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
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
· 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.
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 .
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.