Other notes

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“between geometry and the gesture” Trisha Brown.

The concept of symmetry, central to geometry and to mathematics as a whole, evolved during the 18th century, and is often equated with the emergence of modern mathematics. In common usage, symmetry refers to a correspondence between the parts of a figure or pattern.

A major development in our understanding of symmetry occurred when the correspondence was conceived of as an OPErATION on the figure, rather than a PrOPErTy of it. For example, the equality of the lengths of the sides of a square can be expressed as the invariance of the square under a quarter turn. This represents a crucial shift away from a static, figurative conception of symmetry and towards a dynamic, operative one.

With the notion of a symmetry operation two natural possibilities arise.

1. the possibility of composing two symmetry operations, one after another, and, conversely, decomposing a symmetry operation into simpler components.

2. the possibility of interpreting any operation on a space as a formal symmetry, be it pure, pleasing, or otherwise.

Questions about configurations of incident lines date back to the Greeks, but it wasn’t until much later that the mathematical study of perspectival symmetry, projective geometry, took form. .. For Dehn, these new geometries brought an intelligence to mathematics implicit in sensory experience.

It has become more and more apparent that different systems can be visualized, that, say, different types of spaces are compatible with experience. This is not to imply that the mathematician can now choose his assumptions at will. Not only is such arbitrariness likely to result in developments without beauty, but it is also likely to lead to contradictions that make all of the work an illusion.

"This is surely a different imagination of the postmodern fate of the medium than what other critics have proposed, such as its dematerialization into a “phenomenological vector” of opticality or horizontality or its permanent dismantling into the aggregative apparatus of film or video" See Krauss, “A Voyage on the North Sea.”.

http://en.wikipedia.org/wiki/Symmetry_in_mathematics

http://dgtal.org

http://www.walkerart.org/collections/artworks/agree-to-disagree-online

Philip Ording: A DEFINITE INTUITION

Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them — addition, subtraction, multiplication and division.)

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Another reference on sense experience and human perception is Brian Massumi's reflection on Feeling, Emotion and Affect on his book Parables of the Virtual, Movement, Affect, Sensation. “For Massumi, affect is precisely a matter of how intensities come together, move each other, and transform and translate under and beyond meaning, semantics and fixed systems/cognitions. Part of the assumption here is that, even in the most reactionay circumstances, nothing happens if affective intensity has not already paid us a visit. This refines our understanding of why territory - spatial and temporal - is always “existential territory“. It has as much a territorry that enables movement as something that keeps everything in its place, it is movement itself.” Reading / Feeling - If I Can’t Dance Archives. _

"One night not long ago, I dreamt I was explained the main difference between classical physics and quantum physics. It became perfectly clear to me that where classical physics is deterministic and assumes that the world is constant, ruled by the same laws on all levels, quantum physics looks at statistical, normative movements. In quantum physics there is not one standard way for an object to behave. The state of the phenomenon one wants to observe, take light as an example, can only be determined in the instance it is being observed: Light can be both wave and particle simultaneously. I recognized that in order to explain a single phenomenon, one needs two mutually exclusive theories." http://www.toriljohannessen.no/Nonlocality_page7.html _

"The absence of an empirically identifiable meeting point between the non-physical mind and its physical extension has proven problematic to dualism and many modern philosophers of mind maintain that the mind is not something separate from the body. These approaches have been particularly influential in the sciences, particularly in the fields of sociobiology, computer science, evolutionary psychology, and the neurosciences." at https://en.wikipedia.org/wiki/Mind–body_problem

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Our "shadowy" apprehension of reality in terms of flux, so graphically parabled by Plato in his Republic, has made its greatest impact through Einstein's theories, which demonstrate that energy and mass are interchangeable in the inseparable continuum of space and time. Quantum physics actually refers to subatomic partides, the shadowy stuff of which the universe is made, as “tendencies to exist.” These findings, affirmed by Heisenberg’s principle of “indeterminacy,” indicating how the nature that we seek merely to observe cannot help but be altered by man’s presence, confirm Renaissance man’s largely intuitive recognition of his own personal sense of spatialty, thus making it possible for Leonardo and Bramante to conceive of an art of performance and for us to recognize it as such today. The Art of Performance, A Critical Anthology - 1984 Edited By: Gregory Battcock and Robert Nickas

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Elementary algebra The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is a whole number and hence a rational number and a real number (as well as an algebraic number and a complex number).

The number 0 is neither positive nor negative and appears in the middle of a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).[40] Zero is, however, even.

The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.

Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition. Subtraction: x − 0 = x and 0 − x = −x. Multiplication: x · 0 = 0 · x = 0. Division: 0⁄x = 0, for nonzero x. But x⁄0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule. Exponentiation: x0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0. The expression 0⁄0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)⁄g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)⁄g(x), if it exists, must be found by another method, such as l'Hôpital's rule.

The sum of 0 numbers is 0, and the product of 0 numbers is 1. The factorial 0! evaluates to 1.

http://en.wikipedia.org/wiki/Zero_element

Conditional (CASE) expressions

CASE WHEN n > 0 
         THEN 'positive' 
    WHEN n < 0 
         THEN 'negative'
    ELSE 'zero'
END