Torus and Sphere and π and Φ

 

Article (revised and updated) from collected materials of the 3rd International Research and Application Conference “Tore Technologies” held November 23-24, 2006 at the Irkutsk State Technical University, Russia, pp. 131-143.

(Translation from Russian)

 

TORUS AND SPHERE ARE THE “PARENTS” OF PI (π), PHI

AND «7» BEING THE PRINCIPLES OF MATTER

STRUCTURING IN NATURE

Dr. Valeriy Shikhirin

E-mail: info@elastoneering.com, Website: www.elastoneering.com.

 

Principles of PI numbers

Let us consider “non-flat” versions of PI computation, for example, the relation between volumes, torus and sphere surface areas etc., inscribed into each other; the result of these relations might be integral or fractional PI number (Fig. 1).

Fig. 1 «Evolution of spherical π and toric π

Here, if:

·        PI «forms» circle, round, sphere or ball, then this PI is “spherical” π (πSP, Sphere).

·        PI «forms» tore, and then PI is “toric” π (πT,).

In tore shaping process spherical π is the first to act.

·        PI “forms” active torus knots automatically appearing during tore shaping, and then this PI is “nodal” π (πK, Knot). The number of nodal π is defined by number q – number of coils around tore’s length.

·

During shaping torus knots spherical and toric PIs are the first and the second to act respectively etc.

Numerical meaning of all functional PI is equal to 3,141 592 …

The author thinks it absolutely useless to calculate infinitely how many signs follow a comma in a PI.

Spherical π

We know that the relation of circumference to radius length produces 2π, but this is rather double π than a “single” one; it was obtained experimentally and was observed since great antiquity, at least from the 2nd millennium before our era (Fig.2).

Fig. 2 shows the sequence of simultaneous disintegration of a circle inscribed into sphere, “circumference” in a circle and radius in a circle and what such disintegration results in.

 

This seems to be a unique “direct” computation of double π, in particular: 2π = 2πR/R, and this one is only on a plane.

Besides, it is worth considering some other “sphere/circle” relations, for example the ratio of sphere surface area to surface area of a circle inscribed into sphere that produces “number” 4 or 4 “colors” (?) etc.

«Defects» and «deficiencies» characterizing careless and neglectful attitude of officials responsible for correct interpretation of PI are described in [7].

Toric  π (Fig. 3)

1. If sphere is inscribed into torus or a ball rolls or rests inside hollow ring (closed) with the same radius, then:

– Torus surface area and sphere surface area inscribed into torus ratio produces integral “single” π;

– Torus surface area and circle surface area inscribed into torus ratio produces 2π,

– The relation of surface area to surface area of a flat circular ring with internal radius RRin (Ringinternal) equal to 0 and external radius RRex (Ringexternal) equal to radius inscribed into torus produces 2π,

– The relation of torus volume to sphere volume produces 3/2π, or the ratio of three sphere volumes and two torus volumes produces integral π, or torus includes 3/2π of spheres.

2. If tore is inscribed into sphere, then:

– Torus surface area and sphere surface area ratio produces ½π or one torus surface area and two spheres surface area ratio;

– Torus volume and sphere volume ratio produces 3/8 π or 8 volumes of torus and 3 volumes of sphere ratio.

Besides, it is interesting to see other relations between /sphere elements whose formation can be done by pupils in a secondary school.

Fig. 3 shows the sequence of simultaneous disintegration of torus inscribed into sphere and sphere inscribed into torus and what is left after such disintegration.

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