{Cumulative Doubling Instability}[10th July 1989]

[Redbook6:147][19890710:1722]{Cumulative Doubling Instability}[10th July 1989]

19890710.1722

*

[Text extracted from the ms image reproduced above]

0
↓
1	=	1						1
+
2	=	3					1		1
+	/
4	=	7				1		2		1
+	/
8	=	15			1		3		3		1
+	/
16	=	31		1		4		6		4		1
+	/
32	=	63	1		5		10		10		5		1
+	/
64	=	etc.

*

Cumulative addition of the doubling series leads each time to a state which is the next doubling minus 1: this being presumably an unstable state tending strongly towards the stability of 4 or multiples of 4 (at least from 3 onwards), the next change in the series – the next doubling-- follows immediately, and the whole process is repeated.

Why and how this sometimes goes wrong is, presumably, a matter of (deterministic) chaos.

What is curious** is the fact that it cannot start until the cumulative count is 3, assuming that 1 could as easily double to 2 or collapse back to 0.***

*(This is the cumulative count of the Pascal Triangle.)

[In the ms this diagram is in the left margin with the first para to the right of it)]

**(Theologically, that is.)

(But see earlier Vol. [Presumably, [Redbook5:194][19880625:0955b]{Pascal's Triangle}[20^th June 1988] &alff eg [Redbook5:218-239][19880722:2307]{The Sphere}[22nd July 1988] &c])

(& [[Redbook6:152][19890713:1847]{Pascal’s Church}[13th July 1989],] p152)

***{See [[Redbook6:152][19890713:1847]{Pascal’s Church}[13th July 1989],] 152}

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