Spheres

Chapter 5

Spheres

Armillary Sphere

An armillary sphere was an ancient astronomical instrument. It was a spherical model of the universe and consisted of a number of graduated brass rings representing the chief celestial circles, such as the meridian, equator, and the tropics. Armillary spheres were probably invented about 255 bc and were used until the 17th century.

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 (GP-B's gyroscopes are the roundest objects ever made. Engineers at the NASA Marshall Space Flight Center polished them to within 0.01 microns (less than 40 atom-widths) of perfect sphericity.)

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Meniscus

= one concave face interior

= one convex face exterior.

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Collective excilations area mode of oscillation in a many body system in which there is a cooperative motion of the whole system.

Complex conjugate

R cos θ – ir sin Θ         Polar form of z*

Z+z* = 2x, zz* = x^z + y^z

Argand reflection of complex numbers about the real axis.

Coriolis force is a simplified calculation of rotating systems (surface) such as the movement of air over the surface of a rotating object.

The toroid has a major radius of R at a cross-section, self-induction in a circle (A=πr²) L≈μ₀N²A/2πR, N= turns if R>>r.

Torque is the ability of a force to rotate a body about some axis, and is measured by a quantity called the toque T. The torque due to a force of F has the magnitude T=Fd, d=distance.

Alexander Waugh(1999) “ We see in every direction, wheels, cycles revolving cogs in a mechanical clock, the spheres of the earth and the moon, all the planets revolving around the sun, eclipses, spheres endlessly returning in unchanging circles, the hands of a clock going around. The universe is a geometrically four-dimensional sphere.”

The first polyhedron is the one that has one face and yet an infinite number of faces. The sphere! The presence of the sphere, in droplets and bubbles, fruit seeds etc. the shape arises any where growth begins from a point and has no reason not to push outward from that point evenly in all directions. The image of the sphere has been used through the ages to depict physical, biological, and philosophical phenomena. Spherical shells that the surfaces curved in two directions (Laplace’s law) that double curvature give great strength and stiffness for an investment of material. Maximization of internal volume for a given surface compounds the material economy – for that nothing beats a sphere. They are resistant to uniform transmutably pressure differences. A charged metal sphere, Let e be its charge and a radius the surface density w. w=e/4πa³ the potential of this rotational field is φ_a=e/r+k on the sphere it has the constant value φ_a=e/a+k. In order to obtain an electrostatic field that is physically possible, we must assign a terminus or sink for the flux of force issuing from the charged sphere. The second concentric hallow sphere of internal radius encloses the first and is charged with negative electricity. Since the charge –e is distributed uniformly over the sphere the surface density has  the value w=e/4πb ² and the  potential is φ_b=e/b+k. This is called a spherical capacitor. The quotient of the positive charge e by the “voltage” or potential difference  φ _a – φ _b =v  between the positively and negatively charged conductors is call the capacitance of the capacitor. For the spherical capacitor we have: v=φ _a –φ_b=e(1/a – 1/b)=e b-a/ab and its capacitance is c= e/φ_a-φ_b = ab/b-a, by diminishing the distance between the two spheres very large capacitance can be obtained. In the MKSA system the voltage of a spherical capacitor is v=φ_a-φ_b=e/4π_εo(1/a – 1/b) = e(b-a)/4π_εoab, and the capacitance is c=4π_εoab/b-a. Parallel – plate capacitor = distance between spheres of uniform fields. E = φ₁-φ₂/d, d=b-a, s=4πab the surface density w of electricity is w=φ₁-o₂/4πd the separation at plates(spheres) is d and the area is s, c=sw/φ₁-φ₂  = s/4πd.

Ellipsoid of revolution

The equipotent surface of the rotational field is co focal ellipsoids of revolution. The strength of the source-line of length 2c we set = e the potential produced is φ=

 e/2c ∫^te_-e dξ/r in which ξ is the distance of the point of integration from mid-point of the source-line and r is the distance of the field point from the source (point) at mid-point we have r=√[(z-ξ)² + x ² + y ²] and we find φ = -e/2c {1n(z-ξ+r)}^ξ=+e_ξ=-e = e/2c 1n z+c+r₁/z-c+r₂ where r₁ and r₂ are the distances of the field point from the end points designated ξ =-c and ξ =+c of the source line. The elliptical coordinates u and v in the zh-plane. U=1/2(r₁+r₂). V=1/2 (r₁+ r₂) the elementary geometry and the equation of the ellipse z²/a² + h²/b² = 1 the ere ads = 2πh√(dz²+dh²) = 2πb dz√[1-(cz/a²)²] the surface charge density is w=e/ 4πab√[1-(cz/a^z)²] the density has its smallest value Wm=e/4πab on the equator of the ellipsoid (where z=0) and the greatest value W_max=e/4πb² at the poles (z=±a) the field strength E=4πw  On the surface of a charged metallic body increases with increasing curvature (sharp-point effect). The image force e²/4a² the potential energy is =-c²/4c² directed toward the metallic surface. If the surface density is inversely proportional to the cube of the distance from the point charge the polar coordinates (p,Ψ)  in the plane the total charge is ∫wds = ea/2π ∫^∞_o∫^2π_o pdpdΨ/)p²+a²)^3/2 = ea[1/√(p²+a²)]^p=∞_p=o=-e. Dielectric sphere in a uniform field in a vacuum ε₁=ε, ε ₂  =1, the interior field is reduced by a factor of 3/(ε+2), compared to “in air.” In the space outside the sphere the field acts like that of a dipole of moment p=E_o k =E_o a³ ε-1/ε-2. The polarization  p(moment per unit volume) has a value p=E_o x 3/4π ε-1/ε+2 = E_i x ε-1/4π  and the internal field  E _i  produces this polarization is E: =E_o - 4π/3 p  and the conducting sphere behaves like an insulator of infinitely great permittivity. Spherical moment of inertia m 2/5 r² radius any diameter, m 2r²/5 equatorial radius (polar axis), m 2/3 r² shell mean radius, m 2/5 (r₁s-r₂5)/(r₁3-r₂3) r₁=external r₂=internal, α=360^o/n about an axis n=1,2,3,4,or 6. Rotator-reflection axis ñ=ĩ,2^~,3^~ … Poisson equation – spherical charge distribution p(r) and asks for the electric field strength  F(r) at a distance r ₁ from the center of the sphere. The charge outside the sphere of radius r ₁  does not contribute to F,  because the field strength inside a spherical cavity is zero. Q₁ = ∫^ri_o p(r), 4πr² x dr  The distribution of charge fields inside and outside a sphere act as a point charge located in the center of the sphere. F(r₁) = Q₁/4π_εoε x rf =1/4πε_oε x rf x ∫^r1_op(r) x 4πr²dr. F(r₁) is connected with the potential by F(r₁) =-(dφ/dr)_r1 therefore rf x (dφ/dr)_r1 = -1/ε_oε x∫^r1_o p(r) x r² x dr.

Ring theory

Emmy Noether  deals with abstraction of numbers as well as functions and operations “abstract algebra” also symmetries gauge and Noether theory = 100,000 trillion times smaller scale. Symmetry is dependent on time-space and control of the dynamics of physical interactions of matter. Space is considered isotropic, the same in all directions. In a continuum there is no smallest step, an infinite number of possible translational symmetry operation. Our universe is three-dimensional continuous translational symmetry. Symmetry is present when the equations don’t include a special point in space, “the laws of physics are invariant under translations in space” the laws of physics are time-translational invariant. Space has continuous rotational symmetries of the  laws of physics. The sphere is invariant under transformation, there are an infinite number of symmetry operations that can be performed on a sphere. A sphere is a continuous symmetry. The rotational symmetry of a spherical object is intimately connected to the symmetry of space, therefore you can’t distinguish between a rotating spherical object and rotating the entire universe about the spherical object. In a 3 dimensional universe, there are 3 perpendicular directions, in which we can translate a physical system, there must be 3 conserved momenta, one for each direction. Therefore, momentum, position of particles, velocity of a particle, or the force acting on a particle, has both direction in space and a magnitude, called a vector. The total momentum must be conserved. Angular momentum ultimately leads to “spooky” quantum phenomena and bizarre consequences for the behavior of matter in extreme conditions. There is no friction in space. Friction is a consequence of the complexity of the world around us, once accounted for the laws of inertia holds throughout the universe. 1g=10m/s².

38% of g = comfortable acceleration, the magic number of gravity is G_n = 6.673 x 10^-11m³/kgs².

(Note there is an inherent symmetry in the human body and any operation the disturbs that symmetry is not good medicine.)

Aether

Aether “Energy” was a popular field of study in the 1800’s (John Ernest Worrel Keely)

The Characteristics of Aether are 1- A superfluity particulate medium which pervades all space (derivable from fluid mechanic) 2 – A medium, which in its various modes, the building blocks of the physical universe (vibratory physics) 3 – A medium, which in one of its modes is responsible for gravity and inertia (zero-point energy/ sonic stimulation) 4 – A medium which can be controlled by geometric shapes (spherical)

Experiments showed that between December 8^th -15^th and May 8^th-15^th are the heaviest local gravitation and inertial forces.

Tens of unexplainable microscopic and macroscopic effects in natural sciences and especially in physics and biology have been revealed and investigated. A large part of these phenomena were demonstrated by objects having spin or angular momentum. The first researchers who experimentally detected the unusual effects associated with torsion, Russian Professor N.P. Myshkin. In the 1940’s astrophysicist N.A. Kozyrew proposed the connection of rotation with energy output. According to the theory time and rotation are closely interconnected. N.A. Kozyrew detected a change in the weight with the angular velocity and the direction of rotation. The observed effects are explained as being the manifestation of some property of time. According to this theory, every substance has its own “chronal charge” defined by the quantity of chronal particles which were named “chronons.” It is the interaction of these chronons that affect the weight of the object spinning. They generated a field around it generated by spinning masses. A.I. Veinik discovered they were two types of chronons (plus and minus), dependant on the orientation of the spin. In all cases the observed effects of anti-gravitation. To explain the effects of anti-gravitation as a manifestation of torsion fields generated by the spin.

In the course of the latter 50 years there have been numerous reports on anomalous behavior or spin-polarized particles. If a mass ~ 1 kg is rotating with an angular velocity ~ 20,000 rpm then the inner force ~ 30^* 10^-5N. Their chronal fields could not be shielded by the usual screens. Many of these phenomena could be explained as results of the manifestation of long-range fields generated by spin or angular momentum density.

Later experiments conducted by the Institute of Material Research in Russia, It was that the emanation produced by mechanically rotating magnets was able to change the inner structure of any substance (its spin structure) interpreted as torsion radiation. G.I. Shipov based his investigations on the “Theory of physical vacuums” he used geometry of absolute parallelism with 6 additional rotational coordinated 10 movement equations. From Shipov’s vacuum equations every known fundamental physical equation (Einstein, Young-Mills, and Heisenberg) can be deduced in completely geometrized form. Besides the two known long range physical fields, electromagnetic and gravitational – there exists a third long range field possessing significantly richer properties; the torsion field. First the upper limit for the speed of torsion wave is estimated to be not less than 10⁹c (c= speed of light). Second torsion fields are able to propagate not only in the future but in the past as well. Thirdly torsion fields transmit information without transmitting energy. Forth torsion fields are not required to follow the superposition principle.

Since all substances (except amorphous materials) have there own stereo-chemistry, which determines not only the location of atoms in molecules but also determines their mutual spin orientation, the superposition of torsion fields generated by the atomic and nuclear spin of each molecule determines the intensity of the torsion field in the space surrounding each molecule. The superposition of all these torsion fields determines the intensity and spatial configuration of the characteristic torsion field of that substance.

The spin-torsion interaction constant is equal to 10^-5 – 10^-6. This constant is less than the constant of electro magnetic interaction, yet greater than the constant of gravitational interaction. Thus the structure of the torsion field of every object can be changed by the influence of an external torsion field. The new configuration of the torsion field will be fixed as a metastable state and will remain intact after the source of the external torsion field is removed. Since every permanent magnet possesses not only oriented magnetic moments but also classical spin orientation as well, every permanent magnet possesses its own torsion field. Strong torsion fields are generated by high electrical potentials and by devices with organized circular or spiral electromagnetic processes.