{The Sphere}[22nd July 1988]

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

19880722.2307

According to Brian Pippard, reviewing James Gleick's 'Chaos: Making a New Science' in T[imes ]L[iterary ]S[upplement ]880722-28 p800, certain manifestations of the chaotic states show mathematically and in practice development in jumps between powers of 2 – i.e. 1,2,4,8[,] etc. This is entertaining for me as it can be arrived at by Pascal’s “Mystic Triangle”, which I earlier* remarked on as an explanation of how the Circles pattern might have developed. (In [2], +K misses out the second line for some reason:** I was dubious about this at the time.)

[(Table1:) Arithmetical numbers from image shown above:]

							(0)							Line Totals	Cum Totals	Cum Totals +1
							1							1	1	2
						1		1						2	3	4
					1		2		1					4	7	8
				1		3		3		1				8	15	16
			1		4		6		4		1			16	31	32
		1		5		10		10		5		1		32	63	64
	1		6		15		20		15		6		1	64	127	128

[(Table2:) Marginal text etc. (excluding algebraic formulae added later) from image above:]

Possible Circle Application:	(Origin)	(0)	Line Totals	#*Cum. Totals	#*Cum. Totals +1	xy terms	Cartesian #**
Centre {This line omitted per Murphy (&+K!)}		1	1	1	2	(x+y)0 ?
Vertical Polarity		1+1	2	3	4	(x+y)1	x=0
Cardinal Polarities	See***	1+2+1	4	7	8	(x+y)2	x=0 y=0
Diagonals & Cardinals	See ****	1+3+3+1	8	15	16	(x+y)3	x=y x=-y
?Outer & Inner or Diagonals in both Outer & Inner		1+4+6+4+1	16	31	32	(x+y)4	x2 + y2 = r2 ?#
Infilling….)		1+5+10+10+5+1	32	63	64	(x+y)5
↓		1+6+15+20+15+6+1	64	127	128	(x+y)6
			128
			etc.	etc.	etc.
(The ‘1’s on the outside remain				#*This	[column]		#**[=???]
the Vertical Polarity extremes.)		etc.		probably	irrelevant

[(Table3:) Algebraic formulae from image above (with core numbers for reference):]

		(0)	xy terms	Cartesian
		(x0)1(y0)	(x+y)0 ?	#**
		1x1 + 1y1	(x+y)1	x=0
		1x2 + 2xy + 1y2	(x+y)2	x=0 y=0
		1x3 + 3x2y1 + 3x1y2 + 1y3	(x+y)3	x=y x=-y
		1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4	(x+y)4	x2 + y2 = r2 ?#
		1x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1y5	(x+y)5
		1x6 +6x5y1 + 15x4y2 + 20x3y3 + 15x2y4 + 6x1y5 + 1x6	(x+y)6

[Powers and multipliers of 1 are presumably shown only for clarity]

*ref [[Redbook5:194][19880625:0955b]{Pascal's Triangle}[20^th June 1988],] 194

**[Possibly a reference to the same passage quoted in [Redbook5:119-120][19880511:1920]{Revival}[11^th May 1988]: “First there was One, then four… then there were many, the Net.”]

So does Murphy, ‘Additional Mathematics Made Simple’, p140. (This does not seem to be a printing error). <880724>

So does the Tarot Pack? – the 2 ‘Jokers’, at +G & +M. <880727>

[Redbook4:28-33][19870709:2358]{The Invisible Cards}[9^th July 1987]ff,

& esp [Redbook4:31-32][19870710:0855d]{The Invisible Cards [continued(6)]}[10^th July 1987];

& [Redbook4:253][19871221:1955d]{The Tarot Pack [continued (6)]}[21st December 1987]]

***(This stage produces the Circle – see [[Redbook5:206][19880703:0145]{A Mathematical Experience}[3rd July 1988]] pp 206,

[[Redbook5:218-239][19880722:2307c]{The Sphere [continued (3)]}[22nd July 1988],] 220

****

ref e.g. [2]

	+Mk				+K				+C
/		\		/		\		/		\
			xP				xS

#[Ref [Redbook5:206][19880703:0145]{A Mathematical Experience}[3rd July 1988], 1^st fn]

[& cf [Redbook8:145][19901223:1645b]{The Sierpiński Gasket}[23rd December 1990]<20221016>]

[continues]

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