Duality 

The simultaneous co-existence of paired (conjugate)  components such that their interaction forms the necessary criteria for existence of an elementary system. 

Duality in nature is amazingly beautiful, for it is the way nature was created. 
Duality in nature is simply mysterious, for it is the way that nature exists. 
It is beautiful because all things were originally created in a splendid harmonious world.
It is mysterious because different creatures have different patterns of duality. 
If we are not confused very often about the duality of natural phenomena, 
we do not really understand what it is. 
This may be the way that we exist. 

                                                            Duality Principles 

The general principle of duality is undefeatable, 
since the existence of its opponents proves this universal principle.

Duality is a fundamental concept that underlies almost all natural phenomena. Roughly speaking, duality is the product of the interaction that takes place between the paired components of an elementary system. Specifically, in physics these components are position and momentum; displacement  and  force; strain and stress, etc. While these properties take different shapes and forms within each system, they invariably retain their basic nature. 

The two elementary components exist relative only to one another and their causative purpose. For example, within the boundary of each dimension, every position is meaningless without an understanding of their momentum. As well, momentum within the dimension is also meaningless without an understanding of the various positions. 

Duality is the interplay between these dual variables that helps define the nature of elementary interaction. In the process, duality helps to resolve the dilemma within each dimension, framing the nature of its link to its next higher dimension. The next higher dimension is always some collective expression of the its lower dimensional counterpart. This link is that dimension's Causative Purpose. 

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1. Duality and Synthesis in the Music of Bela Bartók, E Lendvai 

    pages 174-193 of Module, Proportion, Symmetry, Rhythm G Kepes (editor), George Brazille, 1966; 

In ancient times, the Greeks of the 5th century BC developed a philosophy that is probably best expressed in English: A sound mind in a sound body. They excelled in developing the mind and body as a totality; they saw no difference in terms of value between the physical and the intellectual. One could actually be beautiful, physically strong, and an intellect! Imagine...  With the advent of Christianity and some of the more esoteric or gnostic religions and philosophies from the Near East, a split developed whereby the physical was perceived to be an evil, or at least, of less importance and value than the spirit. The spirit was imaged as being in prison, trapped by a physical body, longing for union with its aker,
some divine source. The thinking was the more the physical was denied, the closer one could come to the Spiritual Source: hence the advent of the myriad forms of asceticism, mortification of the flesh; beauty became sin, sin being those attitudes which led away from the path of the spirit's return to its divine source. 

The results of this thinking led to a cleavage in understanding the totality of our unified personae. Beauty and intellect, body and soul, faith and science were viewed in such a way that the twain were never to meet. Now, at the end of the 20th century we are beginning to return to an understanding of human identity that integrates the physical with the spiritual and intellectual. 

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Dao of Mathematics

• Duality Principles in General Systems (Book by David Gao)
• Duality in General Mathematical Physics 

(article by David Gao in 
Wiley Encyclopedia of Electrical and Electronical Engineering, Wiley, 1998)
• Duality in variational analysis and global optimization
1. articles  by David Gao in Encyclopedia of Optimization, Kluwer 
1. Mono-duality in convex static systems 
2. Bi-duality in convex dynamical systems and d.c. programming
3. Tri-duality and triality in nonconvex systems 
2. Dual Simplex Machine

Duality in Linear Programming
• Duality in Polyhedra
• Duality in Abstract Mathematical Systems
• Natural Dualities for the Working Algebraist, 

by David M. Clark and Brian A. Davey, Cambridge University Press

Projective geometry has a lattice structure that can be seen by ordering the points, lines, and planes by the inclusion relation. In the projective geometry of the plane, the words "point" and "line" can be interchanged, giving for example the dual statements: "Two points determine a line" and "Two lines determine a point." This last statement, sometimes false in Euclidean geometry, is always true in projective geometry because the axioms do not allow for parallel lines. Sometimes the language of a statement must be modified in order that the corresponding dual statement be clear; the dual of the statement "Two lines intersect in a point" is vague, while the dual of "Two lines determine a point" is clear. Even the statement "Two points intersect in a line," however, can be understood if a point is considered as a set (or "pencil") containing all the lines on which it lies, a concept itself dual to the idea of a line being considered as the set of all points that lie on it.

There is a corresponding duality in three-dimensional projective geometry between points and planes. Here, the line is its own dual, because it is determined by either two points or two planes.

In set theory, the relations "contained in" and "contains" can be interchanged, with the union becoming the intersection and vice-versa. In this case, the original structure remains unchanged, so it is called self-dual.

In symbolic logic there is a similar self-duality if "implied" and "is implied by" are interchanged, along with the logical connectives "and" and "or."

Duality, a pervasive property of algebraic structures, holds that two operations or concepts are interchangeable, all results holding in one formulation also holding in the other, the dual formulation. 

Physics Explanation

Dao of Physics

    The apparant duality of nature, as manifest by modern physics, seems to be a function of our consciousness. I suggest that having adapted a hybrid abstraction system, our study of reality has reached the stage where the hybridization interferes with the discovering of 'REALITY' - the world 'out there', both at the micro and macro levels.

In the hybrid system, wave detection has it's roots in audition, and particle detection ( a whole hierarchy of them) has it's roots in vision. All our instruments, although having degrees of bias, seem to be designed such that both biases are detected thus making things 'incomplete'. The only way that I can see (!) to get around the problem is the recognition of the seperateness of both systems and the adaption of measurement systems totally dedicated to one system or the other.

Reality appears twofold because our abstraction system is twofold, not necessarilly because reality is twofold. The methodologies used attempt to detect a 'one' but we are implicitly using two independant systems to do it. This has been extremely successful on this planet, but at the levels of quantum mechanics, we are dealing with resolutions that can be handled by one system but NOT two at the same time since one will interfere with the other. This interference is NOT necessarily out there, but a possible result of us attempting to use two systems at once. (It is important to note that QM is founded on probability waves not 'real' waves. Using this tool, observation suggests a duality, that, for example, light is made of 'wavicles'.)

At the macro level, the world of Relativity seems bias to geometry, whereas the world of Quantum Mechanics is bias to harmonics. In this context, The EPR paradox and Bell's theorem are more about the 'whole' and harmonics, than about 'particles', and Relativity is more about preserving the whole, where the wave equivalent of E = mc2 (E = nfm ) is more a limit on an object's matter wave. Visually this would protect the 'whole' - matter, manifest as a wave, cannot become 'infinite', and thus lose it's wholeness. Bell's theorem makes the same case, but for QM. It says that splitting an apparent whole into two parts mearly creates harmonics that remain linked; the whole is retained. The apparent problems with the QM model is the intrusion of spatial concepts, concepts that derive from vision:

Preservation (Conservation) of the whole:

          Relativity                        QM

          matter wave cannot be           correlations cannot
          infinite.                       be split.

          or                              or

          matter cannot exceed speed      'cutting' a wave only
          of light.                       creates it's harmonics.

The development of QM suggests that QM is to PROCESS what Relativity is to FORM, and as such the two can never be totally, explicitly, united. Furthermore, the observation of mental oscillations between FORM and PROCESS through time (Bateson) and the bias to new developments in PROCESS and their full manifestation in FORM, suggests that QM, often treated as a complement to Relativity, is infact it's FULL successor.

This suggests that over time Relativistic concepts will become explainable and thus replaced by QM concepts. However sensory hybridization may put a limit on this in that it is difficult to remove the 'whole' concept to be replaced by waves (harmonics) since we still have eyes that see 'wholes'. At the moment it is at the scientific interface with 'reality' that these soughts of extremely one sided biases are required. (harmonic wholes can be made into images (see Jenny (1966), inspired by Ernst Chladni) where, with the use of a resonating surface and sand, patterns appear in the sand. These images change as a function of tonal scale, and seem to oscillate between a static state (FORM) and a dynamic state. (PROCESS))

Lasers and Complementarity.

A picture is a good example of FORM in that it stores a static whole measuring amplitude (energy - colour). A hologram is a good example of PROCESS as it is more strongly dependant on time, since it uses phase interference (waves harmonics) as it's method of storage and requires a reference beam to recall the image. The invention of the laser allows for the creation of visual holograms, the hybridization of audition and vision. However, these holograms are currently lacking in full spectrum colour.

By recognizing the hybrid nature of our sensing system, as suggested by neurology, and by adopting specific tools for a specific sense, as we do in astronomy with different types of telescopes, we can develop systems that, for example, go back to pure wave analysis (what ever that is) and incorporate the ignoring of particle concepts. The uncertainty principle as a universal law may be wrong, and this may be manifest in the dichotomies used to describe uncertainty. 

The most common of these is that of position/momentum which can be classified as a 1:many type dichotomy of contexts. Here, position is of a FORM type and momentum of a PROCESS type and yet the PROCESS componant is 'many' (momentum implies changing movement over differing timeframes) whereas the FORM componant is 'one' (one position over differing timeframes). 

The principle of complementarity FORCES these dichotomies and the thinking arising from their use, - 

"In the microcosmos of quantum mechanics, phenomena abound that fly in the face of common sense. Many of these effects are a consequence of the principle of complementarity. It's most popular manifestation is the wave-particle duality. A microscopic object, such as a photon, an atom or an electron, can appear to behave as a water wave in one instance and as a discrete particle in another. Both features complement one another as a complete description of an object. Since the idea of complementarity was first enunciated more than 70 years ago, a belief common among many physicists has been that it is simply a consequence of the uncertainty relation. According to this rule, two complementary variables, such as position and momentum, cannot simultaneously be measured to less than a fundimental limit of accuracy. The uncertainty relation normally prevents one from learning everything about the behaviour of a quantum object. As a result we can never see the object acting both as a particle and as a wave." p56 (Englert, Scully, and Walther (1994))

Furthermore:
 

"We devised and analyzed both real and thought experiments that bypass the uncertainty relation, in effect, to "trick" the quantum objects under study. Nevertheless, the results always reveal that..complementarity remains intact even when the uncertainty relation takes no role."p56 (Englert, Scully, and Walther (1994))

and :
 

"The principle of complementarity implies that in the microcosmos, complete knowledge of the future in the sense of classical physics, is simply not available. If one of a pair of complementary properties of a quantum object is known for sure, then information about the second complementary property is lost." p57 (Englert, Scully, and Walther (1994))

and :
 

"This complementarity is a fact of life and we have to live with it. The Danish physicist Niels Bohr, more than anyone else, insisted on just that, and he deserves the lion's share of the credit for making us accept complementarity as a fundamental truth. It did not come easily, and the resistance put up by devil's advocates as prominent as Albert Einstein himself was formidable. The thrust of their arguments centred on whether complementary properties could be measured simultaneously."p57 (Englert, Scully, and Walther (1994))

and thus:
 

"The principle of complementarity is certainly more fundamental than is the uncertainty relation." p60 (Englert, Scully, and Walther (1994)). 

QM is highly successful on this planet but to use it in it's dualistic mode as a tool for analysis of the universe beyond this planet may lead to problems. It would be to our benefit to develop 'pure' wave analysis tools and ignore relativistic ties, thus enable a deeper understanding of the whole universe. It would also be to our benefit to perform a closer analysis of the manner in which we generate ideas, since the CR concept introduces levels of predictability not previously thought available.

In summary, the tendancy for humans to use the particle/field model is found in descriptions. In physics, to maintain locality and the concept of invarience, a particle undergoing transformtation was said to be operating in a field. This method of description is infact used in esoteric areas as well, where the astral body is supposed to move in the astral plane, as is the spiritual body in the spiritual plane. It can also be used in microbiology where the field is the DNA (group of information) and the particle the mRNA (specific gene for replication). 

The field/particle concept seems to be a natural and unconscious tool used for description. We can associate the field with wholeness ; prior to conscious distinction everything is balanced, a field of harmony including the future observer. A conscious observation is an act of putting a boarder around something in that to describe something we 'cut it out' of the background; bring it to the fore.

The wave equation.

                    Psi = (p) * (f)

This allows for the so called wave/particle duality. When we observe explicitly, we detect a particle. When we observe implicitly we detect a wave. However, the creation of this may arise from sensory hybridization.

Since we seem to have adapted to the environment by internalizing it, then at the highest level, the level of mind, 'we are one' with it - Level 1 (L1). The moment we choose to analize we automatically drop to Level 2. Reality, therefore, is implied. We can not prove explicitly that we exist, we can only accumulate enough L2 data to imply L1's 'true' state; this is a bit like an Airy pattern when seemingly random bits of information (electrons randomly hitting a photographic plate) lead to the emergence of a wave interference pattern (and therefore another level of information but implicit in form).

L2 seems to be the point reached by many mystics. To experience oneness just stop the mind, stop 'bounding' and just 'blend'; make oneself boundary free. By doing this one returns to harmony and balance. 

One of Von Neumann's Quantum Reality models suggests that the point where the probability wave of QM collapses to something 'classical', as in something we see (known as collapsing the wave function), is in the mind (NOT the brain - the mind; consciousness). Everett introduced the 'many worlds' model to get around the wave collapse. In his model, everything that could happen does; new universes appear originating from descision points. This means that the wave collapse never happens and therefore the measurement 'problem' does not exist. Richard Feynman introduced the sum of histories model as a tool of measurement. In this model everything that could happen tries to happen, and it is wave interference that causes reality. Everett's model seems to be an extension of Feynman. In measurements, Feynman's model works.

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wave-particle duality 

    possession by physical entities (such as light and electrons) of both wavelike and  particle-like   characteristics. On the basis of experimental evidence, the German physicist Albert  Einstein first   showed (1905) that light, which had been considered a form of electromagnetic waves, must also be   thought of as particle-like, or localized in packets of discrete energy. The French physicist  Louis de  Broglie proposed (1924) that electrons and other discrete bits of matter, which until then   had been  conceived only as material particles, also have wave properties such as wavelength and
  frequency.    Later (1927) the wave nature of electrons was experimentally established. An
   understanding of the  complementary relation between the wave aspects and the particle aspects of the same   phenomenon was announced in 1928 (see complementarity principle). 

 
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Information about this topic in other articles:

   duality

      category theory
           The duality principle
           from algebra

      lattice theory
           Duality
           from algebra

      optimization theory
           Optimal filtering and state estimation
           from optimization

      projective geometry
           Duality
           from projective geometry
           Projective geometry
           from mathematics, history of
           The foundations of geometry
           from mathematics, history of

      work of

         Kolmogorov
           Kolmogorov, Andrey Nikolayevich
           from Kolmogorov, Andrey Nikolayevich

         Plcker
           Plücker, Julius
           from Plccker, Julius

         Steiner
           Steiner, Jakob
           from Steiner, Jakob
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1. Evolution
2. Duality-Polyhedra: Article
3. Line-Plane
4.  line-plane
5. Control Theory http://www.signal.uu.se/Publications/abstracts/p931.html
6.  G-Duality for G-String
7.  control2
8. Search duality at Britannica.com:
9. Physics: article
10. Non-Abelian Duality
11. Triality-book
12. Theology

• Development, Duality, & the International Economic Regime: Essays in Honor of Gustav Ranis

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•  Maxima & Minima with Applications: Practical Optimization & Duality
by Kaplan, Wilfred 
•  Classifications of Abelian Groups & Pontrjagin Duality
 Loth, P.,  Buy this Book 
• Duality & Supersymmetric Theories, by David I. Olive (Editor)  Peter C. West (Editor)

Buy this Book
•  Fields, Strings & Duality , World Scientific Publishing Company Incorporated (Editor)
Format: Hardcover, 750pp. 
                          ISBN: 981023144X 
                           Publisher: World Scientific Publishing 
                           Company, Incorporated 
                           Pub. Date: May  1997 
• Linear Programming Duality: An Introduction to Oriented Matroids
                Achim Bachem  Walter Kern    Our Price: $63.00 
                           Format: Hardcover, 216pp. 
                           ISBN: 0387554173 
                           Publisher: Springer-Verlag New York, 
                           Pub. Date: August  1992