US20060135288A1

US20060135288A1 - Great-circle geodesic dome

Info

Publication number: US20060135288A1
Application number: US11/296,350
Authority: US
Inventors: Randell Mills
Original assignee: Individual
Current assignee: Individual
Priority date: 2004-12-22
Filing date: 2005-12-08
Publication date: 2006-06-22

Abstract

The present invention is a structural system and a method of fabricating the system comprised of finite elements called basis elements such as great circles or partial great circles to form a great-circle geodesic sphere, dome, or arch called a great sphere, dome, and arch, respectively. The structure is generated by forming a pattern of the basis element rotated at incremental angles about a rotational axis from an initial position to define a so-called primary component orbitsphere-cvf. The primary component orbitsphere defines a stationary pattern as well as a structural element called a secondary component orbitsphere-cvf that is constructed according to the pattern of the primary orbitsphere-cvf. The secondary component orbitsphere-cvf is initially oriented at the initial position and is incrementally rotated about the rotational axis to form the great-sphere-type structure.

Description

This application claims priority U.S. Provisional Patent Application Ser. No.: 60/651,006, filed 9 Feb. 2005; 60/647,406, filed 28 Jan. 2005; 60/643,149, field 13 Jan. 2005; and 60/637,889, filed 22 Dec. 2004, the complete disclosures of which are incorporated herein by reference.

FIELD OF THE INVENTION

A conventional dome comprising a saddle structure or a conventional geodesic dome made of triangular basis elements have elements under tension; whereas, a perfect dome is essentially or entirely under compression since all of weight is equally distributed. Furthermore, a perfect sphere has no unique position on the surface or unique axis. This invention relates to a structure that more closely replicates a perfect sphere or dome using finite elements derived from great circles. In an embodiment, the surface has a characteristic that axes defined by crossings of great-circle elements are not unique. Such an architectural structure or container is anticipated to be stronger and provide more efficiency of material requirements than in conventional systems such as a structure made of triangular basis elements as in the case of a conventional geodesic dome. The structure is also more esthetic and natural. It further permits the construction of near perfect spherical dishes without machining out a solid material which is a very challenging fabrication problem. Broad application in architecture and industry by one skilled in the Art is anticipated and within the scope of the current Invention.

BACKGROUND OF THE INVENTION

This Invention relates to a geometrical derivation of a means to generate a perfect sphere using great circles. The disclosure and background are given in the book, R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2006 Edition, BlackLight Power, Inc., Cranbury, N.J.; posted at http://www.blacklightpower.com/bookdownload.shtml (“Mills GUT”), which is incorporated by reference in its entirety. The Chapter One and Appendix III are preferred references which are also incorporated by this reference.

SUMMARY OF THE INVENTION

The object of this invention is to provide a sphere or spherical section such as a dome as a universal architectural structure or container, that is a more ideal of a perfect sphere or dome or spherical section using finite elements.
The present invention is a structural system and a method of fabricating the system comprised of finite elements called basis elements such as great circles or partial great circles to form a great-circle geodesic sphere, dome, or arch called a great sphere, dome, and arch, respectively. The structure is generated by forming a pattern of the basis element rotated at incremental angles about a rotational axis from an initial position to define a so-called primary component orbitsphere-cvf. The primary component orbitsphere defines a stationary pattern as well as a structural element called a secondary component orbitsphere-cvf that is constructed according to the pattern of the primary orbitsphere-cvf. The secondary component orbitsphere-cvf is initially oriented at the initial position and is incrementally rotated about the rotational axis to form the great-sphere-type structure. The structure defined as Y₀ ⁰(φ,θ) may formed according to Eqs. (67-71) of the ANALYTICAL EQUATIONS TO GENERATE THE ORBITSPHERE CURRENT VECTOR FIELD AND THE UNIFORM CURRENT (CHARGE)-DENSITY FUNCTION Y₀ ⁰(φ,θ) section.
Another embodiment of the Invention comprises a uniform current density function on a two dimensional surface defined as Y₀ ⁰(φ,θ) and a method and constructing Y₀ ⁰(φ,θ). The current density function may have a defined angular momentum along two orthogonal axes that is determined by the selection of the desired angular momentum of the basis elements and by the selection the rotational matrices that form the Y₀ ⁰(φ,θ) with the desired angular momentum projections. The angular momentum components have corresponding magnetic moments in the embodiment wherein the elements are current loops. A further embodiment comprises a uniform mass-density structure with desired angular momentum components. This embodiment is constructed by using mass-flowing elements rather than current elements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is the element pattern given by Eq. (1) shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 2 is the element pattern of the orbitsphere-cvf component of STEP ONE shown with 6 degree increments of θ from the perspective of looking along the z-axis. The yz-plane great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 3 is the element pattern of the orbitsphere-cvf component of STEP ONE shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the xz-plane is shown as red in accordance with the present invention;
FIG. 4 the element pattern given by Eq. (7) shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the xy-plane is shown as red is in accordance with the present invention;
FIG. 5 is the element pattern for the rotation of the xy-plane great circle about the (−i_x,0i_y,i_z)-axis (Eq. (10)) shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the xy-plane is shown as red in accordance with the present invention;
FIG. 6 is the element pattern of the orbitsphere-cvf component of STEP TWO shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the xy-plane is shown as red in accordance with the present invention;
FIG. 7 is the element pattern of the orbitsphere-cvf component of STEP TWO shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 8 is the element pattern of the orbitsphere-cvf shown with 6 degree increments of θ from the perspective of looking along the z-axis onto which L_R, the resultant vector of the L_xyand L_zcomponents, was aligned in accordance with the present invention;
FIG. 9 is the schematic of the relative dimensions of the component orbitsphere-cvfs (STEP-ONE component shown in blue and STEP-TWO component shown in red) that make-up the orbitsphere-cvf in accordance with the present invention;
FIG. 10 is the element pattern given by Eq. (30) shown with 6 degree increments of θ from the perspective of looking along the z-axis. The great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 11 is the element pattern of the orbitsphere-cvf component given by Eq. (33) that is orthogonal to that of STEP ONE shown with 6 degree increments of θ from the perspective of looking along the z-axis. The yz-plane great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 12 is the element pattern given by Eq. (37) shown with 6 degree increments of θ from the perspective of looking along the z-axis obtained from Eq. (32) by rotation about the (i_x,−i^y,0i_z)-axis by π using Eq. (34). The great circle element that served as a basis element that was initially in the yz-plane is shown as red in accordance with the present invention;
FIG. 13 is the orbitsphere, a two dimensional spherical shell in accordance with the present invention;
FIG. 14 is a representation of the uniform element pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (67)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (30) and 30 degree increments of the rotation of this basis element about the (i_x,i_y,0i_z)-axis corresponding to Eq. (4). The perspective is along the z-axis. The great circle element that served as a basis element that was initially in the plane along the (i_x,−i_y,0i_z)- and z-axes of each secondary component orbitsphere-cvf is shown as red. Note that it is stationary over the convolution due to phase matching in accordance with the present invention;
FIG. 15 is a representation of the uniform element pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (67)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (30) and 30 degree increments of the rotation of this basis element about the (i_x,i_y,0i_z)-axis corresponding to Eq. (4). The great circle element that served as a basis element that was initially in the plane along the (i_x,−i_y,0i_z)- and z-axes of each secondary component orbitsphere-cvf is shown as red. The perspective is transverse to the z-axis in accordance with the present invention;
FIG. 16 is a representation of the uniform element pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (68)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (32) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the yz-plane of each secondary component orbitsphere-cvf is shown as red. Note that it is not stationary over the convolution due to phase matching. It is out of phase with the secondary component orbitsphere by a −π/4 rotation about the z-axis. The perspective is along the z-axis in accordance with the present invention;
FIG. 17 is a representation of the uniform element pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (68)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (32) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the yz-plane of each secondary component orbitsphere-cvf is shown as red. The perspective is transverse to the z-axis in accordance with the present invention;
FIG. 18 is a representation of the uniform element pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (68)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (32) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the yz-plane of each secondary component orbitsphere-cvf is not highlighted. The perspective is along the z-axis in accordance with the present invention;
FIG. 19 is a representation of the uniform current pattern of the Y₀ ⁰(0,O) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (71)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (70) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the plane along the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
—and y-axes of each secondary component orbitsphere-cvf is shown as red. Note that it is not stationary over the convolution due to phase matching. It is out of phase with the secondary component orbitsphere by −π/4 about the y-axis. The perspective is along the z-axis in accordance with the present invention;
FIG. 20 is a representation of the uniform current pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (71)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (70) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the plane along the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
—and y-axes of each secondary component orbitsphere-cvf is shown as red. The perspective is transverse to the z-axis in accordance with the present invention, and
FIG. 21 is a representation of the uniform current pattern of the Y₀ ⁰(φ,θ) orbitsphere shown with 30 degree increments (N=M=12 in Eq. (71)) of the angle to generate the orbitsphere current-vector field corresponding to Eq. (70) and 30 degree increments of the rotation of this basis element about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis corresponding to Eq. (20). The great circle current loop that served as a basis element that was initially in the plane along the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
—and y-axes of each secondary component orbitsphere-cvf is not highlighted. The perspective is along the z-axis in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

An embodiment of the present Invention comprises an architectural system and a method of fabricating the architectural system. The system comprises a structure formed from great-circle elements or sections of such elements to form a sphere or a section of a sphere such as a half shell, a dome, or an arch. The elements may have essentially the same diameter and may be bound together at the crossings of the elements or may be intertwined at the crossings. Ideally the system is constructed to equalize the forces throughout the surface of the structure. Preferably, the structure is two-dimensional with equal weight or pressure distribution.
In an embodiment, the elements are full great circles of the same diameter, and the corresponding structure is a sphere. In another embodiment, the basis elements are half great circles of the same radius, and the corresponding structure is a dome.
In an embodiment, the architectural structure has the form given by at least one of Eqs. (67-71) of the Analytical Equations to Generate the Orbitsphere Current Vector Field and the Uniform Current (Charge)-Density Function Y₀ ⁰(φ,θ) section. In an embodiment, the elements are arranged according to the structure of a primary component orbitsphere-cvf wherein each great-circle element of the primary component orbitsphere-cvf is replaced by a secondary orbitsphere-cvf wherein each great-circle element of each orbitsphere-cvf element comprises a structural element such as a tube, bar, rod, beam, or similar element that is in the form of a great circle or a section of a great circle.
In an embodiment, the curved elements such as great circles or partial sections of great circles are approximated with straight subelements. In an embodiment, an element having an average structure of a curve can be constructed from elements that are not curved such as straight elements. Other such subelements are known to those skilled in the Art.
In any embodiment, the great-circle basis elements may comprise the framing to provide structural strength, and the framing may be covered with a continuous membrane or a tiling that forms a continuous or partially continuous covering of the framing. In other embodiments, the structures may accommodate structures such as doors and windows and other structural elements.
In an embodiment, the number of such secondary component orbitsphere-cvf elements is equal to the number of great-circle elements of the primary component orbitsphere-cvf. However, the number of elements of the secondary component orbitsphere, N, and the primary component orbitsphere, M, do not need be the same. One skilled in the Art could also determine the optimum numbers N and M to achieve a desired goal such as least cost to achieve an application with the desired performance characteristics.
In an embodiment the angles N and m of Eqs. (67), (68), and (71) are integer multiples within the range of ±0.00001% to 50%, preferably within the range of ±0.0001% to 25%; more preferably ±0.01% to 10%, and most preferably ±1% to 5%. The higher the number of elements, N and M, the more closely the structure approximates a perfect sphere or partial sphere corresponding to great circle elements or partial great circle elements, respectively. The tolerance from a perfect sphere or partial sphere depends on the particular application. One skilled in the Art would be able to determine the structure and geometrical comprise between the number of elements and realization of perfect weight or pressure distribution, or physical form.
In an embodiment, the angles n2π/N and m2π/M of Eqs. (67), (68), and (71) are generally represented by nθ₁/N and mθ₂/M wherein each of θ₁and θ₂is an independent angle within the range of 0≦θ_1,2≦2π. One skilled in the Art could select the range to form the desired structure from the basis element such as a great circle or partial great circle.
The architectural system may be used as a building or part of a building such as an arena, stadium, atrium, roof of a structure, a dwelling, or a structural component of a structure such as a support for a bridge or building.
In other embodiments, the invention comprises a container, dish, or vessel formed from great-circle elements or sections of such elements to form a sphere or a section of a sphere such as a half shell, a dome, or an arch. The container may be for solids, liquids, or gases. Spherical dishes are also embodiments. A spherical antenna or mirror for uses such as communications or telescopes are further embodiments. In another embodiment, the structure is used as the fuselage or structural elements of a submarine or pressurized gas container for applications such as transporting liquefied natural gas.
The system may be formed directly from great-circle elements or sections of great-circle elements. Alternatively, a number of secondary component orbitsphere-cvf elements or sections of such elements may be fabricated and these elements may be connected to form the structure. The great-circle basis elements may be welded, bolted, riveted, clamped, glued, or otherwise fastened at their crossings with other such elements to form the architectural structure.
Another embodiment of the Invention comprises a uniform current density function on a two dimensional surface defined as Y₀ ⁰(φ,θ) and a method and constructing Y₀ ⁰(φ,θ). The current density function may have a defined angular momentum along two orthogonal axes that is determined by the selection of the desired angular momentum of the basis elements and by the selection the rotational matrices that form the Y₀ ⁰(φ,θ) with the desired angular momentum projections. The angular momentum components have corresponding magnetic moments in the embodiment wherein the elements are current loops. A further embodiment comprises a uniform mass-density structure with desired angular momentum components. This embodiment is constructed by using mass-flowing elements rather than current elements.
Embodiments of the structures and methods are given in the ANALYTICAL EQUATIONS TO GENERATE THE ORBITSPHERE CURRENT VECTOR FIELD AND THE UNIFORM CURRENT (CHARGE)-DENSITY FUNCTION Y₀ ⁰(φ,θ) section. A specific embodiment to construct a uniform current density function with $L_{xy} = \frac{ℏ}{4} and L_{z} = \frac{ℏ}{2}$
is given. Other angular momentum components are within the scope of the Invention with the use of basis elements of the corresponding angular momentum as would be used by one skilled in the Art. The great dome embodiment is also taught in this section with the use of structural elements rather than current-loop basis elements.
Analytical Equations to Generate the Orbitsphere Current Vector Field and the Uniform Current (Charge)-Density Function Y₀ ⁰(θ,φ)
STEP ONE by the Rotation of a Great Circle About the (i_x,i_y,0i_z)-Axis by 2π
Great Circle in the yz-Plane About the (i_x,i_y,0i_z)-Axis
Following the procedure given in Fowles [1], the orbitsphere-cvf component of STEP ONE is generated by the rotation of a great circle in the yz-plane about the (i_x,i_y,0i_z)-axis by 2π. A first transformation matrix is generated by the combined rotation of a great circle in the yz-plane about the z-axis by π/4 then about the x-axis by θ where positive rotations about an axis are defined as clockwise: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \cos (\frac{π}{4}) & \sin (\frac{π}{4}) & 0 \\ - \sin (\frac{π}{4}) \cos θ & \cos (\frac{π}{4}) \cos θ & \sin θ \\ \sin (\frac{π}{4}) \sin θ & - \cos (\frac{π}{4}) \sin θ & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (1) \end{matrix}$
The transformation matrix about (i_x,i_y,0i_z) is given by multiplication of the output of the matrix given by Eq. (1) by the matrix corresponding to a rotation about the z-axis of −π/4. The output of the matrix given by Eq. (1) is shown in FIG. 1 wherein θ is varied from 0 to 2π. The rotation matrix about the z-axis by −π/4, $zrot (- \frac{π}{4}),$
is given by $\begin{matrix} zrot (- \frac{π}{4}) = [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) & \cos (\frac{π}{4}) & 0 \\ 0 & 0 & 1 \end{matrix}] Thus, & (2) \\ \begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = zrot (- \frac{π}{4}) \cdot \\ [\begin{matrix} \cos (\frac{π}{4}) & \sin (\frac{π}{4}) & 0 \\ - \sin (\frac{π}{4}) \cos θ & \cos (\frac{π}{4}) \cos θ & \sin θ \\ \sin (\frac{π}{4}) \sin θ & - \cos (\frac{π}{4}) \sin θ & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] \end{matrix} & (3) \end{matrix}$
Substitution of the matrix given by Eq. (2) into Eq. (3) gives $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{2} + \frac{\cos θ}{2} & \frac{1}{2} - \frac{\cos θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ \frac{1}{2} - \frac{\cos θ}{2} & \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ \frac{\sin θ}{\sqrt{2}} & - \frac{\sin θ}{\sqrt{2}} & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (4) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} (\frac{1}{2} - \frac{\cos θ}{2}) r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ (\frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ - \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ + \cos θ r_{n} \sin ϕ \end{matrix}] & (5) \end{matrix}$
The orbitsphere-cvf component of STEP ONE that is generated by the rotation of a great circle in the yz-plane about the (i_x,i_y,0i_z)-axis by 2π corresponding to the output of the matrix given by Eq. (5) is shown in FIG. 2 wherein the sign of φ is positive for 0≦θ≦π or and negative for π≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section of Mills GUT,
Great Circle in the xz-Plane About the (i_x,i_y,0i_z)-Axis
Alternatively, STEP ONE comprises the rotation of a great circle in the xz-plane about the (i_x,i_y,0i_z)-axis by 2π. The coordinates of the great circle are given by the matrix: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} r_{n} \cos ϕ \\ 0 \\ r_{n} \sin ϕ \end{matrix}] & (6) \end{matrix}$
The matrix for the rotation about the (i_x,i_y,0i_z)-axis is given by Eq. (4) wherein θ is varied from 0 to 2π and the sign of φ is positive for 0≦θ≦π and negative for π≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section. The current pattern that is equivalent to that shown in FIG. 2 is shown in FIG. 3.
It follows from the results shown in FIGS. 2 and 3 that the component orbitsphere-cvf for STEP ONE can further be generated by the rotation of the linear combination of the basis-element great circles in the yz- and xz-planes about the (i_x,i_y,0i_z)-axis by π using Eqs. (4) and (6) wherein θ is varied from 0 to π. It is a general feature for the generation of the components given in this Appendix that a linear combination of the orthogonal basis-element great circles can used rather than a single element wherein the range of θ is varied from 0 to π rather than from 0 to 2π.
STEP TWO by the Rotation of a Great Circle About the (−i_x,0i_y,i_z)-Axis by 2π Followed by a Rotation About the z-Axis by π/4
Great Circle in the xy-Plane About the (−i_x,0i_y,i_z)-Axis by 2π Followed by a Rotation About the z-Axis by π/4
Following the procedure given in Fowles [1], the orbitsphere-cvf component of STEP TWO is generated by the rotation of a great circle in the xy-plane about the (−i_x,0i_y,i_z)-axis by 2π followed by a rotation about the z-axis by π/4. A first transformation matrix is generated by the combined rotation of a great circle in the xy-plane about the y-axis by −π/4 then about the z-axis by θ; $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \cos (\frac{π}{4}) \cos θ & \sin θ & \sin (\frac{π}{4}) \cos θ \\ - \cos (\frac{π}{4}) \sin θ & \cos θ & - \sin (\frac{π}{4}) \sin θ \\ - \sin (\frac{π}{4}) & 0 & \cos (\frac{π}{4}) \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (7) \end{matrix}$
The transformation matrix about (−i_x,0i_y,i_z) is given by multiplication of the output of the matrix given by Eq. (7) by the matrix corresponding to a rotation about the y-axis of π/4. The output of the matrix given by Eq. (7) is shown in FIG. 4 wherein θ is varied from 0 to 2π.
The rotation matrix about the y-axis by π/4, $yrot (\frac{π}{4}),$
is given by $\begin{matrix} yrot (\frac{π}{4}) = [\begin{matrix} \cos (\frac{π}{4}) & 0 & - \sin (\frac{π}{4}) \\ 0 & 1 & 0 \\ \sin (\frac{π}{4}) & 0 & \cos (\frac{π}{4}) \end{matrix}] Thus, & (8) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = yrot (\frac{π}{4}) \cdot [\begin{matrix} \cos (\frac{π}{4}) \cos θ & \sin θ & \sin (\frac{π}{4}) \cos θ \\ - \cos (\frac{π}{4}) \sin θ & \cos θ & - \sin (\frac{π}{4}) \sin θ \\ - \sin (\frac{π}{4}) & 0 & \cos (\frac{π}{4}) \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (9) \end{matrix}$
Substitution of the matrix given by Eq. (8) into Eq. (9) gives the current pattern for the rotation of the xy-plane great circle about the (−i_x,0i_y,i_z)-axis as shown in FIG. 5: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} & - \frac{1}{2} + \frac{\cos θ}{2} \\ - \frac{\sin θ}{\sqrt{2}} & \cos θ & - \frac{\sin θ}{\sqrt{2}} \\ - \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} & \frac{1}{2} + \frac{\cos θ}{2} \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (10) \end{matrix}$
STEP TWO is then given by a rotation of this result about z-axis by π/4 using $zrot (\frac{π}{4})$
given by $\begin{matrix} zrot (\frac{π}{4}) = [\begin{matrix} \cos (\frac{π}{4}) & \sin (\frac{π}{4}) & 0 \\ - \sin (\frac{π}{4}) & \cos (\frac{π}{4}) & 0 \\ 0 & 0 & 1 \end{matrix}] & (11) \end{matrix}$
Thus, STEP TWO is given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = zrot (\frac{π}{4}) \cdot yrot (\frac{π}{4}) \cdot [\begin{matrix} \cos (\frac{π}{4}) \cos θ & \sin θ & \sin (\frac{π}{4}) \cos θ \\ - \cos (\frac{π}{4}) \sin θ & \cos θ & - \sin (\frac{π}{4}) \sin θ \\ - \sin (\frac{π}{4}) & 0 & \cos (\frac{π}{4}) \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (12) \end{matrix}$
Substitution of the matrices given by Eqs. (8) and (11) into Eq. (12) gives $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} & \frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2} & \frac{- \frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} \\ - \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} & \frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2} & - \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} \\ - \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} & \frac{1}{2} + \frac{\cos θ}{2} \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (13) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} (\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \cos ϕ + (\frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2}) r_{n} \sin ϕ \\ (- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \cos ϕ + (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \sin ϕ \\ (- \frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \end{matrix}] & (14) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{4} r_{n} (\sqrt{2} \cos ϕ + 2 \sin θ (- \cos ϕ + \sin ϕ) + \sqrt{2} \cos θ (\cos ϕ + 2 \sin ϕ)) \\ \frac{1}{4} r_{n} (- \sqrt{2} \cos ϕ - \sqrt{2} \cos θ (\cos ϕ - 2 \sin ϕ) - 2 \sin θ (\cos ϕ + \sin ϕ)) \\ \frac{1}{2} r_{n} ((- 1 + \cos θ) \cos ϕ + \sqrt{2} \sin θ \sin ϕ) \end{matrix}] & (15) \end{matrix}$
The orbitsphere-cvf component of STEP TWO that is generated by the rotation of a great circle in the xy-plane about the (−i_x,0i_y,i_z)-axis by 2π followed by a rotation about the z-axis by π/4 corresponding to the output of the matrix given by Eq. (15) is shown in FIG. 6 wherein the sign of φ is positive for 0≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section. (The eigenvector $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
of the matrix given by Eq. (20) corresponding to a rotation about this axis can also be obtained by phase-shifting φ in Eqs. (14-15)) by π/4. The phase shift is given by a shift of time such that φ is phase-shifted by π/4. Since the current on each loop is continuous and periodic, this mathematical aspect does not change the physical current-density as readily appreciated by comparing FIG. 6 to FIG. 7. However, phase shifts with respect to θ do have an impact. Phase matching in a convolution operation to generate the uniform function Y₀ ⁰(θ,φ) is discussed in the Matching Phase, Angular Momentum, and Orientation section.)
Great Circle in the yz-Plane About the (−i_x,0i_y,i_z)-Axis by 2,T Followed by a Rotation About the z-Axis by π/4
Alternatively, STEP TWO comprises the rotation of a great circle in the yz-plane about the (−i_x,0i_y,i_z)-axis by 2π followed by a rotation about the z-axis by π/4. The coordinates of the great circle are given by the matrix: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (16) \end{matrix}$
The matrix for the rotation about the (−i_x,0i_y,i_z)-axis by 2π followed by a rotation about the z-axis by π/4 is given using Eq. (13) wherein a is varied from θ to 2π and the sign of φ is negative for 0≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section. The current pattern that is equivalent to that shown in FIG. 6 is shown in FIG. 7.
STEP TWO by Rotation of a Great Circle About the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-Axis by 2π
The orbitsphere-cvf component of STEP TWO is also generated by the rotation of a great circle about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis by 2π wherein the basis-element great circle bisects the xy-quadrant and is parallel to the z-axis. The coordinates of the great circle are given by the matrix that rotates a great circle in the yz-plane about the z-axis by π/4: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \cos (\frac{π}{4}) & \sin (\frac{π}{4}) & 0 \\ - \sin (\frac{π}{4}) & \cos (\frac{π}{4}) & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (17) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \sin (\frac{π}{4}) r_{n} \cos ϕ \\ \cos (\frac{π}{4}) r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] = [\begin{matrix} \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ r_{n} \sin ϕ \end{matrix}] & (18) \end{matrix}$
Since STEP TWO is given by the rotation of the yz-plane basis-element great circle (Eq. (16)) about the (−i_x,0i_y,i_z)-axis by 2π followed by a rotation about the z-axis by π/4 using Eq. (13), the equivalent result may be obtained by first rotating the great circle given by Eq. (18) about the z-axis by −π/4, $zrot = (- \frac{π}{4}),$
then applying Eq. (13): $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{\frac{1}{2} + \frac{\cos ϕ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} & \frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2} & \frac{- \frac{1}{2} + \frac{\cos ϕ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} \\ - \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} & \frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2} & - \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2} \\ - \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} & \frac{1}{2} + \frac{\cos θ}{2} \end{matrix}] \cdot zrot (- \frac{π}{4}) \cdot [\begin{matrix} \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ r_{n} \sin ϕ \end{matrix}] & (19) \end{matrix}$
Using $zrot (- \frac{π}{4})$
from Eq. (2) gives $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{4} (1 + 3 \cos θ) & \frac{1}{4} (- 1 + \cos θ + 2 \sqrt{2} \sin θ) & \frac{1}{4} (- \sqrt{2} + \sqrt{2} \cos θ - 2 \sin θ) \\ \frac{1}{4} (- 1 + \cos θ - 2 \sqrt{2} \sin θ) & \frac{1}{4} (1 + 3 \cos θ) & \frac{1}{4} (\sqrt{2} - \sqrt{2} \cos θ - 2 \sin θ) \\ \frac{1}{2} (\frac{- 1 + \cos θ}{\sqrt{2}} + \sin θ) & \frac{1}{4} (\sqrt{2} - \sqrt{2} \cos θ + 2 \sin θ) & \cos^{2} \frac{θ}{2} \end{matrix}] [\begin{matrix} \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ \frac{r_{n} \cos ϕ}{\sqrt{2}} \\ r_{n} \sin ϕ \end{matrix}] & i . (20) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{(1 + 3 \cos θ) r_{n} \cos ϕ}{4 \sqrt{2}} + \frac{(- 1 + \cos θ + 2 \sqrt{2} \sin θ) r_{n} \cos ϕ}{4 \sqrt{2}} + \frac{1}{4} (- \sqrt{2} + \sqrt{2} \cos θ - 2 \sin θ) r_{n} \sin ϕ \\ \frac{(1 + 3 \cos θ) r_{n} \cos ϕ}{4 \sqrt{2}} + \frac{(- 1 + \cos θ - 2 \sqrt{2} \sin θ) r_{n} \cos ϕ}{4 \sqrt{2}} + \frac{1}{4} (\sqrt{2} - \sqrt{2} \cos θ - 2 \sin θ) r_{n} \sin ϕ \\ \frac{(\frac{- 1 + \cos θ}{\sqrt{2}} + \sin θ) r_{n} \cos ϕ}{2 \sqrt{2}} + \frac{(\sqrt{2} - \sqrt{2} \cos θ + 2 \sin θ) r_{n} \cos ϕ}{4 \sqrt{2}} + \cos^{2} \frac{θ}{2} r_{n} \sin ϕ \end{matrix}] & i . (21) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{4} r_{n} (2 \sin θ (\cos ϕ - \sin ϕ) + \sqrt{2} (2 \cos θ \cos ϕ + (- 1 + \cos θ) \sin ϕ)) \\ \frac{1}{4} r_{n} (- 2 \sin θ (\cos ϕ + \sin ϕ) + \sqrt{2} (\cos θ (2 \cos ϕ - \sin ϕ) + \sin ϕ)) \\ \frac{1}{2} r_{n} (\sqrt{2} \cos ϕ \sin θ + (1 + \cos θ) \sin ϕ) \end{matrix}] & i . (22) \end{matrix}$
where the sign of φ is negative for 0≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section of Mills GUT. The current pattern of the orbitsphere-cvf component of STEP TWO that is generated by the rotation of a great circle in the xyz-plane about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis by 2π corresponding to the output of the matrices given by Eqs. (20-22) is equivalent to that shown in FIGS. 6 and 7. In the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-rotational-axis case, the great circle current loop that serves as a basis element and initially and finally bisects the xy-quadrant and is parallel to the z-axis is shown as red in FIG. 7.
The current pattern of the orbitsphere-cvf component of STEP TWO shown in FIG. 6 is also generated by the rotation of a great circle in the xy-plane about the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-axis by 2π. Here, the great circle given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (23) \end{matrix}$
is input to transformational matrix given by Eq. (20), and the sign of φ is positive for 0≦θ≦2π in order to give the angular momentum projections given in the Orbitsphere Equation of Motion for l=0 section of Mills GUT.
Characteristics of the Orbitsphere-cvf
From Eqs. (1.76) and (1.77), angular momentum components of the orbitsphere-cvf are $L_{xy} = \frac{ℏ}{4} and L_{z} = \frac{ℏ}{2} .$
The corresponding resultant angular momentum vector, L_R, has magnitude √{square root over (5)}h/4 along the direction of the spherical-coordinate angles θ=0.4636 rad, $ϕ = \frac{3 π}{4} rad .$
To obtain the view along L_R, the orbitsphere-cvf is first rotated counterclockwise about the vector (i_x,i_y,0i_z) by an angle −0.4636 rad using Eq. (4) or Eq. (1.69) wherein Δα_x′ and Δα_y′. are each −√{square root over (2)}(0.4636) rad to align L_Rwith the z-axis as shown in FIG. 8.
The angular momentum is constant with respect to rotation of the orbitsphere-cvf about the axis of the resultant angular momentum vector, L_R. In this case, the corresponding component angular momentum L_xyis rotationally constant about the xy-axis, and the corresponding L_xand L_ycomponents are rotationally constant about the x- and y-axes, respectively. The component L_zis further rotationally constant about the z-axis. The constancy of the angular momentum with respect to rotation of the orbitsphere-cvf about each of the principal axes determines that the corresponding rotational symmetry of each axis is C_∞ for the corresponding component even though the spatial symmetry of the current distribution is less. The angular-momentum axis of each component orbitsphere-cvf corresponding to either STEP ONE or STEP TWO is also a C_∞-axis. Each component is comprised of great circles, and each great circle has a spatial and angular-momentum C_∞-axis perpendicular to the plane it defines. In addition, each component as well as the orbitsphere-cvf has the origin as a spatial inversion center (C_i) as shown in FIGS. 1.10A-F of Mills GUT.
Each component orbitsphere-cvf has an infinite number of spatial C₂-axes that lie in a symmetry plane (σ_ν-plane) with a perpendicular spatial C_∞-axis. Consider that the C₂- and C_∞-axes shown in FIG. 9 are defined as the z-axis and the y-axis of the STEP-ONE component, respectively. The x-axis is perpendicular to the y- and z-axes. Then, the σ_ν-plane is the xz-plane. The parameters of the width a from the center at the position of the σ_ν-plane to the edge of the STEP-ONE component of the orbitsphere-cvf and the distance b from the edge to the apex of a circle defined by the orthogonal STEP-TWO component orbitsphere-cvf are shown in FIG. 9. Also shown is the angle θ between the C_∞-axis (y-axis) of the STEP-ONE component and the intersection of the edge of the two orthogonal component orbitsphere-cvfs. By symmetry, this is also the angle between the σ_ν-plane and the edge. These parameters can be related to the radius r_nof the orbitsphere-cvf. From FIG. 9, the relationship between the radius and the width is
a ² +a ² =r _n ² (24)
Thus, the width is $\begin{matrix} a = \frac{r_{n}}{\sqrt{2}} Since & (25) \\ a + b = r_{n} & (26) \end{matrix}$
the distance from the edge of the STEP-ONE component to the to the apex at the STEP-TWO component using Eq. (25) is $\begin{matrix} b = r_{n} (1 - \frac{1}{\sqrt{2}}) & (27) \end{matrix}$
The angle θ is then $\begin{matrix} θ = \cos^{- 1} \frac{\frac{r_{n}}{\sqrt{2}}}{r_{n}} = \frac{π}{4} & (28) \end{matrix}$
As shown in FIG. 9, the current density of the STEP-ONE component orbitsphere-cvf is constant along the C₂-axis (z-axis) and increases from the origin to the edge along the C_∞-axis. Since the current is on a sphere, the corresponding polar coordinates along this span are from $ϕ = 0 to ϕ = \frac{π}{4} .$
It can be appreciated from FIG. 2 that at φ=0, the great circles are initially at an angle of $θ^{'} = \frac{π}{4}$
relative to the C₂-axis as defined in FIG. 9 and are at an angle $θ^{'} = 0 at ϕ = \frac{π}{4} .$
Thus, the differential area per loop is given by the cosine of the angle $ϕ + \frac{π}{4},$
and the density D is given by the inverse of the area function: $\begin{matrix} D = \sec (ϕ + \frac{π}{4}) & (29) \end{matrix}$
The Uniform Current (Charge)-Density Function Y₀ ⁰(θ,φ)
Boundary Constraints
The further constraint that the current density is uniform such that the charge density is uniform, corresponding to an equipotential, minimum energy surface is exactly satisfied by using the orbitsphere-cvf as a basis element to generate Y₀ ⁰(θ,φ). Utilizing the symmetry properties of each component of the orbitsphere-cvf corresponding to either STEP ONE or STEP TWO and the orthonormality of the trigonometric functions that generate the orbitsphere-cvf, a convolution operator comprising an autocorrelation-type function [2] gives rise to the spherically-symmetric current density, Y₀ ⁰(θ,φ). The operator comprises the convolution of each great circle current loop of the orbitsphere-cvf designated as the primary orbitsphere-cvf with a second orbitsphere-cvf designated as the secondary orbitsphere-cvf. The angular momenta of the convolved elements are matched. The elements are also orientation matched by rotation of the secondary about the appropriate axis (axes), and the elements are phase matched using a rotation of each secondary orbitsphere-cvf element about its C_∞-axis. The convolution is over the angular span θ=0 to θ=2π corresponding to the rotation of the basis-current loop which generated the primary orbitsphere-cvf. The angular momenta of the secondary elements project onto the resultant angular momentum axis, L_R-axis, of the primary orbitsphere- cvf equivalently to those of its great circles. The resulting exact uniform current distribution obtained from the convolution has the same angular momentum distribution, resultant, L_R, and components of $L_{xy} = \frac{ℏ}{4} and L_{z} = \frac{ℏ}{2}$
as those of the orbitsphere-cvf used as a primary basis element.
Properties of the Orbitsphere-cvf Permissive to Generate Y₀ ⁰(θ,φ)
First, consider the symmetry properties of the each of the two orthogonal components the orbitsphere-cvf corresponding to STEP ONE and STEP TWO. As shown in the previous sections and the Orbitsphere Equation of Motion for l=0 section, a basis-element great circle is rotated by 2π to generate the same current pattern as that generated over the surface by rotation of two orthogonal great circle current loops by π using Eqs. (1.69) and (1.71). The resulting component orbitsphere-cvf is always perpendicular to the 2π-axis of rotation used to generate the component from the great circle. This 2π-rotational axis defines a unique C_∞-axis which serves as a vector to characterize the angular momentum of the orbitsphere-cvf great circles. The resultant angular momentum of any given nth pair of great-circle elements defined by the rotational angle of the nth great circle at θ=θ_nand its orthogonal partner at θ=θ_n+π may be along the C_∞-axis or perpendicular to it depending on the current direction of the great circles for 0≦θ≦π and π≦Θ≦2π. In the case that the angular momentum is perpendicular to the C_∞-axis, the angular momentum vector rotates about the C_∞-axis by π as a function of θ of Eq. (5) or as a function of θ of Eqs. (1.72-1.73) as shown in FIG. 1.11 of Mills GUT. Also, shown in FIG. 1.11 is an example of a stationary angular momentum vector that results when the angular momentum of each pair of great circles is along the C_∞-axis.
Component Orbitsphere-cvf Orthogonal to that of STEP ONE be the Rotation of a Great Circle About the (i_x,−i_y,0i_z)-Axis by 2π
As a further example of a stationary orbitsphere-cvf angular momentum vector, consider the case of STEP ONE using the great circle current loops shown in FIG. 1.4 of Mills GUT as the basis elements. The resultant angular momentum vector of magnitude $ℏ$ $\frac{}{2 \sqrt{2}}$

shown in FIG. 1.4 is stationary throughout the rotations of the orthogonal-great-circle basis set about the (i_x,−i_y,0i_z)-axis by an angle π or one of the great circles by 2π that transform the axes as given in Table 1 during the generation of the component of the orbitsphere-cvf. As shown by comparing FIGS. 2 and 11, the resulting component orbitsphere-cvf called STEP ONE ⊥ is orthogonal to that of STEP ONE given by rotation of the basis-element(s) great circle(s) about the perpendicular axis of (i_x,−i_y,0i_z).

TALE 1


Summary of the results of the rotation of the two orthogonal current loops
in the xz- and yz-planes about the (i_x, −i_y, 0i_z) -axis to generate a
component orbitsphere-cvf with a stationary angular momentum vector.

Step	Initial Direction	Final Direction	Initial to Final	L_x	L_y
	of Angular	of Angular	Axis
	Momentum	Momentum	Transformation
	Components	Components
	({circumflex over (r)} × {circumflex over (K)})^a	({circumflex over (r)} × {circumflex over (K)})^a

$1 ⊥$	{circumflex over (x)}, −ŷ	{circumflex over (x)}, −ŷ	+x′ → −y	$\frac{ℏ}{4}$	$- \frac{ℏ}{4}$
			+y′ → −x
			+z′ → −z

\begin{matrix} ^{a} K is the current density, r is the polar vector of the great circle, and \\ “ \hat{} ” denotes the unit vectors \hat{u} \equiv \frac{u}{\langle u \rangle} . \end{matrix}

Great Circle in the yz-Plane About the (i_x,−i_y,0i_z)-Axis
Following the procedure given in Fowles [1], the component orbitsphere-cvf that is orthogonal to that of STEP ONE (STEP ONE ⊥) is generated by the rotation of a great circle in the yz-plane about the (i_x,−i_y,0i_z)-axis by 2π. A first transformation matrix is generated by the combined rotation of a great circle in the yz-plane about the z-axis by $- \frac{π}{4}$
then about the x-axis by θ where positive rotations about an axis are defined as clockwise: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) \cos θ & \cos (\frac{π}{4}) \cos θ & \sin θ \\ - \sin (\frac{π}{4}) \sin θ & - \cos (\frac{π}{4}) \sin θ & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (30) \end{matrix}$
The transformation matrix about (i_x,−i_y,0i_z) is given by multiplication of the output of the matrix given by Eq. (30) by the matrix corresponding to a rotation about the z-axis of $\frac{π}{4} .$
The output of the matrix given by Eq. (30) is shown in FIG. 10 wherein θ is varied from 0 to 2π.
The rotation matrix about the z-axis by π/4, $zrot (\frac{π}{4}),$
is given by Eq. (1 1). Thus, $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = zrot (\frac{π}{4}) \cdot [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) \cos θ & \cos (\frac{π}{4}) \cos θ & \sin θ \\ - \sin (\frac{π}{4}) \sin θ & - \cos (\frac{π}{4}) \sin θ & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (31) \end{matrix}$
Substitution of the matrix given by Eq. (11) into Eq. (31) gives $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{2} + \frac{\cos θ}{2} & - \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ - \frac{1}{2} + \frac{\cos θ}{2} & \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ - \frac{\sin θ}{\sqrt{2}} & - \frac{\sin θ}{\sqrt{2}} & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (32) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} (- \frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ (\frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ - \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ + \cos θ r_{n} \sin ϕ \end{matrix}] & (33) \end{matrix}$
The component orbitsphere-cvf that is orthogonal to that of STEP ONE that is generated by the rotation of a great circle in the yz-plane about the (i_x,−i_y,0i_z)-axis by 2π corresponding to the output of the matrix given by Eq. (33) is shown in FIG. 11 wherein the sign of φ is positive for 0≦θ≦π and negative for π≦θ≦2π in order to give the currents directions shown in FIG. 1.4 of Mills GUT. The angular momentum vector is stationary along the (i_x,−i_y,0i_z)-axis as shown in FIG. 1.4 of the Orbitsphere Equation of Motion for l=0 section of Mills GUT. The same result is given by the rotation of a great circle in the xz-plane about the (i_x,−i_y,0i_z)-axis by 2π using Eqs. (6) and (32).
Matching Phase, Angular Momentum, and Orientation
For STEP ONE ⊥, the resultant angular momentum vector, L_R, is along (i_x,−i_y,0i_z). The angular momentum is constant for any rotation about the axis; thus, it is a C_∞-axis relative to the angular momentum. However, rotation about this axis does change the phase (coordinate position relative to the starting position) of the component orbitsphere-cvf. For example, a rotation by π about the (i_x,−i_y,0i_z)-axis using Eq. (32) causes the basis-element great circle to rotate by $\frac{π}{2}$
about the z-axis as shown in FIG. 12. The rotation matrix about the (i_x,−i_y,0i_z)-axis by π, (i_x,−i_y,0i_z)rot(π), is given by $\begin{matrix} (i_{x}, - i_{y}, 0 i_{z}) rot (π) = [\begin{matrix} \frac{1}{2} + \frac{\cos π}{2} & - \frac{1}{2} + \frac{\cos π}{2} & \frac{\sin π}{\sqrt{2}} \\ - \frac{1}{2} + \frac{\cos π}{2} & \frac{1}{2} + \frac{\cos π}{2} & \frac{\sin π}{\sqrt{2}} \\ - \frac{\sin π}{\sqrt{2}} & - \frac{\sin π}{\sqrt{2}} & \cos π \end{matrix}] = [\begin{matrix} 0 & - 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix}] Thus, & (34) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = (i_{x}, - i_{y}, 0 i_{z}) rot (π) \cdot [\begin{matrix} \frac{1}{2} + \frac{\cos θ}{2} & - \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ - \frac{1}{2} + \frac{\cos θ}{2} & \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ - \frac{\sin θ}{\sqrt{2}} & - \frac{\sin θ}{\sqrt{2}} & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (35) \end{matrix}$
Substitution of the matrix given by Eq. (34) into Eq. (35) gives the current pattern for the $\frac{π}{2}$
phase shift relative to the z-axis corresponding to a rotation of the component orbitsphere-cvf given by Eq. (32) about the (i_x,−i_y,0i_z)-axis by π as shown in FIG. 12: $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \sin^{2} \frac{θ}{2} & - \cos^{2} \frac{θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ - \cos^{2} \frac{θ}{2} & \sin^{2} \frac{θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ \frac{\sin θ}{\sqrt{2}} & \frac{\sin θ}{\sqrt{2}} & - \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (36) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} - \cos^{2} \frac{θ}{2} r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ \sin^{2} \frac{θ}{2} r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ - \cos θ r_{n} \sin ϕ \end{matrix}] & (37) \end{matrix}$
In general, for the stationary-angular-momentum-vector case, the angular momentum vector along the C_∞-axis is always at an angle of $\frac{π}{4}$
relative to the plane of the basis-element great circle that generated the component orbitsphere-cvf. Thus, the plane of the great circle relative to the xyz coordinate system can be rotated over a span of $\pm \frac{π}{4}$
rotation about L_R. The rotation of the basis-element great circle corresponds to changing the phase of the component orbitsphere-cvf in Eq. (38). The phase of the secondary component orbitsphere-cvf can be matched to that of the basis-element great circle of a primary component orbitsphere-cvf by rotation about L_R. Furthermore, each component orbitsphere-cvf is comprised of great circles, and each great circle has a C_∞-axis perpendicular to the plane. This feature further permits the phase within the great circles of the secondary component orbitsphere-cvf to be matched to that of the basis element great circle of the primary.
Since the angular momentum vector is stationary and is a C_∞-axis, the secondary component orbitsphere-cvf can be made to match the angular momentum of the basis-element great circle of any primary component orbitsphere-cvf by rotations that align the vector of the former with that of the latter. In addition to phase and angular momentum, the orientation of the secondary component orbitsphere-cvf is matched to that of the great circles elements of the primary, by rotation about the appropriate axis that aligns the angular momenta and orientation of the secondary component orbitsphere-cvf with the basis-element great circle of the primary. Thus, the secondary component orbitsphere-cvf serves as a basis element in a convolution operator as shown in the Convolution Operator section.
Convolution Operator
The orbitsphere-cvf comprises two components corresponding to each of STEP ONE and STEP TWO. A uniform current-density function can be generated from each component independently having the same resultant angular momentum as the corresponding component, and the two uniform functions can be superimposed to give Y₀ ⁰(θ,φ).
Consider the component STEP ONE ⊥ generated in the Component Orbitsphere-cvf Orthogonal to that of STEP ONE by Rotation of a Great Circle about the (i_x,−i_y,0i_z)-axis by 2π section. The component is generated by rotation of the two orthogonal great circles about the (i_x,−i_y,0i_z)-axis by π or the rotational of one of the loops about this axis by 2π. In the former case, the resultant angular momenta of the pair of orthogonal great circle current loops and that of the component orbitsphere-cvf are along this rotational axis, and in the latter case, the resultant angular momentum of the component orbitsphere-cvf is equivalent to that of the former.
When the resultant angular of the component orbitsphere-cvf is along the axis about which the basis elements or element are rotated to generate the component orbitsphere-cvf, the angular momentum of the basis elements or elements in conserved in the superposition of the rotated basis elements or element. In the case that one basis element is used, the rotational axis is the 2π-rotational axis of the Properties of the Orbitsphere-cvf Permissive to Generate Y₀ ⁰(θ,φ) section. Thus, in the general case that the resultant angular momentum of the component orbitsphere-cvf is along the 2π-rotational axis, a secondary nth component orbitsphere-cvf can serve as a basis element to match the angular momentum of any given nth great circle of a primary component orbitsphere-cvf. The replacement of each great circle of the primary orbitsphere-cvf with a secondary orbitsphere-cvf of matching angular momentum, orientation, and phase comprises an autocorrelation-type function [2] that exactly gives rise to a uniform current density having the same resultant angular momentum as the corresponding primary component. The uniform distributions for STEP ONE and STEP TWO can be generated independently and superimposed to give Y₀ ⁰(θ,φ).
The orbitsphere-cvf shown in FIGS. 1.10A-G of Mills GUT and 8 and 9 comprises the superposition or sum of the components corresponding to STEPS ONE (FIGS. 2 and 3) and STEP TWO (FIGS. 6 and 7). Thus, the convolution is performed on each component designated a primary component. The convolution of a secondary component orbitsphere-cvf element with the each great circle current loop of each primary orbitsphere-cvf is designated as the convolution operator, Y₀ ⁰(θ,φ), given by $\begin{matrix} \begin{matrix} A (θ, ϕ) = \frac{1}{2 r_{n}^{2}} \lim_{{Δθ}_{2} \to 0} \sum_{m^{'} = 1}^{m^{'} = \frac{2 π}{\langle {Δθ}_{2} \rangle}} \lim_{{Δθ}_{1} \to 0} \sum_{m = 1}^{m = \frac{2 π}{\langle {Δθ}_{1} \rangle}} 2 °O (θ, ϕ) \otimes \\ (\begin{matrix} 1_{1}^{°} 0 (θ, ϕ) δ (θ - m {Δθ}_{1}, ϕ - ϕ^{'}) \\ + 1_{2}^{°} 0 (θ, ϕ) δ (θ - m^{'} {Δθ}_{2}, ϕ - ϕ^{″}) \end{matrix}) \\ = \frac{1}{2 r_{n}^{2}} \lim_{{Δθ}_{2} \to 0} \sum_{m^{'} = 1}^{m^{'} = \frac{2 π}{\langle {Δθ}_{2} \rangle}} \lim_{{Δθ}_{1} \to 0} \sum_{m = 1}^{m = \frac{2 π}{\langle {Δθ}_{1} \rangle}} 2 °O (θ, ϕ) \otimes \\ (\begin{matrix} {GC}_{STEP ONE} (m {Δθ}_{1}, ϕ^{1}) \\ + {GC}_{STEP TWO} (m^{'} {Δθ}_{2}, ϕ^{″}) \end{matrix}) \end{matrix} & (38) \end{matrix}$
wherein (1) the secondary component orbitsphere-cvf that is matched to the basis element of the primary is defined by the symbol 2°O(θ,φ), (2) the primary component orbitsphere-cvf of STEP M is defined by the symbol 1°_MO(θ,φ), (3) each rotated great circle of the primary component orbitsphere-cvf of STEP M is selected by the Dirac delta function δ(θ−mΔθ_M,φ−φ′,); the product 1°_MO(θ,φ)δ(θ−mΔ_M,φ−φ′) is zero except for the great circle at the angle θ=mΔθ_Mabout the 2π-rotational axis; each selected great circle having 0≦φ′≦2π is defined by GC_STEPM(mΔθ_M,φ′), and $(4) \frac{1}{2 r_{n}^{}}$
is the normalization constant. In Eq. (38), the angular momentum of each secondary component orbitsphere-cvf is equal in magnitude and direction as that of the current loop with which it is convolved. Furthermore, the orientations and phases of the convolved elements are matched by rotating the secondary component orbitsphere-cvf about the appropriate principle axis (axes) and about the C_∞-axis along its angular momentum vector, respectively. With the magnitude of the angular momentum of the secondary component orbitsphere-cvf matching that of the current loop which it replaces during the convolution and the loop then serving as a unit vector, the angular momentum resulting from the convolution operation is inherently normalized to that of the primary component orbitsphere-cvf.
The convolution of a sum is the sum of the convolutions. Thus, the convolution operation may be performed on each of STEP ONE and STEP TWO separately, and the result may be superposed in terms of the current densities and angular momenta. $\begin{matrix} A (θ, ϕ) = \frac{1}{2 r_{n}^{}} (\begin{matrix} \lim_{Δθ} \sum_{m^{'} = 1}^{m^{'} = \frac{2 π}{\langle {Δθ}_{2} \rangle}} 2 °O (θ, ϕ) \otimes {GC}_{STEP ONE} (m {Δθ}_{1}, ϕ^{1}) \\ \lim_{Δθ} \sum_{m = 1}^{m = \frac{2 π}{\langle {Δθ}_{1} \rangle}} 2 °O (θ, ϕ) \otimes {GC}_{STEP TWO} (m^{'} {Δθ}_{2}, ϕ^{″}) \end{matrix}) & (39) \end{matrix}$
Factoring out the secondary component orbitsphere-cvf which is a constant at each position of GC_STEPM(mΔθ_M,φ′) gives $\begin{matrix} A (ϕ, θ) = \frac{1}{2 r_{n}^{}} 2 °O (θ, ϕ) (\begin{matrix} \lim_{Δθ} \sum_{m^{'} = 1}^{m^{'} = \frac{2 π}{\langle {Δθ}_{2} \rangle}} {GC}_{STEPONE} (m {Δθ}_{1}, ϕ^{1}) \\ + \lim_{Δθ} \sum_{m = 1}^{m = \frac{2 π}{\langle {Δθ}_{1} \rangle}} {GC}_{STEP TWO} (m^{'} {Δθ}_{2}, ϕ^{″}) \end{matrix}) & (40) \end{matrix}$
The summation is the operator that generates the primary component orbitsphere-cvf of STEP M, 1°_MO(θ,φ). Thus, the current-density function is given by the dot product of each primary orbitsphere-cvf with itself. The result is the scalar sum of the square of each of the STEP ONE and STEP TWO primary component orbitsphere-cvfs: $\begin{matrix} A (θ, ϕ) = \frac{1}{2 r_{n}^{}} ({(1_{1}^{°} O (θ, ϕ))}^{2} + {(1_{2}^{°} O (θ, ϕ))}^{2}) & (41) \end{matrix}$
where the dot-product scalar is valid over the entire spherical surface.
Component Orbitsphere-cvf Squared for STEP ONE Using the Rotation of a Great Circle About the (i_x,i_y,0i_z)-Axis by 2π
The convolution of the secondary orbitsphere-cvf with the primary orbitsphere-cvf gives the dot product of the primary orbitsphere-cvf with itself over the entire spherical surface. From Eq. (5), the corresponding scalar equation for the component orbitsphere-cvf squared for STEP ONE is given by $\begin{matrix} x^{2} + y^{2} + z^{2} = [\begin{matrix} {((\frac{1}{2} - \frac{\cos θ}{2}) r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ)}^{2} \\ + {((\frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ)}^{2} \\ + {(- \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ + \cos θ r_{n} \sin ϕ)}^{2} \end{matrix}] & (42) \end{matrix}$
Multiplying out the squared terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = [\begin{matrix} {(\frac{1}{2} - \frac{\cos θ}{2})}^{1} r_{n}^{} \cos^{2} ϕ + \frac{\sin^{2} θ}{\sqrt{2}} r_{n}^{} \sin^{2} ϕ \\ - 2 (\frac{1}{2} - \frac{\cos θ}{2}) r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ + {(\frac{1}{2} + \frac{\cos θ}{2})}^{2} r_{n}^{} \cos^{2} ϕ + \frac{\sin^{2} θ}{2} r_{n}^{} \sin^{2} ϕ \\ + 2 (\frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ + \frac{\sin^{2} θ}{2} r_{n}^{} \cos^{2} ϕ + \cos^{2} θ r_{n}^{} \sin^{2} ϕ \\ - 2 \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕcos θ r_{n} \sin ϕ \end{matrix}] & (43) \end{matrix}$
Further multiplying out the squared terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{} [\begin{matrix} (\frac{1}{4} - \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4}) \cos^{2} ϕ + \frac{\sin^{2} θ}{2} \sin^{2} ϕ \\ - 2 (\frac{1}{2} - \frac{\cos θ}{2}) \cos ϕ \frac{\sin θ}{\sqrt{2}} \sin ϕ \\ + (\frac{1}{4} + \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4}) \cos^{2} ϕ + \frac{\sin^{2} θ}{2} \sin^{2} ϕ \\ + 2 (\frac{1}{2} + \frac{\cos θ}{2}) \cos ϕ \frac{\sin ϕ}{\sqrt{2}} \sin ϕ \\ + \frac{\sin^{2} θ}{2} \cos^{2} ϕ + \cos^{2} θ \sin^{2} ϕ \\ - 2 \frac{\sin θ}{\sqrt{2}} \cos ϕ \cos θsin ϕ \end{matrix}] & (44) \end{matrix}$
Combining terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{} [\begin{matrix} (\frac{1}{2} + \frac{\cos^{2} θ}{2} + \frac{\sin^{2} θ}{2}) \cos^{2} ϕ \\ + (\sin^{2} θ + \cos^{2} θ) \sin^{2} ϕ \\ + 4 (\frac{\cos θ}{2}) \cos ϕ \frac{\sin θ}{\sqrt{2}} \sin ϕ \\ - 2 \frac{\sin θ}{\sqrt{2}} \cos ϕ \cos θ \sin ϕ \end{matrix}] & (45) \end{matrix}$
Using the trigonometric identity: $\begin{matrix} \sin^{2} θ + \cos^{2} θ = 1 gives & (46) \\ x^{2} + y^{2} + z^{2} = r_{n}^{} [(\frac{1}{2} + \frac{1}{2}) \cos^{2} ϕ + \sin^{2} ϕ] & (47) \end{matrix}$
By using the trigonometric identity of Eq. (46) for φ, Eq. (47) becomes
x ² +y ² +z ² =r _n ² (48)
which is the equation of a uniform sphere.
Component Orbitsphere-cvf Squared for STEP TWO Using the Rotation of a Great Circle About the (−i_x,0i_y,i_z)-Axis by 2π Followed by a Rotation About the z-Axis by π/4
From Eq. (14), the equation for the component orbitsphere-cvf squared for STEP TWO is given by $\begin{matrix} x^{2} + y^{2} + z^{2} = {\begin{matrix} {((\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \cos ϕ + (\frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2}) r_{n} \sin ϕ)}^{2} \\ + {((- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \cos ϕ + (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n} \sin ϕ)}^{2} \\ + {((- \frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ)}^{2} \end{matrix}} & (49) \end{matrix}$
Multiplying out the squared terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = {\begin{matrix} {(\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2})}^{2} r_{n}^{2} \cos^{2} ϕ + {(\frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2})}^{2} r_{n}^{2} \sin^{2} ϕ + \\ 2 (\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2}) r_{n}^{2} \cos ϕsinϕ + \\ {(- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2})}^{2} r_{n}^{2} \cos^{2} ϕ + {(\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2})}^{2} r_{n}^{2} \sin^{2} ϕ + \\ 2 (- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) r_{n}^{2} \cos ϕsinϕ + \\ {(- \frac{1}{2} + \frac{\cos θ}{2})}^{2} r_{n}^{2} \cos^{2} ϕ + \frac{\sin^{2} θ}{2} r_{n^{2}} \sin^{2} ϕ + \\ 2 (- \frac{1}{2} + \frac{\cos θ}{2}) \frac{\sin θ}{\sqrt{2}} r_{n}^{2} \cos ϕsinϕ \end{matrix}} & (50) \end{matrix}$
Further multiplying out the squared terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{2} {\begin{matrix} ({(\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}})}^{2} - (\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}}) \sin θ + \frac{\sin^{2} θ}{4}) \cos^{2} ϕ + \\ (\frac{\cos^{2} θ}{2} + \frac{\cos θsin θ}{\sqrt{2}} + \frac{\sin^{2} θ}{4}) \sin^{2} ϕ + \\ 2 (\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) \cos ϕsinϕ + \\ ({(\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}})}^{2} + \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} \sin θ + \frac{\sin^{2} θ}{4}) \cos^{2} ϕ + \\ {(\frac{\cos^{2} θ}{2} - \frac{\cos θsinθ}{\sqrt{2}} + \frac{\sin^{2} θ}{4})}^{2} \sin^{2} ϕ + \\ 2 (- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) \cos ϕsinϕ + \\ (\frac{1}{4} - \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4}) \cos^{2} ϕ + \frac{\sin^{2} θ}{2} \sin^{2} ϕ + \\ 2 (- \frac{1}{2} + \frac{\cos θ}{2}) \frac{\sin θ}{\sqrt{2}} \cos ϕsinϕ \end{matrix}} & (51) \end{matrix}$
Combining terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{} {\begin{matrix} (2 {(\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}})}^{2} + \frac{\sin^{2} θ}{2} + (\frac{1}{4} - \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4})) \cos^{2} ϕ + \\ (\cos^{2} θ + \frac{\sin^{2} θ}{2} + \frac{\sin^{2} θ}{2}) \sin^{2} ϕ + \\ 2 (\frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} + \frac{\sin θ}{2}) \cos ϕsinϕ + \\ 2 (- \frac{\frac{1}{2} + \frac{\cos θ}{2}}{\sqrt{2}} - \frac{\sin θ}{2}) (\frac{\cos θ}{\sqrt{2}} - \frac{\sin θ}{2}) \cos ϕsinϕ + \\ 2 (- \frac{1}{2} + \frac{\cos θ}{2}) \frac{\sin θ}{\sqrt{2}} \cos ϕsinϕ \end{matrix}} & (52) \end{matrix}$
Using the trigonometric identity from Eq. (46) and multiplying out the trigonometric cross terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{} {\begin{matrix} (\begin{matrix} \frac{1}{4} + \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4} + \frac{\sin^{2} θ}{2} + \\ (\frac{1}{4} - \frac{\cos θ}{2} + \frac{\cos^{2} θ}{4}) \end{matrix}) \cos^{2} ϕ + \sin^{2} ϕ + \\ 2 (\begin{matrix} \frac{\cos θ}{4} + \frac{\cos^{2} θ}{4} - \frac{\sin θcosθ}{2 \sqrt{2}} + \frac{\sin θ}{4 \sqrt{2}} + \frac{\sin θcosθ}{4 \sqrt{2}} - \\ \frac{\sin^{2} θ}{4} - \frac{\cos θ}{4} - \frac{\cos^{2} θ}{4} - \frac{\sin θcosθ}{2 \sqrt{2}} + \frac{\sin θ}{4 \sqrt{2}} + \\ \frac{\sin θcosθ}{4 \sqrt{2}} + \frac{\sin^{2} θ}{4} - \frac{\sin θ}{2 \sqrt{2}} + \frac{\sin θcosθ}{2 \sqrt{2}} \end{matrix}) \cos ϕsinϕ \end{matrix}} & i . (53) \end{matrix}$
Combining terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = r_{n}^{} {\begin{matrix} (\frac{1}{2} + \frac{\cos^{2} θ}{2} + \frac{\sin^{2} θ}{2}) \cos^{2} ϕ + \sin^{2} ϕ \\ + 2 (0) \cos ϕ \sin ϕ \end{matrix}} & (54) \end{matrix}$
By using the trigonometric identity of Eq. (46) for θ and φ, Eq. (54) becomes
x ² +y ² +z ² =r _n ² (55)
which is the equation of a uniform sphere.
Component Orbitsphere-cvf Squared for STEP TWO by the Rotation of a Great Circle About the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-Axis by 2π
From Eq. (22), the equation for the component orbitsphere-cvf squared for STEP TWO is given by $\begin{matrix} x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{} {\begin{matrix} {(2 \sin θ (\cos ϕ - \sin ϕ) + \sqrt{2} (2 \cos θ \cos ϕ + (- 1 + \cos θ) \sin ϕ))}^{2} \\ + {(- 2 \sin θ (\cos ϕ + \sin ϕ) + \sqrt{2} (\cos θ (2 \cos ϕ - \sin ϕ) + \sin ϕ))}^{2} \\ + 4 {(\sqrt{2} \cos ϕsin θ + (1 + \cos θ) \sin ϕ)}^{2} \end{matrix}} & (56) \end{matrix}$
Multiplying out the terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} θ {(\cos ϕ - \sin ϕ)}^{2} + 2 {(2 \cos θcos ϕ + (- 1 + \cos θ) \sin ϕ)}^{2} + \\ 4 \sqrt{2} \sin θ (\cos ϕ - \sin ϕ) (2 \cos θcosϕ + (- 1 + \cos θ) \sin ϕ) + \\ 4 \sin^{2} θ {(\cos ϕ + \sin ϕ)}^{2} + 2 {(\cos θ (2 \cos ϕ - \sin ϕ) + \sin ϕ)}^{2} - \\ - 4 \sqrt{2} \sin θ (\cos ϕ + \sin ϕ) (\cos θ (2 \cos ϕ - \sin ϕ) + \sin ϕ) + \\ 8 \cos^{2} {ϕsin}^{2} θ + 4 {(1 + \cos θ)}^{2} \sin^{2} ϕ + \\ 8 \sqrt{2} \cos ϕsin θ (1 + \cos θ) \sin ϕ \end{matrix}} & (57) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} θ (\cos^{2} ϕ - 2 \cos ϕsin ϕ + \sin^{2} ϕ) + \\ 2 (4 \cos^{2} {θcos}^{2} ϕ + 4 \cos θcosϕ \\ (- 1 + \cos θ) \sin ϕ + {(- 1 + \cos θ)}^{2} \sin^{2} ϕ) + \\ 4 \sqrt{2} \sin θ (\cos ϕ - \sin ϕ) (2 \cos θ \cos ϕ + (- 1 + \cos θ) \sin ϕ) + \\ 4 \sin^{2} θ (\cos^{2} ϕ + 2 \cos ϕsinϕ + \sin^{2} ϕ) + \\ 2 (\begin{matrix} \cos^{2} {θ (2 \cos ϕ - \sin ϕ)}^{2} + \\ 2 \cos θ (2 \cos ϕ - \sin ϕ) \sin ϕ + \sin^{2} ϕ \end{matrix}) - \\ 4 \sqrt{2} \sin θ (\cos ϕ + \sin ϕ) (\cos θ (2 \cos ϕ - \sin ϕ) + \sin ϕ) + \\ 8 \cos^{2} {ϕsin}^{2} θ + 4 (1 + 2 \cos θ + \cos^{2} θ) \sin^{2} ϕ \\ 8 \sqrt{2} \cos ϕsin θ (1 + \cos θ) \sin ϕ \end{matrix}} & ii . (58) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} {θcos}^{2} ϕ - 8 \sin^{2} θcosϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 8 \cos^{2} {θcos}^{2} ϕ - 8 \cos θcos ϕsinϕ + 8 \cos^{2} θcosϕsinϕ + \\ 2 \sin^{2} ϕ - 4 \cos {θsin}^{2} ϕ + 2 \cos^{2} {θsin}^{2} ϕ + \\ 4 \sqrt{2} \sin θ (\cos ϕ - \sin ϕ) (2 \cos θcosϕ - \sin ϕ + \cos θsin ϕ) + \\ 4 \sin^{2} {θcos}^{2} ϕ + 8 \sin^{2} θcosϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 2 \cos^{2} θ (4 \cos^{2} ϕ - 4 \cos ϕsinϕ + \sin^{2} ϕ) + \\ 8 \cos θcosϕsinϕ - 4 \cos {θsin}^{2} ϕ + 2 \sin^{2} ϕ - \\ (\cos ϕ + \sin ϕ) \\ (8 \sqrt{2} \sin θcosθcosϕ - 4 \sqrt{2} \sin θcosθsinϕ + 4 \sqrt{2} \sin θsinϕ) + \\ 8 \cos^{2} {ϕsin}^{2} θ + 4 \sin^{2} ϕ + 8 \sin^{2} ϕcosθ + 4 \sin^{2} {ϕcos}^{2} ϕ + \\ 8 \sqrt{2} \sin θcosϕsinϕ + 8 \sqrt{2} \sin θcosθcosϕsinϕ \end{matrix}} & ii . (59) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} {θcos}^{2} ϕ - 8 \sin^{2} θcosϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 8 \cos^{2} {θcos}^{2} ϕ - 8 \cos θcos ϕsinϕ + 8 \cos^{2} θcosϕsinϕ + \\ 2 \sin^{2} ϕ - 4 \cos {θsin}^{2} ϕ + 2 \cos^{2} {θsin}^{2} ϕ + \\ 8 \sqrt{2} \cos θ \sin {θcos}^{2} ϕ - \\ 4 \sqrt{2} \sin θcosϕsin ϕ + 4 \sqrt{2} \cos θsin θcosϕsinϕ - \\ 8 \sqrt{2} \cos θ \sin θcos ϕsinϕ + \\ 4 \sqrt{2} \sin {θsin}^{2} ϕ - 4 \sqrt{2} \cos θsin {θsin}^{2} ϕ + \\ 4 \sin^{2} θ \cos^{2} {ϕ8sin}^{2} θcos ϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 8 \cos^{2} {θcos}^{2} ϕ - 8 \cos^{2} θcos ϕsinϕ + 2 \cos^{2} {θsin}^{2} ϕ + \\ 8 \cos θcosϕsinϕ - 4 \cos {θsin}^{2} ϕ + 2 \sin^{2} ϕ - \\ 8 \sqrt{2} \cos θ \sin {θcos}^{2} ϕ + \\ 4 \sqrt{2} \cos θsinθcos ϕsinϕ - 4 \sqrt{2} \sin θcosϕsinϕ - \\ 8 \sqrt{2} \cos θ \sin θcos ϕsin ϕ + \\ 4 \sqrt{2} \cos {θsinθsin}^{2} ϕ - 4 \sqrt{2} \sin {θsin}^{2} ϕ + \\ 8 \sin^{2} {θcos}^{2} ϕ + 4 \sin^{2} ϕ + 8 \cos {θsin}^{2} ϕ + 4 \cos^{2} {θsin}^{2} ϕ + \\ 8 \sqrt{2} \sin θcosϕsinϕ + 8 \sqrt{2} \cos θsinθcosϕsinϕ \end{matrix}} & iv . (60) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} {θcos}^{2} ϕ - 8 \sin^{2} θcosϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 8 \cos^{2} {θcos}^{2} ϕ - 8 \cos θcos ϕsinϕ + 8 \cos^{2} θcosϕsinϕ + \\ 2 \sin^{2} ϕ - 4 \cos {θsin}^{2} ϕ + 2 \cos^{2} {θsin}^{2} ϕ + \\ 8 \sqrt{2} \cos θ \sin {θcos}^{2} ϕ - \\ 4 \sqrt{2} \sin θcosϕsin ϕ + 4 \sqrt{2} \cos θsin θcosϕsinϕ - \\ 8 \sqrt{2} \cos θ \sin θcos ϕsinϕ + \\ 4 \sqrt{2} \sin {θsin}^{2} ϕ - 4 \sqrt{2} \cos θsin {θsin}^{2} ϕ + \\ 4 \sin^{2} θ \cos^{2} ϕ + 8 \sin^{2} θcos ϕsinϕ + 4 \sin^{2} {θsin}^{2} ϕ + \\ 8 \cos^{2} {θcos}^{2} ϕ - 8 \cos^{2} θcos ϕsinϕ + 2 \cos^{2} {θsin}^{2} ϕ + \\ 8 \cos θcosϕsinϕ - 4 \cos {θsin}^{2} ϕ + 2 \sin^{2} ϕ - \\ 8 \sqrt{2} \cos θ \sin {θcos}^{2} ϕ + \\ 4 \sqrt{2} \cos θsinθcos ϕsinϕ - 4 \sqrt{2} \sin θcosϕsinϕ - \\ 8 \sqrt{2} \cos θ \sin θcos ϕsin ϕ + \\ 4 \sqrt{2} \cos {θsinθsin}^{2} ϕ - 4 \sqrt{2} \sin {θsin}^{2} ϕ + \\ 8 \sin^{2} {θcos}^{2} ϕ + 4 \sin^{2} ϕ + 8 \cos {θsin}^{2} ϕ + 4 \cos^{2} {θsin}^{2} ϕ + \\ 8 \sqrt{2} \sin θcosϕsinϕ + 8 \sqrt{2} \cos θsinθcosϕsinϕ \end{matrix}} & v . (61) \end{matrix}$
Combining terms gives $\begin{matrix} x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 4 \sin^{2} θ \sin^{2} ϕ + 4 \sin^{2} θ \sin^{2} ϕ \\ + 4 \sin^{2} θ \cos^{2} ϕ + 8 \sin^{2} θ \cos^{2} ϕ + 4 \sin^{2} θ \cos^{2} ϕ \\ + 4 \cos^{2} θ \sin^{2} ϕ + 2 \cos^{2} θ \sin^{2} ϕ + 2 \cos^{2} θ \sin^{2} ϕ \\ + 8 \cos^{2} θ \cos^{2} ϕ + 8 \cos^{2} θ \cos^{2} ϕ \\ + 2 \sin^{2} ϕ + 2 \sin^{2} ϕ + 4 \sin^{2} ϕ \\ - 8 \sin^{2} θ \cos ϕ \sin ϕ + 8 \sin^{2} θ \cos ϕ \sin ϕ \\ + 8 \cos^{2} θ \cos ϕ \sin ϕ - 8 \cos^{2} θ \cos ϕ \sin ϕ \\ - 4 \cos θ \sin^{2} ϕ - 4 \cos θ \sin^{2} ϕ + 8 \cos θ \sin^{2} ϕ \\ - 8 \cos θ \cos ϕ \sin ϕ + 8 \cos θ \cos ϕ \sin ϕ \\ - 8 \sqrt{2} \cos θ \sin θ \cos^{2} ϕ + 8 \sqrt{2} \cos θ \sin θ \cos^{2} ϕ \\ - 4 \sqrt{2} \cos θ \sin θ \sin^{2} ϕ + 4 \sqrt{2} \cos θ \sin θ \sin^{2} ϕ \\ - 4 \sqrt{2} \sin θ \cos ϕ \sin ϕ - 4 \sqrt{2} \sin θ \cos ϕsin ϕ + 8 \sqrt{2} \sin θ \cos ϕ \sin ϕ \\ - 8 \sqrt{2} \cos θ \sin θ \cos ϕ \sin ϕ + 8 \sqrt{2} \cos θ \sin θ \cos ϕ \sin ϕ \\ \begin{matrix} + 4 \sqrt{2} \cos θ \sin θ \cos ϕsin ϕ + 4 \sqrt{2} \cos θ \sin θ \cos ϕ \sin ϕ - 8 \sqrt{2} \cos θ \sin θ \cos ϕ \sin ϕ \\ + 4 \sqrt{2} \sin θ \sin^{2} ϕ - 4 \sqrt{2} \sin θ \sin^{2} ϕ \end{matrix} \end{matrix}} vi . & (62) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{2} {\begin{matrix} 8 \sin^{2} θ \sin^{2} ϕ \\ + 16 \sin^{2} θ \cos^{2} ϕ \\ + 8 \cos^{2} θ \sin^{2} ϕ \\ + 16 \cos^{2} θ \cos^{2} ϕ \\ + 8 \sin^{2} ϕ \end{matrix}} & (63) \end{matrix}$
By using the trigonometric identity of Eq. (46) for θ and φ, Eq. (63) becomes $\begin{matrix} x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{} {\begin{matrix} 8 \sin^{2} ϕ (\sin^{2} θ + \cos^{2} θ) \\ + 16 \cos^{2} ϕ (\sin^{2} θ + \cos^{2} θ) \\ + 8 \sin^{2} ϕ \end{matrix}} & (64) \\ x^{2} + y^{2} + z^{2} = \frac{1}{16} r_{n}^{} {16 (\sin^{2} ϕ + \cos^{2} ϕ)} & (65) \\ x^{2} + y^{2} + z^{2} = r_{n}^{2} & (66) \end{matrix}$
which is the equation of a uniform sphere.
Orbitsphere-cvf Squared
Eqs. (48). (55), and (66) are each the equation of a uniform sphere. The superposition of the uniform distributions from STEP ONE and STEP TWO is the uniform current density function Y₀ ⁰(θ,φ) that is an equipotential, minimum energy surface shown in FIG. 13. The angular momentum is identically that from the superposition of the primary component orbitsphere-cvfs of the orbitsphere-cvf, $L_{xy} = \frac{ℏ}{4} and L_{z} = \frac{ℏ}{2} .$
The spatially uniform electron current having the orthogonal angular momentum components given by Eqs. (1.76-1.77) of Mills GUT can then be considered conceptually from two viewpoints regarding the basis element of the orbitsphere-cvf which is a two-dimensional vector field comprised of an infinite number of one-dimensional great circles having zero-dimensional crossings. The electron current, Y₀ ⁰(θ,φ), is a continuous uniform superposition of secondary orbitsphere-cvfs onto and over the two-dimensional surface wherein each secondary orbitsphere-cvf of equivalent angular momentum, orientation, and phase replaces a corresponding great-circle current loops of the primary orbitsphere-cvf. Or, equivalently, the primary orbitsphere-cvf is the compression of a secondary two-dimensional orbitsphere-cvf into each of the infinite number of one-dimensional great circles such that L_R, the orientation, and the phase of the former element matches that of the latter over a two-dimensional spherical shell to form a primary two-dimensional vector field.
Matrices to Demonstrate the Convolution to Generate the Uniform Current (Charge)-Density Function Y₀ ⁰(θ,φ)
Either one of the orthogonal basis-element great circles generates the component orbitsphere-cvf for STEP ONE and STEP TWO as given in the STEP ONE by the Rotation of a Great Circle about the (i_x,i_y,0i_z)-Axis by 2π section and the STEP TWO by Rotation of a Great Circle About $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-Axis by 2π section, respectively. Thus, either of the STEP ONE and the STEP TWO components can serve as the primary component orbitsphere-cvf for the convolution operation given by Eq. (38). Similarly, either one of the orthogonal basis-element great circles generates the component orbitsphere-cvf having a stationary angular momentum vector associated with STEP ONE and STEP TWO as given in the Orbitsphere-cvf Orthogonal to that of STEP ONE by the Rotation of a Great Circle about the (i_x,−i_y,0i_z)-Axis by 2π section and the STEP TWO by Rotation of a Great Circle About the $(- \frac{1}{\sqrt{2}} i_{x}, \frac{1}{\sqrt{2}} i_{y}, i_{z})$
-Axis by 2π section, respectively. Thus, either of the STEP ONE and the STEP TWO-associated components can serve as the secondary component orbitsphere-cvf for the convolution operation given by Eq. (38). Computer modeling of the analytical equations to generate the orbitsphere current vector field and the uniform current (charge) density function Y₀ ⁰(θ,φ) is available on the web [3].
STEP ONE Matrices to Visualize the Currents of Y₀ ⁰(θ,φ)
Consider the case that the STEP-ONE primary component orbitsphere-cvf is given by Eqs. (4) and (5) and the STEP-ONE-associated secondary component orbitsphere-cvf is given by Eqs. (30-33). The basis-element great circle of the primary component orbitsphere-cvf is in the yz-plane as shown in FIG. 2, and the current is counterclockwise. Thus, the angular momentum is along the x-axis. The angular-momentum-and-orientation-matched secondary component orbitsphere-cvf is shown in FIG. 10 and is generated by Eq. (30). In this case, the secondary component orbitsphere-cvf is aligned on the yz-plane and the resultant angular momentum vector, L_R, of the secondary component orbitsphere-cvf is also along the x-axis.
Then, the uniform current distribution is given from Eq. (38) as a infinite sum of the convolved elements comprising the secondary component orbitsphere-cvf given by Eq. (30) rotated according to Eq. (4), the matrix which generated the primary component orbitsphere-cvf. The resulting constant function is exact as given by Eq. (48). A representation that shows the current elements can be generated by showing the basis-element secondary component orbitsphere-cvf as a sum of great circles using Eq. (30) and by showing the continuous convolution as a sum of discrete incremental rotations of the position of the secondary component orbitsphere-cvf using Eq. (4). In the case that the discrete representation of the secondary component orbitsphere-cvf comprises N great circles and the number of convolved secondary component orbitsphere-cvf elements is M, the representation of the uniform current density function showing current loops shown in FIGS. 14 and 15 is given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = \sum_{m = 1}^{m = M} [\begin{matrix} \frac{1}{2} + \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} + \frac{\cos (\frac{m 2 π}{2})}{2} & \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & \cos (\frac{m 2 π}{M}) \end{matrix}] \cdot \sum_{n = 1}^{n = N} [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \cos (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \sin (\frac{n 2 π}{N}) \\ - \sin (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & - \cos (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & \cos (\frac{n 2 π}{N}) \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (67) \end{matrix}$
The solution for the discrete form of Y₀ ⁰(θ,φ), the uniform, minimum-energy, equipotential-energy function of the electron, given by Eq. (67) is the basis of the uniform current-density structure and architectural structure of the present Invention comprised of great-circle elements. The latter is called a great dome wherein the stresses are preferably equally distributed over the surface under pure compression.
STEP TWO Matrices to Visualize the Currents of Y₀ ⁰(θ,φ)
Discrete Convolution with a Secondary Component Orbitsphere-cvf in a Plane Along the (i_x,i_y,0i_z)- and z-Axes (xyz-Plane)
The resultant angular momentum vector, L_R, of the secondary component orbitsphere-cvf given by Eq. (32) is along (i_x,−i_y,0i_z), corresponding to a basis-element great circle in the yz-plane having a counterclockwise current. The angular momentum direction is reversed by reversing the direction of the current to clockwise. Consider the case that the STEP-TWO primary component orbitsphere-cvf is given by Eqs. (20-23). Further, consider the case that the STEP-TWO-associated secondary component orbitsphere-cvf is given by Eq. (32). The basis-element great circle of the primary component orbitsphere-cvf shown in FIG. 7 is in the xyz-plane, and the current is clockwise. Thus, the angular momentum is along the (−i_x,i_y,0i_z)-axis. The angular-momentum-and-orientation-matched secondary component orbitsphere-cvf is shown in FIG. 11 and is generated by Eq. (32). In this case, the secondary component orbitsphere-cvf is aligned on the xyz-plane and the resultant angular momentum vector, L_R, of the secondary component orbitsphere-cvf is also along the (−i_x,i_y,0i_z)-axis.
Then, the uniform current distribution is given from Eq. (38) as a infinite sum of the convolved elements comprising the secondary component orbitsphere-cvf given by Eq. (32) rotated according to Eq. (20), the matrix which generated the primary component orbitsphere-cvf. The resulting constant function is exact as given by Eq. (66). A representation that shows the current elements can be generated by showing the basis-element secondary component orbitsphere-cvf as a sum of great circles using Eq. (32) and by showing the continuous convolution as a sum of discrete incremental rotations of the position of the secondary component orbitsphere-cvf using Eq. (20). In the case that the discrete representation of the secondary component orbitsphere-cvf comprises N great circles and the number of convolved secondary component orbitsphere-cvf elements is M, the representation of the uniform current density function showing current loops shown in FIGS. 16-18 is given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = \sum_{m = 1}^{m = M} [\begin{matrix} \frac{1}{4} (1 + 3 \cos (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} - 1 + \cos (\frac{m 2 π}{M}) + \\ 2 \sqrt{2} \sin (\frac{m 2 π}{M}) \end{matrix}) & \frac{1}{4} (\begin{matrix} - \sqrt{2} + \sqrt{2} \cos (\frac{m 2 π}{M}) - \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) \\ \frac{1}{4} (\begin{matrix} - 1 + \cos (\frac{m 2 π}{M}) - \\ 2 \sqrt{2} \sin (\frac{m 2 π}{M}) \end{matrix}) & \frac{1}{4} (1 + 3 \cos (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} \sqrt{2} - \sqrt{2} \cos (\frac{m 2 π}{M}) - \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) \\ \frac{1}{2} (\frac{- 1 + \cos (\frac{m 2 π}{M})}{\sqrt{2}} + \sin (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} \sqrt{2} - \sqrt{2} \cos (\frac{m 2 π}{M}) + \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) & \cos^{2} \frac{(\frac{m 2 π}{M})}{2} \end{matrix}] \cdot \sum_{n = 1}^{n = N} [\begin{matrix} \frac{1}{2} + \frac{\cos (\frac{n 2 π}{N})}{2} & - \frac{1}{2} + \frac{\cos (\frac{n 2 π}{N})}{2} & \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} \\ - \frac{1}{2} + \frac{\cos (\frac{n 2 π}{N})}{2} & \frac{1}{2} + \frac{\cos (\frac{n 2 π}{N})}{2} & \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} \\ - \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} & - \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} & \cos (\frac{n 2 π}{N}) \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (68) \end{matrix}$
Discrete Convolution with a Secondary Component Orbitsphere-cvf in the xy-Plane
Consider the case that the STEP-TWO primary component orbitsphere-cvf is given by Eqs. (20-23). Further, consider the case that the STEP-TWO associated secondary component orbitsphere-cvf is from Eq. (7). The basis-element great circle of the primary component orbitsphere-cvf is in the xy-plane as shown in FIG. 6, and the current is counterclockwise. Thus, the angular momentum is along the z-axis. The angular-momentum-and-orientation-matched secondary component orbitsphere-cvf is shown in FIG. 4 and is generated by Eq. (7). In order to match phase, Eq. (7) must be rotated about the z-axis by $\frac{π}{4}$
using $zrot (\frac{π}{4})$
using Eq. (11). In this case, the secondary component orbitsphere-cvf is aligned on the xy-plane and the resultant angular momentum vector, L_R, of the secondary component orbitsphere-cvf is also along the z-axis. It is given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = zrot (\frac{π}{4}) \cdot [\begin{matrix} \cos (\frac{π}{4}) \cos θ & \sin θ & \sin (\frac{π}{4}) \cos θ \\ - \cos (\frac{π}{4}) \sin θ & \cos θ & - \sin (\frac{π}{4}) \sin θ \\ - \sin (\frac{π}{4}) & 0 & \cos (\frac{π}{4}) \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (69) \end{matrix}$
Substitution of Eq. (11) into Eq. (69) gives $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{\cos θ}{2} - \frac{\sin θ}{2} & \frac{\sin θ}{\sqrt{2}} + \frac{\cos θ}{\sqrt{2}} & \frac{\cos θ}{2} - \frac{\sin θ}{2} \\ - \frac{\cos θ}{2} - \frac{\sin θ}{2} & - \frac{\sin θ}{\sqrt{2}} + \frac{\cos θ}{\sqrt{2}} & - \frac{\cos θ}{2} - \frac{\sin θ}{2} \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (70) \end{matrix}$
Then, the uniform current distribution is given from Eq. (38) as a infinite sum of the convolved elements comprising the secondary component orbitsphere-cvf given by Eq. (70) rotated according to Eq. (20), the matrix which generated the primary component orbitsphere-cvf using Eq. (23) with Eq. (20). The resulting constant function is exact as given by Eq. (66). A representation that shows the current elements can be generated by showing the basis-element secondary component orbitsphere-cvf as a sum of great circles using Eq. (70) by showing the continuous convolution as a sum of discrete incremental rotations of the position of the secondary component orbitsphere-cvf using Eq. (20). In the case that the discrete representation of the secondary component orbitsphere-cvf comprises N great circles and the number of convolved secondary component orbitsphere-cvf elements is M, the representation of the uniform current density function showing current loops shown in FIGS. 19-21 is given by $\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = \sum_{m = 1}^{m = M} [\begin{matrix} \frac{1}{4} (1 + 3 \cos (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} - 1 + \cos (\frac{m 2 π}{M}) + \\ 2 \sqrt{2} \sin (\frac{m 2 π}{M}) \end{matrix}) & \frac{1}{4} (\begin{matrix} - \sqrt{2} + \sqrt{2} \cos (\frac{m 2 π}{M}) - \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) \\ \frac{1}{4} (\begin{matrix} - 1 + \cos (\frac{m 2 π}{M}) - \\ 2 \sqrt{2} \sin (\frac{m 2 π}{M}) \end{matrix}) & \frac{1}{4} (1 + 3 \cos (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} \sqrt{2} - \sqrt{2} \cos (\frac{m 2 π}{M}) - \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) \\ \frac{1}{2} (\frac{- 1 + \cos (\frac{m 2 π}{M})}{\sqrt{2}} + \sin (\frac{m 2 π}{M})) & \frac{1}{4} (\begin{matrix} \sqrt{2} - \sqrt{2} \cos (\frac{m 2 π}{M}) + \\ 2 \sin (\frac{m 2 π}{M}) \end{matrix}) & \cos^{2} \frac{(\frac{m 2 π}{M})}{2} \end{matrix}] \cdot \sum_{n = 1}^{n = N} [\begin{matrix} \frac{\cos (\frac{n 2 π}{N})}{2} - \frac{\sin (\frac{n 2 π}{N})}{2} & \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} + \frac{\cos (\frac{n 2 π}{N})}{\sqrt{2}} & \frac{\cos (\frac{n 2 π}{N})}{2} - \frac{\sin (\frac{n 2 π}{N})}{2} \\ - \frac{\cos (\frac{n 2 π}{N})}{2} - \frac{\sin (\frac{n 2 π}{N})}{2} & - \frac{\sin (\frac{n 2 π}{N})}{\sqrt{2}} + \frac{\cos (\frac{n 2 π}{N})}{\sqrt{2}} & - \frac{\cos (\frac{n 2 π}{N})}{2} - \frac{\sin (\frac{n 2 π}{N})}{2} \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{matrix}] [\begin{matrix} r_{n} \cos ϕ \\ r_{n} \sin ϕ \\ 0 \end{matrix}] & (71) \end{matrix}$

Claims

1. A geodesic structure comprising:

a plurality of circle elements defining a geodesic structure, the circle elements having substantially the same diameter bound to one another such that the forces are equalized on the geodesic structure.

2. A geodesic structure according to claim 1, wherein the circle elements are half circles and the geodesic structure comprises a dome.

3. A geodesic structure according to claim 1, wherein the circle elements are full circles and the geodesic structure comprises a sphere.

4. A geodesic structure according to claim 1, wherein the geodesic structure comprises an arch.

5. A geodesic structure according to claim 1, wherein no more than two circle elements are aligned on an axis.

6. A geodesic structure according to claim 1, wherein the circle elements are aligned according to Eqs. (1-71).

7. A geodesic structure according to claim 1, wherein the circle elements all have the same diameter.

8. A geodesic structure according to claim 1, wherein the circle elements overlap one another.

9. A geodesic structure according to claim 1, wherein the circle elements are intertwined with one another.

10. A geodesic structure according to claim 1, wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf wherein each circle element of the primary component orbitsphere-cvf is replaced by a secondary orbitsphere-cvf.

11. A geodesic structure according to claim 10, wherein each circle element of each orbitsphere-cvf element comprises a structural element.

12. A geodesic structure according to claim 11, wherein the structural element comprises at least one a tube, bar, rod, or beam.

13. A geodesic structure according to claim 12, wherein the structural elements are welded, bolted, riveted, clamped, glued, or otherwise fastened at their crossings with other such elements to form the architectural structure.

14. A geodesic structure according to claim 1, wherein the architectural structure has the form or a portion of the form provided by at least one of Eqs. (67-69).

15. A geodesic structure according to claim 1, wherein the geodesic structure comprises a container, dish, or vessel formed from the circle elements.

16. A geodesic structure according to claim 1, wherein the circular elements comprise framing to provide structural strength.

17. A geodesic structure according to claim 16, wherein the framing is covered with a continuous membrane or a tiling that forms a continuous or partially continuous covering of the framing.

18. A geodesic structure according to claim 10, wherein a number of secondary component orbitsphere-cvf elements is equal to the number of great-circle elements of the primary component orbitsphere-cvf.

19. A uniform current-density structure comprising a plurality of great-circle element loops wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf wherein each circle element of the primary component orbitsphere-cvf is replaced by a secondary orbitsphere-cvf to form Y₀ ⁰(φ,θ).

20. The uniform current-density function having a magnetic moment along the z-axis that is twice the magnitude of the magnetic moment in the xy-plane.

21. The geodesic structure according to claim 10, wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf represented by

[\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \frac{1}{2} + \frac{\cos θ}{2} & \frac{1}{2} - \frac{\cos θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ \frac{1}{2} - \frac{\cos θ}{2} & \frac{1}{2} + \frac{\cos θ}{2} & \frac{\sin θ}{\sqrt{2}} \\ \frac{\sin θ}{\sqrt{2}} & - \frac{\sin θ}{\sqrt{2}} & \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} (\frac{1}{2} - \frac{\cos θ}{2}) r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ (\frac{1}{2} + \frac{\cos θ}{2}) r_{n} \cos ϕ + \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ - \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ + \cos θ r_{n} \sin ϕ \end{matrix}]

22. The geodesic structure according to claim 21, wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf wherein each circle element of the primary component orbitsphere-cvf is replaced by a secondary orbitsphere-cvf represented by

[\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = \sum_{m = 1}^{m = M} [\begin{matrix} \frac{1}{2} + \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} + \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & \cos (\frac{m 2 π}{M}) \end{matrix}] \cdot \sum_{n = 1}^{n = N} [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \cos (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \sin (\frac{n 2 π}{N}) \\ - \sin (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & - \cos (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & \cos (\frac{n 2 π}{N}) \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}]

23. The uniform current-density structure of claim 20 wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf represented by

\begin{matrix} [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} \sin^{2} \frac{θ}{2} & - \cos^{2} \frac{θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ - \cos^{2} \frac{θ}{2} & \sin^{2} \frac{θ}{2} & - \frac{\sin θ}{\sqrt{2}} \\ \frac{\sin θ}{\sqrt{2}} & \frac{\sin θ}{\sqrt{2}} & - \cos θ \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}] & (36) \\ [\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} - \cos^{2} \frac{θ}{2} r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ \sin^{2} \frac{θ}{2} r_{n} \cos ϕ - \frac{\sin θ}{\sqrt{2}} r_{n} \sin ϕ \\ \frac{\sin θ}{\sqrt{2}} r_{n} \cos ϕ - \cos θ r_{n} \sin ϕ \end{matrix}] & (37) \end{matrix}

24. The uniform current-density structure of claim 23 wherein the circle elements are arranged according to the structure of a primary component orbitsphere-cvf wherein each circle element of the primary component orbitsphere-cvf is replaced by a secondary orbitsphere-cvf represented by

[\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = \sum_{m = 1}^{m = M} [\begin{matrix} \frac{1}{2} + \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{1}{2} - \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{1}{2} + \frac{\cos (\frac{m 2 π}{M})}{2} & \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} \\ \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & - \frac{\sin (\frac{m 2 π}{M})}{\sqrt{2}} & \cos (\frac{m 2 π}{M}) \end{matrix}] \cdot \sum_{n = 1}^{n = N} [\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) & 0 \\ \sin (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \cos (\frac{π}{4}) \cos (\frac{n 2 π}{N}) & \sin (\frac{n 2 π}{N}) \\ - \sin (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & - \cos (\frac{π}{4}) \sin (\frac{n 2 π}{N}) & \cos (\frac{n 2 π}{N}) \end{matrix}] [\begin{matrix} 0 \\ r_{n} \cos ϕ \\ r_{n} \sin ϕ \end{matrix}]