of the universe springs from the Pythagorean Tetractys and each individual can grow and use Tetractys in different ways toward some new solution.

    The mystical part is that Tetractys works for some but does not work for others. It happens that the person encounters a wall and will need to take a break. You will penetrate "the veil of secrecy" with patience and possibly with some background studies. If you apply Tetractys toward some useful solution, (you can publish if you so desire, but) you want to return Tetractys to its original form.

If you wish to study (or be called upon to), you will need to overcome the possibly genuine frustration, obfuscation, reduction, denial, or corruption of Tetractys by others.

One more thing. I've read that some cabalists meditating upon Tetractys had gone insane. You will note that Tetractys is about creation, not corruption.

 The Tetractys of Pythagoras -- The virgin form

Pythagoreans do not dwell on and do not fancy descriptions of reality because the Pythagorean way is about the creation of reality. The making of reality is not about pretensions but it is about the understanding of pretensions. Pythagoreans do not claim that some parts of the universe are delusions, either. The creation of real and stable and objective systems, then, requires a thorough understanding of the virtual domain that deals with the infinite superposition and relation of virtual energies, each of which brings in a measure of relevance. Conflicts resolution such as those stemming from a war, global warming, or financial markets instability call for stable solutions that -- short of outright destruction (usually possible) -- are not obvious or straightforward. From among the myriads of plausible answers Pythagoreans find the ones that are executable and, in the end, tractable.

    * If you are familiar with incommensurable/irrational numbers and how their discovery created consternation among some non-Pythagoreans, consider that it is the sum that yields the hyperstate solution. The individual components (addends) are integers, near-integers, rational numbers, and reduced (truncated, rounded, fractioned) irrational numbers that is, the individual components of the hyperstates are real numbers. (Quantum mechanical aspects in the Quantum Pythagoreans book deal with yet another mechanism besides truncation and rounding that of the ancient Egyptian fractions.)

    Perhaps the best way of starting on irrational numbers is by taking a look at the golden ratio with one original application, or It cannot be exact from March, 2005, DSSP topic [advanced].


What is behind The Numbers

Hyperstates framework is difficult, though not impossible, to figure out. The thing about hyperstates is that in the process of figuring it out you will find the answer to something that is important to you. Your solution may be personal or it may have wide applications. While it is possible to teach "everything" about hyperstates, the idea is that your solution is the most important solution.

If you look at the missing axial parameters as a mystery, you may be able to figure out the parameters and thus the maxim. The secret behind the secret is not the lock that would be in front of the secret, for once you figure it out you will know there is no better lock than the engagement of the mystery. In fact, there is no lock but an interlock that creates order out of the competition of the triad.

The Hyperstates interlock is so robust it cannot be stolen or given away, so extensive it cannot be memorized, so unique you will know right away, so practical it can be applied every day, and so logical that even a wrong answer bespeaks of a correct concept.

Should you come upon a person who claims the earth is flat and square while showing you a triangle, you can laugh and enjoy the conversation while, maybe, you will be able to appreciate there is more than one way to the center of the maze. You may want to move from things and into relationships. There are benefits in a discourse on square portholes when it comes to building ships and transforming new energy sources.

Overall, Pythagoreans may find HyperStates a significant yet natural extension of Tetractys. Perhaps you can wear it as an amulet if you feel the organizing power of Tetractys.

 Enumerated Tetractys of Pythagoreans -- Hyperstates



One App: The Barbury Castle Tetrahedron {July 17, 2011}

 Picture credit:  Illustration credit: This 1991 crop circle has thrilled many people, myself included. In it I saw the triangle of the Pythagorean Tetractys but at that time it was not extended into the 3rd dimension. The Secret Teachings of All Ages by Manly P. Hall has sections on the Pythagorean math and that helped a lot.

I am returning to this topic {on July 17, 2011, 20 yr anniversary} as I am working on my second book. I am not soliciting input -- it is a bit late for that. I'll be commenting on what's on the Internet about the The Barbury Castle Tetrahedron because some people are close.

If we don't count generalizations ("great shift," "warning," "new age") or simplifications ("tree of life" made of a bunch of triangles), there isn't much remaining on the Internet about this great event. Are there no Pythagoreans on this planet who'd care to comment? (Pythagoreans who didn't work for the govt in a similar area and can talk?) There was a couple of people who seemed to have stopped communicating about their experiment (Chris and Jean) after they built the tetrahedron and got some lift out of it -- and so there is an aura of mystery here as well.

 Enumerated Tetractys of Pythagoreans -- Hyperstates  Picture credit: On this web page the Hyperstates illustration (on left) is also a tetrahedron and it is not difficult to see that one facet of it makes the wonderful Tetractys. This tetrahedron symbol is explained in the Quantum Pythagoreans book in all dimensions including 0D. The axes are identified in the book and metrics included as well, even though the metrics are bit different than what you might be used to, for they are the degrees of independence.

The Barbury Castle Tetrahedron has a circle at each of its four corners (vertices) and no English to describe the axial parameters. Here is the first and easy hurdle. The symbol inside the circle at each of the corners describes the qualitative state (or qualitative function) of that point. The whole thing is a logical drawing and it doesn't have to be drawn exact (and it isn't). Well, I shouldn't have said this is easy because the interpretations I read speak of circles as physical spheres and axes as physical tubes. Suddenly the situation seems to have gotten complicated because the two guys (who are quiet) have built the tetrahedron from tubes and spheres, pumped microwave energy into it at one end point -- and got a lift out of it, too. So there is no way I could talk you out of the impression that they are onto something real and really interesting. But this site is more into true knowledge integration (using your right brain) rather than proving or disproving something. The important part is that but one corner of the physically built tetrahedron got the lift. That also tells me these two guys really built it and the picture I saw is not just a mockup. In fact, the corner that got the lift is the one that is on top of the tetrahedron in the picture on right (it's a six-ray swirl, which is really the ± 3D Cartesian coordinates under rotation at the origin).

I found but a couple of people who were or are working the Tetrahedron on the Internet. Here are my comments:

Quote by Harold Stryderight:
"The symbolic meaning of its "ratchet spiral" seems less clear, but could mean perhaps that some internal energy is quantized as one approaches the vortex centre." Bingo! Oh, if you think gravity is QM based, don't say 'space-time' unless you want to tell me you are confused.

Quotes by Jeremy Stride:
" .. the watersphere is often shown as a full circle with two rings .. .." Here, Jeremy is talking about the center circle with the two rings. If you were to substitute 'water sphere' with 'ether,' you'd be spot on. Then again, water has been used in the past as a word for ether ("spider water," "fire into water") and I think many people will understand it just right. Think about the two rings some more, particularly as they are marking the axes.

" .. the watersphere goes at the center of the large tetrahedron structure!" Ne-eh. It's right at the top at the fourth point of the tetrahedron. It's infinite and 0D at the same time. Regardless, I like what I read. Think about the degrees of independence.

In summary so far, the Tetrahedron geometry (based on the number 3) deals the our world's reality. This includes gravitation, atomic construction, and "takeoffs and landings" of interstellar travel. The four sided pyramid geometry (based on the number 4) deals mostly with interstellar travel and the balance between 3 and 4, which is the balance between the real and virtual domains. It's the same thing as talking about Yang-Yin balance but I prefer English. The way the balancing act works is suggested by our pentagonal pyramid.

Living with Numbers

Pythagoras and the Pythagorean tradition puts numbers first. This may seem difficult to some and indeed Aristotle had a field day poking fun at Pythagoreans. The basic aspect to 'All is number' is that each number can be constructed -- that is, each number can become. Each number, then, can be actualized. A number is not at the core, it is the core.

A number can be written on a piece of paper and then the number is a representation of something real, irrational, or transcendental. (Irrationals and transcendentals enable the formation of virtual variables.) Yet numbers can be applied to actually come alive and that is the meaning of 'All is number.'

Pythagoreans not only use numbers to measure somebody else's creation -- they create new stable and alive entities with numbers. A good question is: What is the number or numbers the human is made of? [Actually, it is a root of a number.]

1) Real numbers
represent real variables that come from real (tangible) things. Real numbers are all positive numbers that are naturally finite in magnitude and unbounded in multitude. Some applications need to differentiate between unbounded [masculine] and infinite [feminine] and real numbers are unbounded but not infinite. (Riemann is an excellent source on this and without the Pythagorean male-female esoterica [yet, you want to be well grounded before applying Riemann's North and South poles, as both Cantor and Gödel got sucked in].) For real numbers, zero is not necessary but is included for zero's benefits prevail over formalism. [You will know you are over-doing it when you talk mostly about things you don't have.] Real variables result from a measurement of real entities. In reverse, real numbers create and constrain real things. The result becomes a real value that is, an exact magnitude. Real numbers relate to other (mostly real) numbers with operations such as +, -, *, and ÷. Commutative property holds and is applied through the (real) rules of arithmetic geometry can do the operations but for real numbers arithmetic and algebra are sufficient.

Real numbers also spawn the degrees of independence (some say degrees of freedom). Pythagoras' Tetractys also includes the representation of the degrees of independence as four levels: Top dot is 0D (a geometric point), two dots on the next level is 1D (a line), three dots on yet another level is 2D (a plane), and four dots on the last level is 3D (a volume). Pythagoras' Tetractys also creates geometry through geometric constructs of degrees of independence. The most powerful finding here is that each level of increasing freedom provides a different context within geometry. This aspect is not presently understood, as all mathematicians try to treat geometry uniformly and "as a whole" where dimensions are "just trivial extensions." They are not. For example, squaring of a circle is a quandary that's right between 1D and 2D and is still waiting integration into mainstream.

Pythagorean geometric concepts provide powerfully simple answers to very complex problems. The question "What is and where is the difference between a point and a line?" seems almost impossible to answer objectively. This is the same question as "what is the minimum separation between two points so that we can connect them and call it a line?" Yet there exists a real answer that also yields insight on the smallest possible separation between atoms. Yes, infinitesimals had to wait until the 17th Century to bridge the 0D to 1D.

 Improved Tetractys of Pythagoreans -- Hyperstates

In the generally commensurable world of Euclid, the size of one thing can be used to determine -- that is to measure -- the size of another thing. The real world of Euclid is, therefore, commensurable. All measures in this world then also have finite (or repeating) mantissa. The Pythagorean discovery of incommensurables also points to "another world," which was aptly, albeit partially (and negatively), described by Proclus.

 5 pointed star and pyramid in Salt Of The Earth design  5 point star - Golden Eye Pythagoreans liked some say revered the five pointed star of the pentagram. Relax with our designs, all centered on the five-point star. View our select designs or visit our store at Zazzle (.com/Mike_Geo) and actually relax in them. This particular design on the right constructs the pentagon and the great pyramid from circles that are in the golden proportion. All our designs come from nature that is, they are actualized in nature.

2) Incommensurable numbers
form a class of numbers in addition to real and virtual numbers. Incommensurables are a tough crowd to understand. First, there is a separation of incommensurables into irrational and transcendental, which is based on their origin; irrationals arising from straight (1D) geometry while transcendentals arise from curving (2 or 3D) geometry. Among irrationals there is also a separation based on applications. Incommensurables have an infinite sub-unity portion of a number (mantissa) and, furthermore, numbers in their mantissa do not go on repeating, individually or as a group.

Mainstream math guys got it mixed up and to them the transcendentals and irrationals are the same. However, transcendentals are incommensurables that exist on a curve or, by the same token, they exist in 2D or 3D. Pi is the prime example of a transcendental number. Irrationals are incommensurables that are straight that is, they exist in 1D. Construction-wise, transcendentals need a pyramid for their actualization while all irrationals are constructible through the Pythagorean Theorem.

3) Virtual numbers
represent virtual (intangible) variables. Virtual variables enjoy exclusive existence in the space of incommensurable numbers (much more on this in the Quantum Pythagoreans book because it leads to working with geometric constructs such as the pyramid). The virtual variable comes from (or creates) a virtual entity just as a real variable is associated with a real entity. Virtual numbers center about zero and include zero, positive, negative as well as imaginary (i-based) numbers. Virtual variables exist in infinite superposition and each virtual variable is processed as a whole (in one, two, or three-dimensional representation) by such operations as projection or reduction, but most operations establish the leadership (independence, "strength") of a variable. Presently, however, the projection, leadership and to great extent reduction operations are not understood very well and operations among virtual numbers are limited to matrix arithmetic (where the commutative property does not hold).

The group theory can be applied to establish transition operations between real and virtual numbers because the operation of transformation deals with both the variance and invariance of number's properties. (Pythagorean even-odd grouping of numbers was a good start.) Virtual numbers' positive and negative values are generally (but not always) subjective. Virtual variables "fold in about zero" when these transform (reduce) into real numbers. This is analogous to folding-in of a hand held fan while the pivot (zero) becomes excluded.

A scientist has difficulty understanding the virtual variables because the QM wavefunction is treated mathematically as but a technical parameter while its actual (though virtual) existence is denied.

In the Pythagorean tradition via Aetius, the virtual number is most likely the 'undefined dyad' and I interpret the characterization "undefined" as 'nonlocal' or 'spread out' in the present day quantum mechanical context of a wavefunction having even symmetry -- such as a photon. The 'dyad' is the even wavefunction based on the number 2 of the even (axial) symmetry.

4) Circumpositional numbers
Here is where geometry shines. The only way to explain and prove harmonious and disharmonious tones is through these numbers. Circumpositional numbers divide a circle exactly and they are treated in introductory fashion on our pentagon page. Complete explanation and a formula for tones that are either harmonious or not are in the Quantum Pythagoreans book.


transcendentals create closed -- that is atomic -- topologies. Applications of irrationals are 'good' or 'bad,' and absolutely so. Some irrationals such as the SQRT(5) form the golden proportion and are life-supporting. Other irrationals may not be agreeable to humans. Incommensurables are not well understood throughout this planet's (written) history, other than that the incommensurables are important in transformations and extraction of cosmic energy from ether (Viktor "Cool" Schauberger, Reiki). Proclus' description of irrationals is probably the best description so far, although he focuses on the undesirable aspects of irrationals.

The Western preoccupation with reality handicaps incommensurables' applications. The free energy (zero-point energy) effort, however, got a good start in the US and may yet rebound. In the East, incommensurables are used mostly for personal empowerment and healing. Incommensurables can be actualized (come alive) only through its construction. So, the square root of two is an irrational number that is, however, not actualized per se. Moreover, some people think 1.41421356 is the same thing as, or close enough to, the square root of two, but such number is a real number and cannot be actualized as an irrational number as it no longer carries the infinite mantissa. Incommensurables are in the virtual domain but their means of construction can be real, as Pythagoras discovered. There are other aspects to the actualization and this introduces yet another subset of irrationals. Overall, a very intriguing number group.

The squaring of the circle attempts to resolve the differences between curving and straight geometries -- that is transcendentals and reals -- think atomic construction with adaptive orbitals.

Incommensurables consist of transcendentals and irrationals. Another way of differentiating transcendentals from irrationals is that transcendentals do not come out as a solution from an algebraic equation. For examples and additional discussion see the section on (in)commensurables inside the article on the golden proportion.

Complex numbers is a compound group composed of real (say, a) and virtual (say, b) numbers. Popularized by Gauss, the general format is a + ib, where i is the SQRT(-1). The i is shorthand for ccw rotation by 90 degrees, which facilitates transformation from the real to the virtual domain. Numerically, i signifies that the transformation has taken place but such operation is a paper operation. Reportedly, to be actually able to facilitate the transformation between the real and the virtual domains (and do so reversibly) calls for unusual skills or technologies. Anyway, the utility of complex numbers and its notation is that real and virtual numbers (a and b) exist side-by-side but do not freely interact -- much like apples + oranges remain apples + oranges. The summation (superposition in general) of real and virtual numbers remains as summation and does not advance. Yet, certain operators such as multiply do mix it up and a transformation takes place.

Real numbers are numbers having a naturally finite mantissa. Virtual entities are comprised of incommensurable numbers with infinite mantissa and have no real -- that is, tangible, existence. Virtual entities exist as two kinds of energy [advanced].

As a Pythagorean, you want to understand the meaning of all operators such as multiplication as they work in nature.

Mathematicians' understanding of the incommensurables via Dedekind is incorrect because these numbers' mantissa must be infinite if the incommensurable number is to remain (or become) incommensurable. If bounded in the magnitude of the mantissa (if the mantissa is cut), incommensurables irreversibly become real (commensurable, finite) numbers.

Dedekind could easily earn the label of the dumbest math guy of all time. Yet, he also serves to show the basic derailment of math today -- intractability. Intractability is well understood as the property that does not offer solutions in real-time. Intractability, once encountered, is something that should serve as the dead-end sign. Intractability means that a solution is available eventually, but not now and not anytime soon. In an example, code-breaking that uses exhaustive methods does not yield a solution anytime soon -- perhaps in a thousand or a million years. In an intractable situation the exhaustive methods are, by themselves, useless. Dedekind does one up on that. He uses a procedure in his proof that is intractable and the proof becomes valid in the infinite time in the future. That should be sufficient to have Dedekind declared a winner of a prize such as "The Deep Pocket That Was" [all governments would make a short list on that].

On a practical level, a manager (hopefully not yours) will talk about the future .. .. and the next time about the future as well. He or she need not talk about accomplishments because accomplishments are also in the future. This is your classical intractable manager and sooner you leave his organization the better for you.

Relating Numbers
While there are plenty of differences between animate and inanimate entities, the organizing principles hold for both. People will continue to emphasize whole numbers as well as their sums, ratios, spatial (geometric) relations, and logical combinations. You may note that Pythagoras uses one for a point, two for a line, three for a plane, and four for a volume (solid or irreducible entity). Actually, a point is zero dimensional, line is one dimensional, .. Get your reference straight when someone talks about four or three or..

Numerology is not strict and, for example, two entities produce one relationship. Numerology can be context-dependent when, for example, the interaction among three variables results in tractable matter while three bodies are in general chaotic -- the number three can stand for both stability and chaos.* Numbers, then, do not exist only in standalone fashion because numbers also spawn the operators (relationships) and degrees of independence. Perhaps the best example of the operator is in the definition of Pi.

Once a group of numbers reaches a stable system, Pythagoras calls it the Monad. Monad is 'one-sum', a unique summing sequence or grouping of numbers that relate through the operators. The simplest monad is a triad -- that is, you need at least three numbers or three variables to make something lasting out of it. Indeed, three parameters build the whole real universe. The mystical aspects are treated in alchemy, which deals with transformations and invariance -- that is, a transforming or "becoming" monad has some of its numbers variant and some invariant (see group theory). Monad is synonymous with 'object,' 'entity,' or 'conglomerate.' A Monad is always a real entity that is commensurable with any other monad. (Self-Test:-) If your ears perked up on this paragraph, you are doing well. Monad is also the first counting real number one issuing from the first real thing.

The numerology (coming up) has a qualitative division on the interpretation of numbers as these apply to the real and the virtual domain. You may note that letters issue from numbers in that the vowels have even symmetry**. In all Latin vowels the even symmetry survives, although in the letter 'E' the even symmetry survives via the horizontal axis.

    * Religion and mythology deals with this through the multiple talent of the Personalities, or Aspects, of gods. Shiva can be creator at times and destroyer at times.

    ** Odd symmetry is a symmetry about a point (origin) while even symmetry is a symmetry about a line (axis). Symmetry contains reflected duplication about axis or rotated duplication about a point. All Latin vowels preferentially carry even symmetry. Pythagoreans call even numbers feminine and "inclusive" while odd numbers are masculine and "exclusive." The Tibetan alphabet is highly developed along both symmetries.

We apply incommensurables/irrationals in an article on free energy because geometries have a pivotal role there. When you hear 'pivotal,' think spin. We also have a page on ether.

Pythagorean College Numerology

Number or Shape

Real Domain [ He said ]

Virtual Domain [ She intuits ]

0 (zero)

Naught, Nothing (not a thing), Empty, Vacant, Symmetry about a point, Point (tip) of the short sword or spear, Dot (molecule, atom but think smaller). Indivisible or irreducible point (geometry), 0D. Transitions to infinity in the virtual domain [advanced].

Center, Balance, Infinity, Neutrality, Locus, Crosspoint, Intersection, Air (loose atoms, i.e., smoke, perfume), Into zero (banish operation), 0D (point)

The sword (tip) of Tarot.

1 (one)


For Pythagoreans, Unit 1 is the Great Divide between cosmic and atomic

Unit 1 in geometry: the shortest length (or distance) in a particular construction

The Monad, the first real thing and, therefore, the first counting number. (Monad itself is composed of other numbers -- hence the differentiation between cosmic and atomic.)

One thing, unity, single. Edge of the long sword (one-dimensional object), Solid line, Cut into two using the edge, 1D

Unity, entity, conglomeration. Symmetry about a virtual line (an empty slit is a virtual line). Single joint (or a joining operation such as force) between two (axle, wavy wand, baton, snake, string), Drawing down, Lightning across ("fire" operation), 1D

The wand of tarot

2 (two)


Cutting into two

Movement, Growth, Spin-off, Coin, Cover (blanket), (X-Y) plane, Shield, 2D

Real and Virtual duality (3 vs. 4)

Squaring circle duality (1D vs. 2D). Circle on top. (Circle is masculine.)

Yin-Yang duality with Yang above. Two points in the Tao symbol well suited for dantien (hara in Japanese, couplex in English), a "mystical" point between heaven and earth (or yin and yang, or virtual and real)

Doubling by addition of the same. (This is important. The best way of seeing this is in the biology application of cell "division." The doubling operation "divides" the cell about the virtual axis and two cells arise. Contrast this with the male division, which is a real operation and the cell is physically halved.) Tangents here on the golden proportion, left-right brain separation, energy symmetry..

Amulet, Relationship. Table (anything flat), Plate. Earth. 2D

The pentacle of Tarot

Virtual and Real duality (4 vs. 3)

Squaring circle duality (1D vs. 2D). Square on top. (Square is feminine.)

Yin-Yang duality with Yin above

3 (three)


Monad. Stability. Volume. Three-sided pyramid (tetrahedron). Cube [cube is not representative of number four -- the volume is its principal character here]. Realization, manifestation, actualization. Projection (into real). 3D

Three is very real despite the "spiritual" constructions of Trinities. Monad is the first real thing that, moreover, consists of at least three entities.

Also, once you accept there exists a duality, the next question is, "if duality forms and does not merge into one, what's keeping it separated?" The separation is not only about isolation but also about the transformation from one to the other (and vice versa). The transformative element becomes the third element resulting in the trinity (also think diagonal).

Cauldron. Water (incompressible but adaptable volume). Containment. 3D

Cups of Tarot. Tarot is working the right brain if you use the classic square spread -- think pyramid base and then move up to 3D.

Holly grail (missing from the Bible and so it had to be reintroduced via some back door).

Intelligence is in 3D

4 (four)


Square, Rectangular, Ratio-metric, Cartesian coordinates.

Mandala: By definition based on number four. Visualization of right brain functioning (usually women dominate here but [most] men use the right brain as well). Its okay to put stars with different point count at the center but if such point count does not divide in a circle exactly geometrically, dysfunction is likely. These numbers divide a circle perfectly.

Incommensurable squares (very special Pythagorean concept, pyramid construct)

Non-causal (relational). Informal

Four corners, Cardinal directions, Four-sided pyramid, Stability, Computability determination (tractability) [very complex, different levels plus infinite superposition], Two pairs (X vs. Y) of two virtual variables (with positive and negative values), Quadrature, Infinite superposition, Projection (virtual, square pyramid).


5 (five)


Pentalpha, Pentacle, Pentagon. Friendship, marriage (dual ring, pentagonal pyramid). Health. Seal (protection), Root of life creation (pyramid guts).

Division of a circle by five is ancient, exact, and originally a secret [Pythagoreans declassify things, too.]

(Square) root of five takes you to the golden proportion -- a non-counting aspect(s) of numbers.

Pentacle. Safety, closure (interlock), bonding. {Marriage and bonding is multi-faceted topic and the best article on that is Venus-Earth merged orbit

6 (six)


Two triangles, one pointing up (say light blue) and one pointing down (say navy blue). Front triangle dominates. Triangles cannot merge: May be confused with dantien, which we gave the English name couplex and which is located near the navel. (There is much work that would have to be done for the so-called Star of David to have the functionality of the couplex. As it is, the Star of David is closer to being earth-earth (earth-bound) than being about self-organization. The diagonal is there but the count is incorrect and the point of transference is not there.)

Several overlays and perspectives with hexagonal geometry can result in confused symmetry (hence hex or hexing), which needs to be understood to be avoided or neutralized through the understanding of all (and finite) possibilities.

Because a circle can be divided by six exactly, you can draw a six-pointed star in a circle. However, a six-pointed star does not harmonize and is not derived from planetary orbits (see right column). As a single star -- derived from 3-pointed star -- the symmetrical six sided arrangement in a single circle is (could be) associated with potentially unbounded energy accumulation within the circle using 2D and possibly 3D Fourier -- important to know what form of energy we are dealing with -- it can blow, too. (On the back cover of Quantum Pythagoreans book, the drawing called 'The Heart Of The Sail' hints at this issue.)

Heart charka/chakra is not a six pointed star but a quadrant-based mandala or cross or Cartesian coordinates or locus of the virtual domain. The exchange (transformations) of earthly and divine energies happens through dantien, aka couplex. [Buddhists need to separate numbers three and four, and revisit charkas' symbols. You do not want to merge three and four. Having said that, Buddhists figured out the derived ("by convention") existence of time and maybe they think they do not need to tinker with numbers.]

While six is reported to be "a perfect number," the experience does not support it. One can make number six unlucky because it is "behind seven, nine, eleven, and thirteen," and none of these numbers harmonize with 6. (13 passes the octave for 6.) Number 15 and all even numbers cannot help because the duo reduces the hexagon to a triangle. A six-pointed star does not derive from planetary orbits. This means that the six pointed star cannot be made in the dual circle.

There is no 7/6 ratio on the musical scale but there is 6/5 ratio (G/E) [try it together to see if it sounds harmonious -- it should be]. Much fun can be had with planetary orbit ratios and the tones and the multi-pointed "orbit stars" they make.

7 (seven)


Seven notes (steps) in between the octave

Circle is not divisible exactly by seven -- important just for this attribute. Seven and nine are the only numbers between 2 and 10 that cannot divide a circle into geometrically-equal segments. Seven-spoke wheel should be the strongest!

Seven is the first number that does not divide a circle. 7 may be "implicated" in force creation.

3 (Real) + 4 (Virtual) = 7 parameters for self-organization: All there is. Whole, holistic

3 vs. 4 of the alchemical tradition. While separate, both exist together

Also, 3 + 4i can be seen as "seven" even though the real and virtual always remain separate but a transformation between the two can be had

8 (eight)


Octave, musical. Earth-Mars orbits interlock in an octagon, which is musical (harmonious).

Check to see if the construction of eight is: incremental (octagon), skipping one point (two squares), skipping three points, or skipping five points (octacles). [Muslims have a clue but they do not know the root. One of these is disharmonious.]

Four pairs of inner states in the four-sided pyramid (on third level).

Cardinal and semi-cardinal directions. [Union Jack's center can be seen as Couplex (Dantien).]

Semi-cardinal (diagonal) direction is about transformation. [Catholics think cardinals and semi-cardinals are about the Sun and possibly Venus but the doctrine cannot let them see transformations -- particularly individual-empowered salvation.]

9 (nine)


Nine is a good candidate for a so-so number. [Nein? Okay, how about tic-tac-toe.]

Need to differentiate between the areas of squares and the diagonals of squares.

Nine is a composite number but cannot "decompose" (divide) circle evenly and thus nine is not musical. Nine can have hexing (confusing) qualities because it has both "forbidden" and composing qualities and it is a square number. Metaphysically, the number nine is a "traitor" or "spoiler" [Chinese emperors might disagree].

Grid for cardinal and semi-cardinal directions.

10 (ten)


Tetractys, hyperstates. Projection into 3D (becoming real). Multiple and tractable realizations. Real plane. Transition between dimensions (0-3) as well as transformation. Universal frequency multiplier 43210 (personal theory). Tractability.

Calling number 10 the perfect number detracts from its powerful properties.

Whole numbers spawn rational numbers and are at the heart of the real world. In addition, whole numbers also construct irrational numbers -- that is, integers actualize irrationals in 1D through the Pythagorean Theorem. However, transcendental numbers deal with curvature and with infinite addition (superposition) of numbers. For transcendentals, then, straight numerology is inadequate and the following categories are offered for the transcendentals.

Many to One

Focus. Receive. Crystal, feather (to stem) -- energy intake.
Omega to Alpha (with transformation in between), Buddhist Thunderbolt, Zeus Thunderbolt. Closing a circle -- with transformation

Wand notched on receiving end, Talisman. Bushy or hairy wand [yep, this includes broom]. Infinite superposition followed by instant action.

One to Many

Transmit. Crystal, feather (from stem), Wand pointed on sending end [yep, this includes horns]
Alpha to Omega, Buddhist Bell, Emperor's apple



Does not and cannot have real representation. No magnitude is associated with these numbers. [And for 2000 years Greeks were looking for magnitudes.]

Potentially dangerous -- see Proclus. Life promoting -- see Pythagoreans. Potentially energy releasing -- see Schauberger


Geometric operation. Rotation of an object in space. Point symmetry (multiplication by -1)

Need to differentiate between a number (male, this column) and operator (female, in the right column)

Transformation (if 90 degrees) to/from virtual. Line symmetry (if 180 degrees)

Many geometric representations for this operation: Pentacle, right angle (open), square (close), etc.


Line with 45 (or so) degrees. Mathematical conjugation in matrix arithmetic (very promising with Hermit). Dantien. Energy transformation, Couplex

See Hanub-Ku [every person needs to work this one out. A diagonal form needs to make it to the American flag in some fashion.] Healing

The semicardinal lines (the X) is one of the geometric constructs having even and odd symmetry. It is a line where masculine and feminine meet as you go around the circle


Rotation with changing radius

Virtual energy release or absorption. Magic (and company) -- see Schauberger or Reiki or ..


Infinity. Transcendental, transformative (Complex subject: See Circle and Pi). Alchemical dragon

Delimiter of context. Enticement or invitation to stay, permanence (circle segment, bowl). Protection


Numbers that divide a circle exactly: 2, 3, 5, 15, and 17; including all of their doubles. Harmony

Protection and healing via tones

Universal Harmony


While many authors speak of the harmony of the universe or about the universal balance, the basic idea behind harmony is that it takes two sounds before these can be called harmonious or disharmonious. Any two notes of the Pythagorean (Western) octave are for the most part harmonious. The difficult part is that -- while we can agree on harmonious or disharmonious sounds -- there is no written procedure or mathematical logic that would allow us to determine ahead of time if the tones will be one way or the other. However, in the article on the five pointed star orbit, which is made from two orbits (Venus and Earth), there is enough disclosure to begin to appreciate what it takes to be in harmony. To build the universe, harmony is a requirement in that it makes lasting planetary orbits or galactic structures -- and atomic orbitals as well. You do not have to make it harmonious but then the system will be rudimentary, degenerates, or comes apart.

The book Quantum Pythagoreans explains what makes two numbers -- or two orbits or two frequencies -- harmonious or disharmonious via a formula. You will then be able to predict which notes are harmonious before you play them. Harmony is then also expressed geometrically for the reader as the stars. For hamony in the musical context, the note you (wish to) play can now be instanly associated with the harmonious or disharmonious notes. Ditto for chords. This relates not only to pleasing sounds but to your health as well. As always, harmonious and disharmonious sounds are about dual use -- supportive/creative or detractive/destructive -- and so you may want to know what makes the diff.

The golden proportion is a unique pair of two numbers -- one incommensurable (irrational) and one rational. Because the ratio and other relations of these two numbers also have interesting arithmetic and geometric properties, they should be included in the Pythagorean style harmony analysis. We have a page on the golden proportion with new application and (of course) a relationship to the Great Pyramid. The unreduced real number of the golden proportion is 2, and it is likely the number 2 (or ½) is representative of the octave, for octave doubles the parameters such as frequency or spatial distances and deals with (relates to) square numbers.

The Pythagorean Y


Pythagoreans were engaged in proper living and proper conduct. Today we could call it character building but an additional objective is the salvation of the soul. Pythagoreans were following a path, the Way, in which mathematics is applied to cosmology and harmony, which in turn is about music in general and health in particular and friendship all around. It is likely the geometric shape of the letter Y was used symbolically for a person to choose a particular path for its soul. You come to a "fork in the road" and you want to make the decision that is right for you.

Yet, I would not agree with particular interpretations of the Pythagorean Y as symbolizing the choice between the right road (difficult but happy in the end) and the wrong road (easy and unhappy in the end). [Yes, there is hell and it is a very unhappy and a wholly oppressive domain. It takes a lot of work on this plane not to return to it – but again, it is your choice and your responsibility rather than a question of right or wrong in absolute terms.] Pythagoras understands the triune nature of the soul: That of a merchant making a profit, that of a soldier standing up for honor, and that of the seeker of the truth. Everybody has to make decisions on what is right-or-wrong and do so every day. The Pythagorean way includes the rejection of the wrongs but it is in the framework of the free will. The seeker of the truth, a Pythagorean, gains power in a different way. The ever enlarging context of your knowledge of the truth gives you the power to create and as you do so you may gain the power to uncreate as well – and there is then a necessary right-but-humble aspect to it. Yet, a Pythagorean pursuit of the truth requires toughness, discipline, resilience, and honesty toward self (which is usually more difficult than honesty toward others).

There exists the duality of the real and the virtual – the visible real universe and the spirit – that are intertwined yet separate. The Pythagorean Y could be about this very duality particularly as these are qualitatively different. Here is a sentence from Aetius regarding Pythagoras:

"And again he includes the monad and the undefined [indefinite, nonlocal, spread out] dyad among the first principles; and for him one of the first principles tends toward the creative and form-giving cause, which is intelligence, that is god, and the other tends toward the passive and material cause, which is the visible universe."

Aetius, Placita, 1st Century CE

The real (material) and the virtual (godly, spiritual) components of duality, being not only intertwined but actually interacting while remaining separate, also suggest the letter Y. Moreover -- and even if you choose one over the other -- the components of the duality will always engage each other. The bottom line on this is that one cannot be both, for the duality cannot be merged.

Another Y

I'd love to have this Y associated with Pythagoras:

 Instead of looking at the black ink outline, as we do with all letters, look at the white wedge gap inside the letter and imagine the narrowing end spiraling toward a point at the bottom. Yes, it is about the magic of waves and there is a bit more on this in one of our DSSP topics.


The Pythagorean Theorem

The Theorem relates the squares made from the sides of any right-angle triangle. The Theorem is very old and it is then easy to speculate about what really happened back then. Regardless, it is not difficult to show the Theorem originated with Pythagoras or Pythagoreans, for the discovery of the irrational numbers is linked to Pythagoras without much disputation. But first we look at one conclusion in this excerpt from John Burnet's book Early Greek Philosophy:

It is easy to see how this triangular way of representing numbers would suggest problems of a geometrical nature. The dots, which stand for pebbles, are regularly called "boundary-stones" (horoi, termini, "terms"), and the area they mark out is the "field" (chôra). This is evidently an early way of speaking, and may be referred to Pythagoras himself. Now it must have struck him that "fields" [2D] could be compared as well as numbers, and it is likely that he knew the rough methods of doing this tradition in Egypt, though certainly these would fail to satisfy him. Once more the tradition is helpful in suggesting the direction his thoughts must have taken. He knew, of course, the use of the triangle 3, 4, 5 in constructing right angles. We have seen (§ XI) that it was familiar in the East from a very early date, and that Thales introduced it to the Hellenes [Greeks], if they did not know it already. In later writers it is actually called the "Pythagorean triangle." Now the Pythagorean proposition par excellence is just that, in a right-angled triangle, the square on the hypotenuse is equal to the squares on the other two sides, and the so-called Pythagorean triangle is the application of its converse to a particular case. The very name "hypotenuse" (hupoteinousa) affords strong confirmation of the intimate connection between the two things. It means literally "the cord stretching over against," [1D] and this is surely just the rope of the "arpedonapt." It is, therefore, quite possible that this proposition was really discovered by Pythagoras, though we cannot be sure of that, and though the demonstration of it which Euclid gives is certainly not his [Euclid's].

The previous paragraph is a scholarly method for researching the past. But there is another and a simple way to link the Pythagorean Theorem to Pythagoreans.

It is not possible to discover, differentiate, and characterize the irrational numbers without using the Theorem. Not only is the Theorem valid for both the rational and irrational numbers, but you also need the Theorem to differentiate between the two families of rational and irrational numbers. The Theorem leads to the root and then an analysis is required to show that the root of some numbers is rational and yields a finite sub-unity mantissa, while the root of other numbers is irrational and yields an infinite (or nonverbal or unspeakable or unprintable) sub-unity mantissa. Today we would say that the Theorem is the enabling technology for the discovery and characterization of irrationals. Today's mathematician could also say the Theorem is necessary and sufficient for the discovery of irrationals.

Also, because one cannot get to irrationals directly via rationing, one must first engage the square numbers (today's area or 2D). Pythagoreans were [and are] very much into square numbers and that opened the path toward the Theorem. In other words, one starts in 1D, moves to 2D, and returns to 1D via the root before being able to get to the irrationals. There is no other way to get to irrationals exactly and in finite time (tractably).

The Pythagorean Theorem works with addition or subtraction (superposition) of curved areas, areas also known as lunes. The Theorem relates the squares among the sides of any right angle triangle and it is then easy to erect a square on the triangle sides. A circle's area is also proportional to its diameter squared and so the Theorem works with circle areas erected on the sides as well. This particular rendering of the Pythagorean Theorem may reveal powerful properties (read energy) when dealing with 2D geometry. The illustration is from the back cover of the Quantum Pythagoreans book.

The Pythagorean School

Historically, writings about Pythagoras appear to emphasize authenticity by fully capitalizing the reference – such as when saying "It belonged to HIM." But I think the full capitalization references The Pythagorean School, and when saying " .. it is HIS .. " means it belongs to The School. In addition to authenticity and ownership, the oral tradition of Pythagoras teaching is expressed with the usual preamble: HE himself said it .. . The oral tradition is not inconsequential and does not get weaker with time. The present day scholars sometimes muse at the absence of Pythagoras' own writing. They do agree, however, that such writings do not exist – rather than being physically lost through the ages, say. This is an advanced topic that calls for understanding of ether's properties.

A good start on etheric knowledge is through a passage in the Pythagorean Oath: " .. who transmitted to our soul the tetractys .. "

We would be hard pressed to find the equivalent of the Pythagorean School today. The School did not seek funding from outside, although it is likely the School accepted donations. However, all knowledge such as proofs and breakthroughs belonged to the School. The School enforced its own secrecy provisions (of trade secrets of today), but with methods one would call magical. It is likely some Martial arts schools operate on such basis today.

You, A Pythagorean

If you think you are a Pythagorean, chances are you are. It is not easy to arrive at such designation without joining a real organization. But you know that by truth you know conceit and that is how the truth prevails.

The Pythagorean way of learning, teaching, and building does not deal with suffering, for conflicts exist as the imbalance that leads to the next level of the truth. The building part, moreover, establishes the truth in the objective realm. Enjoy.

You might not think yourself a Pythagorean but others may think you are. Galileo would not think of being a Pythagorean but his discoveries and his mathematical -- as well as apolitical -- thinking earned him a Pythagorean label from no other than the members of the Inquisition. Enjoy.

Finally, if you could think of a person who started wearing a new piece of clothing that would catch on as a fashion statement for over two thousand years, would you think of Pythagoras? Yes, Pythagoras wore trousers and most of the world today wears trousers. Being a Pythagorean today is about the truth and you can pursue it looking good, too.


Mother Goose of Tetractys

Presently, the Pythagorean Theorem mnemonic exists as "Pythagoras' Trousers," which is a tie-in, or perhaps a pun, between Pythagoras' attire (unusual at the time) and his theorem in its geometric form. It is quite likely the Pythagorean numerology of small numbers (1-10, say) was designed to introduce Pythagorean concepts to larger audiences -- not unlike the verses of Mother Goose that combine the poetic and magical qualities of English: "..and the cow jumped over the moon." In the next step the numerical compositions are put together to make stable creations (monads).

While the 'tetractys' is a particular method of composing a poem, it is not taken that way here. Rather, when you are engaged in a particular and difficult pursuit, which is usually full of conflicts, you just might come upon a solution or a resolution. The elation and euphoria then results in composing a short and personal poem as a sort of a whoopee icing on the cake.

It is about the atom staying together
Organization of matter going on forever

Something nice but don't know what
Forgot to put the eye on top

It is the physical plane of emerging matter
It explains galactic clusters being made better

Whimsical extension of Pythagoras' Tetractys
Perhaps for those who think of abyss

Reminds me of the ancient alchemy of sulphur, salt, and mercury
With the winged shoes there is no hurry

Something separated and joined at the same time --
gives me the creeps
It has everything to do with quantum mechanics

One, two, three, four
Cannot count dimensions four.
Zero, one, two, three,
Thanks for making house for free


Book by Mike Ivsin




Tetractys of Pythagoras organizes the universe at all scales. Degrees of independence play the role that up to now has not been disclosed.

Quantum Pythagoreans book describes numbers, their properties, and their ability to make reality through geometric constructs such as the pyramid. The Pythagorean way is the road to reality, for the creation of reality in the form of a new atom is the only proof you need. Continue ..


















Added paragraphs on Irrationals and Becoming July 2005.

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Last update May 14, 2013