Changes

Let us now consider the family of <math>4^\text{th}\!</math> gear curves through the extended space <math>\operatorname{E}^4 X = \langle x, dx, d^2 x, d^3 x, d^4 x \rangle.</math>  These are the trajectories that are generated subject to the law <math>d^4 x = 1,\!</math> where it is understood in making such a statement that all higher order differences are equal to <math>0.\!</math>

Let us now consider the family of <math>4^\text{th}\!</math> gear curves through the extended space <math>\operatorname{E}^4 X = \langle x, dx, d^2 x, d^3 x, d^4 x \rangle.</math>  These are the trajectories that are generated subject to the law <math>d^4 x = 1,\!</math> where it is understood in making such a statement that all higher order differences are equal to <math>0.\!</math>

<pre>

Since <math>d^4 x\!</math> and all higher order <math>d^j x\!</math> are fixed, the entire dynamics can be plotted in the extended space <math>\operatorname{E}^3 X = \langle x, dx, d^2 x, d^3 x \rangle.</math> Thus, there is just enough room in a planar venn diagram to plot both orbits and to show how they partition the points of <math>\operatorname{E}^3 X.</math>  As it turns out, there are exactly two possible orbits, of eight points each, as illustrated in Figures&nbsp;16-a and 16-b.  See here:

Since d^4.x and all higher order d^j.x are fixed, the entire dynamics

can be plotted in the extended space E^3.X = <|x, dx, d^2.x, d^3.x|>.

Thus, there is just enough room in a planar venn diagram to plot all

of these orbits and to show how they partition the points of E^3.X.

As it turns out, there are exactly two possible orbits, of eight

points each, as illustrated in Figures 16-a and 16-b.  See here:

DLOG D23.  http://stderr.org/pipermail/inquiry/2003-May/000502.html

:* [http://stderr.org/pipermail/inquiry/2003-May/000502.html DLOG D23]

Zoom^4 ...

<math>\operatorname{Zoom}^4\ \ldots</math>

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==Note 7==

==Note 7==