Magic square in China

Magic Square in Lo Shu

"Magic Square occurred in Lo Shu of the ancient China. I think that represents the core of Chinese culture: Constructivism, Combinatorics. But they never become main stream culture in China." Steve Han

The story of 'Lo Shu' is as follows:

In the ancient time of China, there was a huge flood.
The people tried to offer some sacrifice to the 'river god'
of one of the flooding rivers, the 'Lo' river, to calm his
anger. However, every time a turtle came from the river and
walked around the sacrifice. The river god didn't accept
the sacrifice until one time, a child noticed the curious
figure on the turtle shell. Hence they realized the correct
amount of sacrifice to make.

http://mathforum.org/alejandre/magic.square/loshu.html

Legends attribute China prehistoric personality, Hsia Yu () - he who tamed the Yellow River of floods ( ) to be the one who discovered the Lo Shu.
Yu saw some very interesting markings on the shell of a giant tortoise that emerged from the River Lo in Central China. This became Lo Shu.
http://www.hiakz.com/loshu.asp

Lo Shu, "scroll of the river Lo", dating as early as 2800 BC.
In Chinese, the square is known as Luo Shu (Simplified Chinese: 洛书; Traditional Chinese: 洛書; pinyin: luò shū; Literal: Luo (River) Book) or the Nine Halls Diagram (Simplified Chinese: 九宫图; Traditional Chinese: 九宮圖; pinyin: jiǔ gōng tú).
http://en.wikipedia.org/wiki/Lo_Shu_Square


The Metaphor of Lo Shu
by Doctor  Rudolf Kaehr

The Lo Shu Story is very interesting! I think it supports very much my ideas about the relationship of Chinese writing (logograms) and the design of new mathematics, maybe based on Morphogrammatics.

The metaphor of Lo Shu seems to confirm that Chinese math is radically different from Greek math as we know it in Western mathematics.

1. First it is in a written form, thus it has to be read (and not heard), i.e., it has to be deciphered and this knowledge has to be translated into sentences, i.e., into spoken language and then interpreted.

2. This procedure is not happening to a single Genius, like a Eureka insight, happening in the mind and then translated into spoken language and from there to an inscribed and written form. What happens with Lo Shu is a societal, co-operative and negotiating interpretation of the inscription on the back of the turtle which has to be discovered.

2. It is practical. The calculation which happens after the "child", i.e., a member of the population, not a Genius, discovered the signs on the turtle, i.e., the magic square, is practical (and mythical, sacrifice for the river-god) bridging the gap between culture and nature to stop the flooding.

3. Thus, the sign on the turtle is in fact not a sign but a topological logogram (morphogram), a tabular matrix, a mathematical inscription. But there is no need to identify such a tabular inscribtion with the modern mathematical concept of a matrix.

4. The problem inscribed on the turtle is purely combinatorial, and not in any sense logical, or sentence-based. It is in this sense not a riddle. Also not pronounced by an Oracle, offered to the scholars to interpretation.

5. The "truth" of the Square is given by a the reading (collecting) of it from all possible points of view, i.e., from all positions of the counting process, the sum has to be the same, which is 15. This process is generating an invariance principle as the form of truth.

6. The way of the counting, represented by the chain of numbers, is for all positions different. Each point of view has its own history and rationality.

7. The final result, the number 15, is not depending on the view-points, but is invariant of the single approaches. It is producing a collective mediation of the different positions, and at once generating societal collectivity. Also it appears as a final result, 15, the number is complex by the history of its construction. Each position has its own number 15. But a single result wouldn't convince the river-god. The convincing result is the collective number 15 as a result of mediation.

8. To each position, point of view, a singular number system can be attributed. Thus the result can be seen as a mediation, harmony, of different number systems. There is no information involved which would demand a unique number system like the modern linear arithmetic of natural numbers.

9. Also it is connected with numbers, the numbers are placed in a tabular order, square, grid, matrix and not linearly as a singular succession. The numbers are marked as patterns. Despite the strict positionality of the numbers, there is no zero sign involved.

There are many more interesting aspects in the story, like the fact that the turtle-matrix gives access to Nature. Thus, the writing is a bridge between human culture and nature. The turtle is considered as celestial. It is also producing time, societal time of the people involved. It is local, a gift for the specific river-god, but connected with a global cosmic insight. The Lo Shu is mediating local and global thematizations. 

With this grammatological approach, I tried to avoid to map the Magic Square of Lo Shu to Western binary digitalism and arithmetic as it is usual today. Thus, in strict mathematical terms, the Lo Shu is not a Magic Square. Simply because it is not a mathematical matrix based on natural numbers. Such an interpretation of Lo Shu as a Magic Square is denying its specific Ancient concept of Numbers. This may have far reaching consequences in the context of interpretations of I Ching, Trigrams, Feng Shui and Yin&Yang.

Source

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PART FIVE:
CONCLUSION

What can be learned from the experiences I mentioned above? I think there are two lessons: philosophical and mathematical. The first lesson is that a philosophical concept in our heads evolves in a living dialogue in which we live, including those on the Internet. The dialogue was basically thinking together. In thinking together we get new perspectives. The synthesis of perspectives that is a progression in thinking none other than an internal dialogue. Similarly, the achievement of consensus in external dialogues, internal synthesis which typically generates an intellectual satisfaction.  

Second lesson, for a math world explorer as I am, to be ready to face the collapse of assumptions that had since long time and he thought as an absolute truth. The emergence of non-Euclidean geometries of Riemann and  non-cantorian set theory of Robinson, for example, reflect patterns of destruction that assumption. The emergence of new math or science as a result of a crisis is usually accompanied by the emergence of a new paradigm that is the philosophical foundation of science. So actually personally intellectual crisis will be reflected as a social paradigm crisis.

Well, now I do not know if my personal intellectual crisis will lead to a change or transformation of my integralist thought to be a more comprehensive philosophy again. But clearly it was a new intellectual crisis is mathematical, not philosophical. To analyze this new crisis, perhaps it is better if we examine backward to see how the crisis was solved so far.

Crisis of the early 20th century physics, for example, the emergence of symptoms due to the discovery of radioactivity and the reality of atomic stability. The crisis gave birth to a new physics which is then referred to as the extension of quantum mechanics known as quantum field theory. Quantum field theory is a relativistic version of quantum mechanics. This means that quantum field theory is a synthesis of quantum mechanics and relativistic mechanics.

Quantum field theory is ultimately in the end, having developed into a theory called the Standard Model of elementary particles. The standard model is a gauge field theory for the three fundamental interactions that electric-magnetic field interaction, weak nuclear interaction and the strong nuclear interaction. So Einstein's dream to marry the electric-magnetic interactions and gravitational interactions proved to be too early. Apparently electromagnetic interaction is more suitable when mated with the weak interaction that gave birth to the theory of electro-weak interactions field. Then the electro weak theory is mated with the theory of the strong interaction between quarks birth gauge field theory called the Standard Model.

Later physicists expand the  Einstein's dream by forwarding an elementary particle theory marrying Einstein's theory of gravity, not with Maxwell theory of electromagnetism, but with the  the Standard Model. The marriage lead to the birth of five different string theories, but equally true. Although different they have in common: change the dimensionless particle theory to the theory of one-dimensional strings that vibrate in ten-dimensional physical space where six dimensions curled in a very small rolls. Indeed, there are similarities nonetheless and five different true and equally valid theory. Of course this damage our understanding of logic that only receive only one theory should be correct.

It is similar period at the time of birth of quantum theory. At the time it was discovered two kind of mechaniics that seems to be contradictory to each other: Schrodinger's wave mechanics and Heisenberg's matrix mechanics which talking about particles. Luckily there was a Paul Dirac who can prove that the two theories are different representations for the same theory, namely quantum mechanics.  

This time there was a genius physicist who tried to incorporate all five different theories into one theory of M theory on a vibrating membrane in eleven dimensions of space physics. So the synthesis was done by adding the dimension of fundamental physics objects and add dimension to the room they occupied. Unfortunately that theory has not been getting the perfect shape. He was able to explain all the phenomena microcosmic there, but he predicted the existence of super-particles, as a couple of fundamental particles exist, which until now could not be observed experimentally.

M-theory is indeed a gee-wish theoryusing sophisticated mathematics, but unfortunately as he was ill-fitting theory. Because M-theory is the son of superstring theory predicts both couples for fundamental particles and fields that have never been observed. In addition, M-theory can not explain the physics of the 21st century crisis sparked by the discovery of dark energy and dark matter which makes up more than 90% of the mass-energy of the universe by cosmological observations astronomical disciplines. So on the other hand, M theory as a theory of particle physics expectations, is a theory that very narrow. He predicted things that were not observed on the one hand. On the other hand, he could only explain less than 5% cosmological reality. A feat that is not encouraging.
The fact that is a crisis of the 21st century physics.  

Actually, the crisis of the 21st century physics Biological less I personally, since I've left the study of rock physics and stir into the world of mathematics. But now in the world of mathematics I had a crisis triggered by a mathematical study Dr.Rugerro Maria Santilli. Rugerro Santilli was actually a physicist. He saw the fundamental contradictions between the theory and the theory of electro-weak gavitation.  

Therefore, he revised the two theories with mathematical iso-real number, the alti-real numbers with r = -1 in my terminology, to eliminate the contradiction. He even found two types of new mathematical again called geno-and hyper-mathematics mathematics. Geno-mathematical physics processes irreversible irreversible alias. Hyper-mathematics to biological processes in addition to irreversible also toward the goal is worth double. Unfortunately Rugerro Santilli did not receive a positive response from mainstream physics.

So here's my crisis. I am at a crossroads. Will I be turned my back to follow Rugerro Santili back to physics and then reform it or I will continue to explore the platonic world of mathematics, finding new structures regardless of whether the findings would be useful outside of mathematics? I think I've decided my choice. Most likely, I will continue in the world of mathematics to explore the world that has not been touched people: algebra multicomplex expanded to include number of new structures such as those found Santilli isoreal field or quasi-field  of omni real numbers that I found. Hopefully, I'll be strong enough to continue.

Me, Math and the Internet (4)

PART FOUR:
HIDDEN FIELD OF REAL NUMBERS.

The story of crisis like this. In my search of new physics, I finally found a site of Doctor Rugerro Maria Santili quantum mechanics that have been revised into what he called hadrons mechanics. What is the basis of the revision? Apparently he changed his field of real numbers with unit = 1 be a real number field with unit = -1. How can that be? The answer is simple, try to redefine multiplication of real numbers with multiplication a ** b == - a * b, then clearly this new product meets the properties of commutative, associative and distributive for + and **. And easily proved that this new unit number field is -1. 

 

Thus, with the set of all real numbers we can create a new field that is isomorphic to the usual field of real numbers. Old numbers defined the new so-called isonumbers. Hadrons mechanics is based on the new mathematical construct of  isonumbers where he defines an all new calculus and analysis with the new multiplication.  

 

Well, when I saw a couple iso for the field of real numbers, I'm thinking of looking for another couple to the field of real numbers. I also generalize the multiplication multiplication * where ** a ** b == a * r * b where r is any real number not equal to zero. This new multiplication obviously also commutative, associative and distributive towards +. It can easily be proved that the new unit for multiplication is 1 / r. In other words, for a given r we can make a real number field or a field with a new multiplication ** == * r *. I call this new number alti-real numbers in the hypercomplex eGroup  owned by Jens Koeplinger .

Even in eGroup I redefine real numbers as omni-real numbers in which addition + is redefined ++ == +s+ and the multiplication  * is redefined to ** == *r* . In this case, the new addition is both commutative, and associative and have -s as a unit. It turns out that the two new defined operation are no longer distributive ** to ++. So all omni-real numbers form a quasi-field. As a consequence the field of alti-real numbers is a special case of quasi-field of omni-real numbers with s = 0. Thus we can build again hypercomplex numbers on the basis of quasi-field of omni-real numbers or the field of alti-real numbers. This is the logical consequence of the expansion of the field of real numbers and a huge work is waiting for the examination of it. The work I am facing is gigantic in the future.

Because of this discovery, I have to rewrite the n-color numbers part 3 and I found that the two numbers are the same color can be multiplied with each other if we define how multiplication of two 1 in the same color. Usually we choose the theoretical colored 1 to itself is equal to same colored 1. But it turns out with our new knowledge about the number alti-real, then I can multiply a colored 1 with itself to get colored r which is not equal to zero. Because color is none other than the unit vector in a given direction, then we can define a one-dimensional vector space with a multiplication of two unit vectors is r* unit vector. So I found the alti-vector number.

Later, after reading a paper by a follower of Rugerro Santilli in China, I discovered that the alti-real numbers and alti-vectors numbers, I think I found it, each of which is none other than the type one and type two Santilli numbers. Santilli basically never limit the number r of alti-real multiplication with -1. Even a mathematics professor from China later developed the theory of Numbers to Santillian Number Theory with the new multiplication. He even developed a new theory of cryptography by using two types of field of Santillian finite natural numbers.

But if you read his paper carefully, he seems to restrict to linear codes. I think if the theory be extended to the Marek-Santillian finite polyplex ring, then we can also make more complex Santillian cryptograpy. This is my homework now. I had to rewrite my fourth dialogue about polynomial finite numbers which I will refer to it as popularized by n-colored numbers. Sketch for it is already written, but now the third series of Mathematical Dialogue only new part two, and then jammed. Hopefully I will have the strength to break the deadlock.

Me, Math and the Internet (3)

PART THREE:

LOOKING FOR HYPERCOMPLEX

NUMBERS 

Well, when I joined the mailing list Hypernumbers moderated by Kevin Carmody when I still embrace the integration of schools with only about half integralism same mathematical congruent with the view of social constructivism plus universalism platonic. The moderator then assume that hypernumber by Charles Muses as a generalization of complex numbers is a natural mathematical, comprehensive and closed. 

 In fact I would argue that it's the real math is incomplete, social and open. The Hyper-Numbers version of the Muses it is an extension of complex numbers in stages, but I think the math is not yet an ideal as imagined Plato. In view of the Muses, real numbers and complex numbers is only the first two levels of the whole hyper number has eight levels. Of course, initially I did not tell half integral mathematical view it. 

But when I had to pass the list owner censorship, I had to make a blog hypernumber on blogger.com as an alternative. Unfortunately blogspot was slow to be updated, so I opened eGroup hypernumber at yahoogroups.com accompanying eGroup hypercomplex homemade Jens Koeplinger which is an alternative to egroupnya Kevin Carmody whose name was changed by the moderator to ScienceAndConsciousness discussing science's version of the Transcendental Meditation Maharishi Mahesh Yogi .

Fortunately, I finally found a quick blog space, named multiply.com where I present my thoughts are popular. The views of integralism on the mathematical in the personal version of Jean Piaget to the social version through the dialogue about Magic Square. But then I also has to remember, when the dialogue on the quasicrystal, the path from the universal to the personal a la Plato that I have forgotten. Consequently, the concept of integral mathematics I could be formulated somewhat coherent and complete.

As it goes, I proposed a dialogue on colored numbers which is no other than popularization of mathematics hypercomplex numbers as the extension of complex numbers that is different from the  hypernumbers theory of Charles Muses. In my view, colored numbers or hypercomplex numbers is a much larger space than the space of  numbers as it is imagined by Pythagoras .

Among the number of non-musean hypercomplex numbers are the Marek 14 findings of polyplex number in the Czech Republic. But unfortunately, it does not define what exactly Marek  polyplex number is except the given the multiplication table of its units. The multidimensional polyplex numbers has a multiplication abd addition which are commutative, associative and distributive. But unlike the sometimes complex number we can multiply two nonzero polyples numbers to produce zero. This means that zero can have dividers that is not equal to zero.

Fortunately, when I try to present polyplex numbers as colored numbers, I found that basically polyplex numbers form a ring equivalent to all the polynomial ring modulo a certain finite degree polynomial which is defined by Marek a defining characteristic polynomialfor the multiplication of the polyplex numbers.  

Sadly, despite of the finding of the equality, I still can not prove the Marek conjecture that  fundamental algebra of a polyplex numbers with certain complex characteristic polynomial can be broken down into a direct sum-algebra polyplex number with these simple polynomials which multiplied to the complex caharacteristic polynomial. (Sorry, the actual conjecture of Marek is a bit more complex than that). This is certainly a great deal of work for me.

So, while I defer proving the Marek conjecture, apparently recently I got a severe intellectual crisis. During this time I saw the hypercomplex linear algebra is built on the basis of a linear vector space, which in turn built on the field of real numbers. But now, unexpectedly, my belief in the uniqueness of the real number field as the largest one dimensional number seems to be falling apart.

And I also experienced the greatest intellectual crisis in the history of my life. Intellectual crises previously was when I studied the theory of relativity and quantum mechanics at the time I was a student. At that time, of course there was a guide that help me out of the crisis which breaks down the belief of the absolute Newtonian space and time. The guides are my teachers. Now I have to face it alone. Sorry. I was accompanied by friends like you in the cyberspace.

Me, Math and the Internet (2)

PART TWO:
Integralism OF MATHEMATICS

I would like to tell you a story about a crisis: an intellectual crisis of myself these last few days. The story is like this. When I was retired from the faculty of physics, only then can I wrestle with my hobbies: math. Because I love the philosophy, then I asked where whether the mathematical objects such as numbers and any extension such as functions and mathematical equations is real. Where also lies the geometric space and all geometric shapes in it. In the end I asked where the math world is.

Since I was a theoretical physicist, so I know that the math in nature is only a small part of mathematics. That's why I had the idea that mathematical objects are in a larger world of the living ideas of Plato, which in Islamic philosophy, is then identified with the knowledge of God or the mind of God. Well, the thought of God is realized in the mathematics of nature. The rest of it is in the mind of God, but human beings can try to dive into the mind of God can.

Well, how mathematics is becoming so theological. Sorry, but we could dispose of God and left His Mind as the ideal world of mathematics, and concentrating on the new ideal world can be explored by human. So what is thinking? I got an answer from a Muslim intellectual who has now gone Endang Saifuddin Anshori. According to kang Endang, thinking is arguing with yourself. Usually, we arre arguing with others by using language. The point is the use of language, so the bottom line thinking is having a conversation with yourself.

So, the thinking is the internalization of social discourse. In other words Wittgenstein proposition that mathematics as a language game made sense. As the game of course, there are basic rules of naming and naming-through agreements. The basic rule was none other than logic. Naming was none other than a mutually agreed terms. Naming and regulations are nothing more than social consensus agreement. Thus the real math is socially constructed. This means that the postmodernists have a point.

But postmodernism does not recognize the mathematical ideal as it is assumed by Plato. For those who see mathematics as a social construction, then the whole of mathematics continues to open and flourish elsewhere. But for me the development of constructive mathematics it is directed towards the development of the mathematical ideal of Plato as it's final destination.

This integralism that combines postmodernist relativism with absolutism of the modernist thinkers about the reality of mathematics. The motion of these developments is not an order from a transcendent but a movement of an immanent self-organization. Well, that's the philosophy of integralism about mathematics. But that is not the whole integral philosophy of mathematics.

The movement from the social to the ideal is the continuation of the movement of individual personal to the social. So according integralism, mathematics follows a motion from the social to the personal through a universal ideal. While the personal is the terminal of a cycle from the universal (cosmological) through the collective (biological) to the personal (psychological).

Thus, the overall motion math, in my view, is from the universal to the individual personally, through the collective biological, back to the universal (ontological) through the collective cultural (sociological). Motion back and forth through the collective personal to the universal is the constant process of differentiation /integration which we know as the emanation /creation of the Creator in philosophy integralism.

Me, Math and the Internet (1)

PART ONE:
LEARNING MATHEMATICS IN CYBERSPACE

When I retired as a professor of physics, and so I was very happy but mingled with sadness. Glad that I was free from teaching duties. But sad because if I stop teaching, so my brain will degenerate quickly. Therefore, I learned more in one of the largest universities in the world, namely the Internet. I decided what I have to learn is math. Why did I choose math? Because of when I was a little boy, I was interested in the beautiful images of the fundamental mathematical objects such as regular polyhedron of  Plato in three-dimensional space as it is shown below.



Greater interest in it even after my retirement from physics. Because I found a website from Tony Smith 
, a lawyer who studied physics and then propose a theory about everything that do not require a hypothesis about the fundamental particles as pieces of string. Of course this caught my attention. But the most interesting is the fact that Smith said that the space-time dimension is equal to 8 instead of 4 as we know it in the mainstream of physics such as the theory of relativity and quantum theory.

In the language of mathematics, physics time space was actually a octonion . Well from Tony Smith's website I knew Ben Goertzel   the hypothesized octonion as psychological space, Kent Palmer who actually saw the social space is a octonion. From there I knew sedenion as a 16 dimension hypercompleks numbers which has twice the dimension of octonion. From there anyway I know the two leaders discussed the mathematical number sedenion: Kevin Carmody and Bob de Marrais   Of course I would like to know more about sedenion. That is why I participate in the mailing lists they lead. Apparently mailing oktonion Dialognet led Kent Palmer is not very responsive. While in de Marrais mailing list I was lead the Kevin Carmody list called hypernumber discussing the number system findings of the late Charles Muses which also includes hypercomplex numbers, such octonion and sedenion.

That is why I participate actively in hypernumber mailing list and there I met new friends such as Jens Koeplinger using sedenion numbers in Muses version to the theory of gravity and Marek Eckhart who found polyplex number system. There I proposed also the theory of general  quaternions (4-dimensional hypercomplex numbers). But here I do not want to talk about such wonderful numbers. I want to tell you about myself.

Dialogue on n-Color Numbers (3)

DIALOGUE ON MANY-COLOR NUMBERS

Part Two: The Fascinating 3-Color Numbers

In the last dialogue Ki Algo sees the 2-color number system as a representation of polynomial number system. But we can also see the polynomial numbers or polyplex number system as the representation of colored number system. In the following dialogue we will listen how  Ki Algo teach Si Emo about 3-color number using the concept of polyplex number.

3-Color Number Multiplication table

Ki Algo: Now, let us talk about 3-color number
Si Emo: Do you mean a number like 1 + 2 + 3 ?
Ki Algo: Yes. It can also be represented by polynomial 1 + 2 x + 3 x²
Si Emo: That means it is a rest polynomial when a polynomial is divided by degree 3 polynomial. It is a number that is called 3-plex number or terplex number
Ki Algo: Good, you remember lesson number one perfectly. Now, let us investigate 3-plex or terplex numbers generated by the dividing polynomial x³-1.
Si Emo: That means we have multiplication table for mononomial units like this

*	1	x	x2
1	1	x	x2
x	x	x2	1
x2	x2	1	x

Ki Algo: That's good. Let us represent the mononomial with colored ones. For example x with orange one 1and x2 with green one 1
Si Emo: OK the multiplication table are now like this

*	1	1	1
1	1	1	1
1	1	1	1
1	1	1	1

Ki Algo: How can you use it to find the multiplication of  (1+2-3)*(2-3+4) for example
Si  Emo: I just put the first number 1+2-3 in the leftmost column and the second number 2-3+4 to the uppermost row and multiply each element of the leftmost column with each element of the uppermost row like this

*	2	-3	4
1	2	-3	4
2	1	-6	8
-3	-6	9	-12

and then I add up the elements of the 3x3 white matrix colorwise like this
(+2+8+9) + (-3+1-12) + (+4-6-6) = 19 - 14 - 8.
So  (1+2-3)*(2-3+4)  =  19 - 14 - 8
Ki Algo: Good you're already understand the arithmetic of terplex or 3-color number

Self-Powered 3-Color Number

Ki Algo: There is a 3-color numbers which are powered to themselves
Si Emo: What is the self powered number?
Ki Algo: It is 1/3 + 1/3 +1/3
Si Emo: Let me check it using the multiplication table

*	1/3	1/3	1/3
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9

I will collect the same color numbers and add them up
(1/9 + 1/9+1/9) + (1/9 + 1/9 +1/9) + (1/9 + 1/9 +1/9) = 1/3 + 1/3 +1/3.
Yes, it is self-squared.(1/3 + 1/3 +1/3)² = 1/3 + 1/3 +1/3
So, it is also self-powered (1/3 + 1/3 +1/3)ⁿ = 1/3 + 1/3 +1/3 for any n
 Ki Algo: You can also checked this number 2/3 - 1/3 -1/3
Si Emo: OK

*	2/3	-1/3	-1/3
2/3	4/9	-2/9	-2/9
-1/3	-2/9	1/9	1/9
-1/3	-2/9	1/9	1/9

the same color number will add up like this
(4/9 + 1/9 + 1/9) + (-2/9-2/9+1/9) + (-2/9+1/9-2/9) = 2/3 - 1/3 - 1/3
so it is self-squared
(2/3 - 1/3 - 1/3)² = 2/3 - 1/3 - 1/3
and consequently it is also self-powered
(2/3 - 1/3 - 1/3)ⁿ = 2/3 - 1/3 - 1/3 for any n
Ki Algo: Yes, it is.
Si Emo: Are there any more self-powered 3-color numbers?
Ki Algo: Yes, but they are not interesting because it is so obvious. They are 1 and 0
Si Emo:  Not interesting at all. They inherited from Black Number

3-Color Zero divisors

Ki Algo: Now about your question in the first dialogue. Is terplex number system a field?
Si Emo:  Well, is it?:
Ki Algo: No, it has infinitely many zero divisors.
Si Emo: Show me two of them!
Ki Algo: You can multiply the two self-powered 3-color numbers. Check it up!
Si Emo: I will put  1/3 + 1/3 +1/3  in the leftmost column part and  2/3 - 1/3 -1/3 in the uppermost row part of the 3-color multiplication.

*	2/3	-1/3	-1/3
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9

I'll collect numbers of the same color  and add it up.
(2/9 -1/9 -9) + (-1/9 + 2/9 -1/9) + (-1/9 -1/9 + 2/9) = 0
Wow,
(1/3 + 1/3 +1/3)*(2/3 - 1/3 -1/3)=0
So, self-powered 3-color numbers are zero divisors
Ki Algo: Yes, you can also try to multiply  1/3 + 1/3 +1/3 with a + b + c with a+b+c=0
Si Emo: OK. I'll try it up.

*	a	b	c
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3

I'll collect numbers of the same color  and add it up.
(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)+(c/3 + b/3 + a/3)=(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)1+(c/3 + b/3 + a/3)1=0 because a+b+c=0
Oh my goodnes! There are infinite number of 3-color numbers zero divisors. It's fascinating.
Ki Algo: Yes, 3-color number system is not a field. But like Black-Red or Counter-Complex numbers it is a direct sum of two fields anyway.
Si Emo: Oh yeah? What fields are they?
Ki Algo: Wait until our next dialogue.

Notes on the Dialogue

3-color number system is a generalization of complex number system.

1. It is called 3-polyplex numbers by the Czech Marek 17 in his theory of polyplex numbers in http://tech.groups.yahoo.com/group/hypercomplex/
2. It is called terplex numbers by the British Roger Beresford in his theory of Hoop Algebras in here
3. It is called tricomplex numbers by the Rumanian physicist Silviu Olariu in his theory of n-complex numbers in his paper http://front.math.ucdavis.edu/0008.5120
4. It is called 3-numbers by the Russian group who study the polynumbers  in http://hypercomplex.xpsweb.com/page.php?lang=en&id=148

Dialogue on n-color Numbers (2)

DIALOGUE ON MANY-COLOR NUMBERS

Part Two: The Fascinating 3-Color Numbers

In the last dialogue Ki Algo sees the 2-color number system as a representation of polynomial number system. But we can also see the polynomial numbers or polyplex number system as the representation of colored number system. In the following dialogue we will listen how  Ki Algo teach Si Emo about 3-color number using the concept of polyplex number.

3-Color Number Multiplication table

Ki Algo: Now, let us talk about 3-color number
Si Emo: Do you mean a number like 1 + 2 + 3 ?
Ki Algo: Yes. It can also be represented by polynomial 1 + 2 x + 3 x²
Si Emo: That means it is a rest polynomial when a polynomial is divided by degree 3 polynomial. It is a number that is called 3-plex number or terplex number
Ki Algo: Good, you remember lesson number one perfectly. Now, let us investigate 3-plex or terplex numbers generated by the dividing polynomial x³-1.
Si Emo: That means we have multiplication table for mononomial units like this

*	1	x	x2
1	1	x	x2
x	x	x2	1
x2	x2	1	x

Ki Algo: That's good. Let us represent the mononomial with colored ones. For example x with orange one 1and x2 with green one 1
Si Emo: OK the multiplication table are now like this

*	1	1	1
1	1	1	1
1	1	1	1
1	1	1	1

Ki Algo: How can you use it to find the multiplication of  (1+2-3)*(2-3+4) for example
Si  Emo: I just put the first number 1+2-3 in the leftmost column and the second number 2-3+4 to the uppermost row and multiply each element of the leftmost column with each element of the uppermost row like this

*	2	-3	4
1	2	-3	4
2	1	-6	8
-3	-6	9	-12

and then I add up the elements of the 3x3 white matrix colorwise like this
(+2+8+9) + (-3+1-12) + (+4-6-6) = 19 - 14 - 8.
So  (1+2-3)*(2-3+4)  =  19 - 14 - 8
Ki Algo: Good you're already understand the arithmetic of terplex or 3-color number

Self-Powered 3-Color Number

Ki Algo: There is a 3-color numbers which are powered to themselves
Si Emo: What is the self powered number?
Ki Algo: It is 1/3 + 1/3 +1/3
Si Emo: Let me check it using the multiplication table

*	1/3	1/3	1/3
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9

I will collect the same color numbers and add them up
(1/9 + 1/9+1/9) + (1/9 + 1/9 +1/9) + (1/9 + 1/9 +1/9) = 1/3 + 1/3 +1/3.
Yes, it is self-squared.(1/3 + 1/3 +1/3)² = 1/3 + 1/3 +1/3
So, it is also self-powered (1/3 + 1/3 +1/3)ⁿ = 1/3 + 1/3 +1/3 for any n
 Ki Algo: You can also checked this number 2/3 - 1/3 -1/3
Si Emo: OK

*	2/3	-1/3	-1/3
2/3	4/9	-2/9	-2/9
-1/3	-2/9	1/9	1/9
-1/3	-2/9	1/9	1/9

the same color number will add up like this
(4/9 + 1/9 + 1/9) + (-2/9-2/9+1/9) + (-2/9+1/9-2/9) = 2/3 - 1/3 - 1/3
so it is self-squared
(2/3 - 1/3 - 1/3)² = 2/3 - 1/3 - 1/3
and consequently it is also self-powered
(2/3 - 1/3 - 1/3)ⁿ = 2/3 - 1/3 - 1/3 for any n
Ki Algo: Yes, it is.
Si Emo: Are there any more self-powered 3-color numbers?
Ki Algo: Yes, but they are not interesting because it is so obvious. They are 1 and 0
Si Emo:  Not interesting at all. They inherited from Black Number

3-Color Zero divisors

Ki Algo: Now about your question in the first dialogue. Is terplex number system a field?
Si Emo:  Well, is it?:
Ki Algo: No, it has infinitely many zero divisors.
Si Emo: Show me two of them!
Ki Algo: You can multiply the two self-powered 3-color numbers. Check it up!
Si Emo: I will put  1/3 + 1/3 +1/3  in the leftmost column part and  2/3 - 1/3 -1/3 in the uppermost row part of the 3-color multiplication.

*	2/3	-1/3	-1/3
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9

I'll collect numbers of the same color  and add it up.
(2/9 -1/9 -9) + (-1/9 + 2/9 -1/9) + (-1/9 -1/9 + 2/9) = 0
Wow,
(1/3 + 1/3 +1/3)*(2/3 - 1/3 -1/3)=0
So, self-powered 3-color numbers are zero divisors
Ki Algo: Yes, you can also try to multiply  1/3 + 1/3 +1/3 with a + b + c with a+b+c=0
Si Emo: OK. I'll try it up.

*	a	b	c
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3

I'll collect numbers of the same color  and add it up.
(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)+(c/3 + b/3 + a/3)=(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)1+(c/3 + b/3 + a/3)1=0 because a+b+c=0
Oh my goodnes! There are infinite number of 3-color numbers zero divisors. It's fascinating.
Ki Algo: Yes, 3-color number system is not a field. But like Black-Red or Counter-Complex numbers it is a direct sum of two fields anyway.
Si Emo: Oh yeah? What fields are they?
Ki Algo: Wait until our next dialogue.

Notes on the Dialogue

3-color number system is a generalization of complex number system.

1. It is called 3-polyplex numbers by the Czech Marek 17 in his theory of polyplex numbers in http://tech.groups.yahoo.com/group/hypercomplex/
2. It is called terplex numbers by the British Roger Beresford in his theory of Hoop Algebras in here
3. It is called tricomplex numbers by the Rumanian physicist Silviu Olariu in his theory of n-complex numbers in his paper http://front.math.ucdavis.edu/0008.5120
4. It is called 3-numbers by the Russian group who study the polynumbers  in http://hypercomplex.xpsweb.com/page.php?lang=en&id=148