Coordinates

Introducing coordinates
Pointing is about introducing coordinates. One cannot do too much math with non-coordinated structures. In a combinatorial set, all elements are 'total' conjugates without any other determination. For example, if |A|=3 and |B|=5, all we can write about A and B inside Species Theory may be expressed as 5 ' ' = 3 (Sym5 is a two point extension of Sym3).

To define a (mathematical) function on the (mathematical) set A, first one must point three times A toward •••A, thus obtaining an X.X.X; and then define the function on X.X.X, only after the elements of A got their own identity and they become distinct within respect to any conjugation.

X.X.X...X may be seen as a 'solid' structure, a 'zero-entropy' structure, with no possibility to make confusions between conjugate elements.

In Geometry, the coordination is implicitly introduced when choosing points or quadrangles or whatever. After choosing two points, all other points of an affine line are well determined (modulo the Galois conjugation)(unlike sets, the geometrical structures are efficiently coordinated with less pointing operations; after several pointing operations they became X.X.X...X's). In the corresponding algebraic structures, 0 and 1 are technologically introduced by the very first axioms, and all other numbers become well determined : one element-one sign, without confusions.

The pointing verb, the key to coordination, is 'let it be' (French 'soit'). For example, to totally coordinate the complex field and to eliminate the Galois confusion, one must point the i : let i be such that i.i = 1. Nicolae-boicu (talk) 12:00, 15 January 2013 (UTC)

Labeling the Fano plane
File:Fano plane.svg|thumb|The Fano plane

Let temporarily, just for an easier reading
 * Fano = X7/PSL(2,7)a = the species that correspond to the collineation group of the well knownFano plane

and
 * Klein = P4bic = the species that correspond to the Klein group that fixes a rectangle.

then
 * Fano" = X.Klein

meaning : Since two points determine a line, after labeling any two points in the Fano plane another point is settled. The relabeling liberty for the rest of the four remaining points is described by the Klein Group. By continuing the differentiation, one obtains :
 * Fano"' = X'.Klein + X.Klein' = Klein + X.X.X.X

meaning : After labeling three distinct points in the Fano plane, exactly two situation can occur :
 * - (Either the points are collinear and) the remaining labeling liberty is described by Klein
 * - (Or the points form an oval and) the whole plane is coordinated : X.X.X.X

The corresponding species cycle index is :


 * $$ Z_{\mathrm{Fano}}(x_1, x_2, x_3, x_4, x_7)= {1 \over 168} [ x_1^7 + 21 x_1^3 x_2^2 + 42 x_1 x_2x_4 + 56 x_1 x_3^2 + 48 x_7 ]  $$

After two differentiation with respect to x1 the "Klein" species shows up:


 * $$ Z_{\mathrm{Fano}}'' = x_1.{1 \over 4} [ x_1^4 + 3 x_2^2 ] = x_1Z_{\mathrm{Klein}} $$

The e.g.f. is


 * $$\iint x\cdot6\cdot{x^4 \over 4!} = 30\cdot{x^7 \over 7!}$$

hence there are 30 ways to label the plane. Here 6 represents the six distinct ways of labeling the affine (Klein) corresponding plane.