{Chaotic Determinism (+ Extracts) [continued (12)]}[4th August 1988]

[Redbook5:253-254][19880804:1354b]{Chaotic Determinism (+ Extracts) [continued (12)]}[4th August 1988]

19880804.1354

[continued]

*The onset of chaotic states at c.16** in the basic period-doubling series which Pascal’s Triangle manifests*** (and which seems to start the chaos bifurcations, [–] even if other periods appear, and double, later) can also be applied to the structure of the Circles (set up by Pascal’s Triangle**** when it has reached exactly the same 16/32 Point of Accumulation.

(Incidentally, I don’t know if it is a co-incidence, but Feigenbaum’s number, 4.669, if taken as a Power [of 2], must represent that point on Pascal’s Triangle, ie n = 4.669, between 16 and 32.)#

[Text extracted from ms image shown above:]

					C
				(n=8)→	Attraction
	Simplification				(n=0)	→			Ordination
			(n=7)		1		(n=1)
				128	0	2
G~	Revolution	(n=6)	64		+		4	(n=2)	Outward Action	M~
				32	←	8
			(n=5)		16		(n=3)
	Fragmentation			(n=4.669?)↑	(n=4)				Complication
					Distraction
					A~

The beauty of this is that the Circle-constructing phase breaks up at or after n = 4 (16), which is where the Circles start (on the basis given above) to spin;#* and the spin itself follows the same pattern, producing a break up (or, in psychological terms, a breakdown) at or after 16 (where n = 4.669?).#** What this means is that, because of the results of the ‘experiments’ of Chaotic maths, the choice of 16 (8+8) points at which to complete the ordered stage of Circle construction is

(a) not arbitrary[;] and

(b) in accordance with the structure of the Outer Circle itself.

The Inner Circle, presumably, simply reverses the process, although the qualities are bound to differ because of that difference of direction.

*Points of particular relevance to C[ircles] A[nalysis]. <880807>

**[ie 2⁴]

***[[Redbook5:218-239][19880722:2307]{The Sphere}[22nd July 1988]]

****ref [[Redbook5:222-223][19880724:1443g]{The Sphere [continued (7)]}[24th July 1988]] 223ff

#(I can’t quite see the connection – but then, I don’t yet know exactly how F[eigenbaum]’s [Constant] No. was obtained.)

[See [Redbook5:251][19880803:1949c]{Chaotic Determinism (+ Extracts) [continued (7)]}[3rd August 1988], fn#*]

#*ref [[Redbook5:222-223][19880724:1443g]{The Sphere [continued (7)]}[24th July 1988]] p223ff.

[& probably originated earlier]

#**Is this a fractal dimension?

[See [Redbook5:251][19880803:1949c]{Chaotic Determinism (+ Extracts) [continued (7)]}[3rd August 1988], fn#*;

[Redbook4:271-274][19871230:0017]{The Mandelbrot Set}[30th December 1987] (which does not refer to Feigenbaum or the Constant)]

[continues]

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