{The Sphere [continued (3)]}[22nd July 1988]

[Redbook5:219-220][19880722:2307c]{The Sphere [continued (3)]}[22nd July 1988]

19880722.2307

[continued]

*But I am also intrigued by the loose similarity of the (x+y)ⁿ basis** to the equation for the circle: x² + y² = r²*** (via Pythagoras’ Theorem).****

#

Taking in also the simple equations for the Cardinal Polarities (or Cartesian axes):

Vertical ‘y’ axis/polarity: x=0

Horizontal ‘x’ axis/polarity: y=0

and the diagonals (which may not be necessary):

Top left to bottom right: y=-x (and x=-y)

Bottom left to top right: y=x (and x=y)

(on the convention that positive, +, is right and up) – there perhaps begins to emerge some vague and simplified idea of how the construction and operation of the Circles#* might begin to be mathematically described.

The prospect of linking this in some way with actual experiment in Probability (i.e. Random) and Chaotic states makes this particularly interesting, as these are the two areas of mathematics which seem to have the most success in describing the World we actually experience every day.

*[See two last previous entries]

**NB Distributive Law separates! <880724>#

The obvious relationships are:

(1) Difference of:

2(x²+y²) – (x+y)² = (x-y)²

= 2x² + 2y² -x² -2xy -y²

= x² -2xy +y²

= (x-y)²

– which was arrived at by experiment;

i.e. 2 X (Circle Formula) less (Pascal#** Line 3 (Circle)) = ((Vertical Dimension) less (Horizontal Dimension))^squared

(or vice-versa, presumably, & no matter which is greater of x & y)

&

(2) If x = y (as they would in a circle),

(x+y)² = 2(x²+y²)

i.e. (Pascal#** Line 3#*** formula) = Twice (Circle Formula)

<880724>

***[where r = radius – see [Redbook5:206][19880703:0145]{A Mathematical Experience}[3rd July 1988]]

****[In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c: “a² + b² = c²”, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. (Adapted from Wikipedia)]

#Murphy, Additional Mathematics Made Simple, p67.

#*(or ‘Circle’ – can be viewed (per Pascal’s#** Δ [Triangle]) as simply one circle rotating 2 ways.)

#**[See last previous entry but one, [Redbook5:218-239][19880722:2307]{The Sphere}[22nd July 1988]]

#***[ms has Pascal Line 2 but this must be a mistake for Pascal Line 3]

[continues]

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