Other notes
“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://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.)
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