The Latest Development of 3D-SST Part A: Spacetime properties of a spiral string toryx

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1 The Latest Development of 3D-SST Part A: Spacetime properties of a spiral string toryx Vladimir B. Ginzurg Research & Development Department, International Rolling Mill Consultants, Inc. 6 Driftwood Drive, Pittsurgh, PA Three-Dimensional Spiral String Theory (3D-SST) models the creation of hydrogen atom from quantum vacuum. Part A outlines the spacetime properties of the prime elements of nature, called toryces. Each toryx contains a circularlypropagating leading string and a toroidal trailing string propagating around the leading string at the velocity of light. Toryces are divided into four distinct topological groups ased on specific cominations of inversion states of their strings. Susequent parts of the theory show that elementary particles are formed from toryces which physical properties are directly related to their spacetime properties. Introduction Several models of the structure of elementary particles employ various kinds of spiral structures [-8]. The author of this paper developed a three-dimensional spiral string theory (3D-SST) [9-8] that utilizes a toroidal spiral string element called toryx to model the spacetime and physical properties of the constituents of elementary particles. Toryx contains a circular leading string propagating along its circular path and a toroidal trailing string propagating around the leading string at the velocity of light in vacuum. Toryces are divided into four distinct topological groups ased on specific cominations of inversion states of their strings. Propagation of toryx leading strings oeys a derived toryx law of motion. Physical properties of toryces are directly related to their spacetime properties. Unification of polarized toryces produces stale elementary matter particles y following a proposed toryx conservation law. Toryces can exist at various excitation and oscillation quantum states, and the transitions etween these states are accompanied y either asorption or radiation of the constituents of elementary radiation particles called helyces. The main purpose of the Part A of the 3D-SST descried in this paper is to present main features of toryces in sufficient spacetime terms, allowing one to model elementary matter particle, complex particles, such as nucleons, and, finally, the hydrogen atom. A similar approach is used in other parts of this theory to descrie spacetime and physical properties of helyces that can e either suluminal or superluminal.

2 Spacetime structure and properties of the toryx Figures A A3 show the structure and main spacetime parameters of a asic toryx that does not include its fine structure. The toryx contains a circular leading string with the radius r and a toroidal trailing string with the radius r. Leading string propagates along its circular path at the spiral velocity V, and trailing string propagates along its toroidal spiral path at the spiral velocity V. Complexity of derived spacetime parameters of the toryx is greatly affected y assumptions limiting the toryx s degrees of freedom. This paper outlines one of the simplest solutions ased on the following three assumptions. First assumption The length of one winding of trailing string L is equal to the length of one winding of leading string L : L L (- < r < + ) (A-) Second assumption The toryx eye radius r is constant: r r r const. (- < r < + ) (A-) Third assumption At each point of spiral path of trailing string its spiral velocity V is constant and equal to the velocity of light c in vacuum: V V V c const. t r (- < r < + ) (A-3) In equation (A-3): V t = translational velocity of trailing string and V r = rotational velocity of trailing string. Notaly, although the spiral velocity of trailing string V is equal to the velocity of light, equation (A-3) prevents neither V t nor V r component of spiral velocity of trailing string from eing superluminal with the other component expressed y imaginary numers. Figure A. Isometric view of a asic toryx.

3 It follows from equation (A-) that when the radius of leading string r is equal to the radius of the toryx eye r the radius of trailing string r reduces to zero. Consequently, the toryx transforms into a circular real inversion toryx with the radius r i. r i r (A-4) Figure A. Top view of a toryx. Figure A3. Cross-section of a toryx. 3

4 In the case of V r = and, according to equation (A-3), the real inversion string propagates at the constant velocity of light c with the frequency f i equal to: f i c r i (A-5) To provide synchronous motion of trailing string with leading string, the translational velocity of trailing string V t at its point a (Fig. A3) must e equal to the spiral velocity of leading string V (Fig. A4). V t V (A-6) Considering that the toryx parameters r i, c and f i are constant, we can express all toryx spacetime parameters in terms relative to these parameters. In Tale A, relative spacetime parameters of the toryx are expressed as a function of the relative radius of leading string = r /r i. Tale A. Toryx relative spacetime parameters as a function of the relative radius of leading string at the point a (Fig. A3). Parameter Leading string Trailing string Radius r r (A-7) r r i Wavelength (A-9) r i r i Length of one winding The numer of windings l L (A-) l ri ri w (A-3) w i L (A-8) (A-) (A-) (A-4) Translational velocity Rotational velocity Spiral velocity Frequency Cycle time V t t c V r r c V (A-5) (A-7) V t t c V r r c (A-9) c f f f i 4 f 4 i (A-) (A-3) c V f f i f 5 i f 5 (A-6) (A-8) (A-) (A-) (A-4) 4

5 Figure A4. Spacetime parameters of the toryx strings at the middle point a of trailing string (Fig. A3). The relative radius of leading string is related tot the toryx peripheral radius y the equation: (A-5) The analysis of the toryx spacetime equations requires us revise several conventional mathematical terms and concepts. Infinility versus a conventional zero Conventional zero () is a symol representing a complete asence of any quantity or magnitude [9,]. As a numer the zero is defined as the integer immediately preceding in a conventional numer line. The conventional zero is commonly used for counting of non-divisile entities, ut it represents nothing that can possily exist in nature which entities can e infinitesimally small ut never reaching the asolute zero. Figure A5. Infinility ( ), infinity ( ) and unity ( ). 5

6 The 3D-SST replaces the conventional zero () with a new term called the infinility ( ) that stands for the infinite nil (Fig. A5). The infinility is an inverse of infinity. One may approach the infinity y multiplying y,,, etc.: Step ( ) Step ( ) (A-) Step ( ) Step Similarly, one may approach infinility y dividing y,,, etc.: Step : : ( ) Step : : ( ) Step 3 3 : 3 : ( ).... Step (A-) Notaly, it requires the same numer of inversely-symmetrical steps to approach infinity and infinility. The infinility and infinity are defined in a similar way. Positive infinity ( ) is a positive quantity that is larger than any given positive quantity. Similarly, positive infinility ( ) is a positive quantity that is smaller than any given negative quantity. Thus, in application to oth positive infinility ( ) and the negative infinilitiy ( ) we otain: (A3-3) 3 String and infipoint vs conventional line and point According to the conventional geometry, a line has neither width nor depth ut can e extended on and on forever in either direction. The term a line is usually applied to the straight line while the term a curved line is used if the line is not straight. The 3D-SST replaces oth aove terms with a single term called a string. The string is ased on the proposed Spiral principle: Every line is a helical spiral. This means is that what we see as either a conventional line or curved line turns out to e, upon a closer examination, a helical spiral (Fig. A6). The string corresponds to the extreme case, when the radius of the helical spiral r approaches either positive or negative infinility ( ). 6

7 Figure A6. String and infipoint versus conventional line and point. According to the conventional geometry, a point has neither length nor width ut indicates position (Fig. A6). The 3D-SST replaces the conventional point with the proposed term infipoint. The infipoint is a particular case of the string which length approaches either positive or negative infinility ( ). Consequently, the infipoint appears as a circle with the radius approaching either positive or negative infinility ( ). 4 Toryx trigonometry vs conventional trigonometry Conventional trigonometry is ased on the relationships etween sides of a right triangle. Figure A7 shows the relationship etween the main trigonometric functions and the lengths of the sides x and y for the case when the length of the hypotenuse is equal to. Figure A7. Three main conventional trigonometric functions. The definitions of the conventional trigonometric functions are ased on the transformations of the right triangle as a function of the non-right angle shown in Figure A8. The main features of these transformations are: The triangles located at the two left quadrants are mirror images of the triangles located at the two right quadrants. Similarly, the triangles located 7

8 at the two ottom quadrants are mirror images of the triangles located at the two top quadrants. When the length of the hypotenuse of the triangles is equal to, the ranges of the lengths of its sides x and y are etween and. Figure A8. Appearances of a right triangle corresponding to the conventional trigonometry. The 3D-SST uses the toryx trigonometry that employs the following relationship etween the toryx trigonometric function costand the numer that extends from negative infinity ( ) to positive infinity ( ) as the angle changes from to 36 : cost x ( 36 ) (A4-) The letter t in the symols of the toryx trigonometric functions differentiates the toryx trigonometric function from the conventional one. Figure A9 shows the transformations of the right triangle as a function of the angle according to Eq. (A4-). The main features of these transformations are (Tale A): When the angle is etween and 8 the appearances of the right triangles in the toryx trigonometry are the same as in the conventional trigonometry. Thus, within this range of the angle the conventional and toryx trigonometry are identical. 8

9 When the angle is etween 8 and 36, the right triangle of the toryx trigonometry ecomes outverted. Consequently, the length of its horizontal side x ecomes greater than, while the length of the other side y is expressed with imaginary numers. When the angle approaches 7 from a smaller angle, the length of its horizontal side x approaches positive real infinity ( ), while the length of the other side y approaches positive imaginary infinity ( i). When the angle approaches 7 from a greater angle, the length of its horizontal side x approaches negative real infinity ( ), while the length of the other side y approaches negative imaginary infinity ( i). When the angle approaches 36 from a smaller angle, the length of its horizontal side x approaches, while the length of the other side y approaches tnegative imaginary infinility ( i). Figure A9. Appearances of a right triangle corresponding to the toryx trigonometry. Tale A. Relationship etween toryx and conventional trigonometric functions. Toryx Conventional ( 36 ) ( 8 ) ( 8 36 ) cos t ( ) cos( ) sec( ) sint ( ) sin( ) i tan( ) 9

10 5 Toryx numer line vs conventional numer line Figure A shows a conventional numer line as a set of real numers extended along a straight line from the zero to the right side towards positive infinity ( ) and to the left side towards negative infinity ( ). Figure A. Conventional numer line. Figure A. Toryx numer line. Unlike the conventional linear numer line the toryx numer line is circular. As shown in Figure A, in the toryx numer line, a set of real numers is extended counterclockwise along a circle from positive infinity ( ) to negative infinity ( ). The relationship etween the numers and the polar angle on the toryx numer line is ased on the toryx trigonometry. Thus, we otain from Eq. (A4-):

11 cost ( 36 ) cost (A5-) Considering the relationship etween conventional and toryx trigonometric functions shown in Tale A, we can present Eq. (A5-) in terms of conventional trigonometry. cos ( 8 ) (A5-) cos cos (8 36 ) cos (A5-3) The toryx numer line is divided into two domains, the infinity domain and the infinility domain occupying equal spaces on the numer line. The infinity domain contains the numers extending clockwise from the positive unity ( ) to the negative unity ( ). It occupies oth top and ottom right quadrants of the numer line. Within the top right quadrant ( 9 ) the numers increase from the positive unity ( ) to positive infinity ( ), while within the ottom right quadrant ( 7 36 ) the numers decrease from the negative unity ( ) to negative infinity ( ). The infinility domain contains the numers extending counterclockwise from the positive unity ( ) to the negative unity ( ). It resides in two left quadrants of the numer line. Within the top left quadrant (9 8 ) the numers decrease from the positive unity ( ) to positive infinility ( ), while within the ottom left quadrant ( 8 7 ) the numers decrease from negative infinility ( ) to the negative unity ( ). Notaly, negative infinity ( ) merges with positive infinity ( ) when / 36, while positive infinility ( ) merges with negative infinility ( ) when 8. There are two kinds of polarization etween the numers of the toryx numer line, inverse and reverse: Inverse polarization - The magnitudes of the numers located in the left quadrants are inversed in respect to the magnitudes of the numers located in the right quadrants. Thus, (A5-4) Reverse polarization - The numers located in the top quadrants and the ottom quadrants have same magnitudes ut reversed signs. Thus, (A5-5)

12 6 Velocities of the toryx trailing string Figure A shows transformations of a hodograph of velocities of trailing string at the middle point a of trailing string (Fig. A3) as its steepness angle φ increases from to 36. The hodograph forms a right triangle which sides represent the relative translational velocity β t and the relative rotational velocity β r of trailing string, while its hypotenuse represents the relative spiral velocity β = of this string. Notaly, the hodograph of velocities of trailing string uses the toryx trigonometry. Figure A. Transformations of a hodograph of velocities at middle point a (Fig. A3). To maintain spacetime integrity of the toryx, as its trailing string propagates at a constant relative spiral velocity β =, the relative translational and rotational velocities, β t and β r, must vary in a very specific way. As shown in Fig. A3 the translational velocity at each point of the trailing string must e proportional to the distance of said point from the toryx center. At the same time, the rotational velocity of the trailing string must vary to satisfy equation (A-3), according to which the trailing string must propagate at a constant relative spiral string velocity.

13 Figure A3. Relative peripheral velocities of the trailing string. Tale A3 shows equations for relative peripheral translational and rotational velocities of the trailing string. Within the ranges of (A6-5) and (A6-6) the in inner and outer peripheral translational velocities t and out t of the trailing string exceed the velocity of light. Tale A3. Relative peripheral velocities of the trailing string. Velocities at the point a in t in r 4 (A6-) (A6-3) out r out t Velocities at the point a ( 4 ) ( 3 ) 3 (A6-) (A6-4).544. (A6-5). 6. (A6-6) Figure A4 shows the variation of the relative instant translational and i rotational velocities, t and i r, for the case when =. Notaly, etween.5 i and.75 portions of the cycle of the trailing string, t exceeds the velocity of light while is expressed with imaginary numers. i r 3

14 i Figure A4. Variation of instant velocities t and i r of the trailing string within one cycle of trailing string. 7 Classification of toryces We use two integrated spacetime parameters of toryces to define their main classification: the toryx vorticity V and the toryx reality R. The toryx vorticity V is equal to the ratio of the radius of trailing string r to the radius of leading string r with the opposite sign. V r r cos t (A7-) Figure A5 shows a circular diagram of the toryx vorticity V as a function of the steepness angle of trailing string φ. Toryces with positive vorticity V are called positive and with negative vorticity V, negative. This diagram can e treated as a proposed toryx numer line for the real numers V. The circular diagram of the toryx vorticity V is divided into two domains, infinity domain and infinility domain, each occupying equal sectors of the diagram. The infinility domain occupies two top quadrants; it contains the values of V extending clockwise from the positive unity ( ) and passing through infinility ( ) to the negative unity ( ). The infinity domain resides in two ottom quadrants; it contains the values of V extending counterclockwise from the positive unity ( ) and passing through infinity ( ) to the negative unity ( ). Notaly, positive infinility ( ) merges with negative infinility ( ) at 9, while negative infinity ( ) merges with positive infinity ( ) at 7. There are two kinds of symmetrical polarization etween the toryx vorticities V of the toryces that elong to the four quadrants of the circular diagram, inverse-symmetrical and reverse-symmetrical. 4

15 Figure A5. Toryx vorticity V as a function of the steepness angle of trailing string φ. The toryces that elong to the top and ottom quadrants are inversesymmetrically polarized y their vorticity V, ecause the magnitudes of their vorticities V are symmetrically inversed while the signs of V are the same. The toryces that elong to the right and left quadrants are reverse-symmetrically polarized y their vorticity V, ecause the signs of their vorticities V are symmetrically reversed while the magnitudes of V are the same. The toryx reality R is equal to the toryx relative peripheral radius. R cos t cost (A7-) Figure A6 shows a circular diagram of the toryx reality R as a function of the steepness angle of trailing string φ. In real toryces the toryx reality R is positive, while in imaginary toryces the toryx reality R is negative. This diagram can e treated as a proposed toryx numer line for numers R. The circular diagram of the toryx reality R is divided into infinility and infinity domains occupying left and right quadrants respectively. There are two kinds of symmetrical polarization etween the values of the toryx reality R of the toryces that elong to the four quadrants of the circular diagram, inverse-symmetrical and reverse-symmetrical. The toryces that elong to the right and left quadrants are inversesymmetrically polarized y their reality R, ecause the magnitudes of their realities R are symmetrically inversed while the signs of R are the same. The toryces that elong to the top and ottom quadrants are reverse-symmetrically 5

16 polarized y their reality R, ecause the signs of their realities R are symmetrically reversed while the magnitudes of R are the same. Figure A6. Toryx reality R as a function of the steepness angle of trailing string φ. The toryces are divided into four main groups according to their vorticity V and reality R: Real negative toryces ( V, R ), ( 9 ) Real positive toryces ( V, R ), (9 8 ) Imaginary positive toryces ( V, R ), (8 7 ) Imaginary negative toryces ( V, R ), (7 36 ). 8 Trends of spacetime parameters of toryces Below are graphical presentations (Figs. A7 A4) of trends of the toryx spacetime parameters as a function of the steepness angle of the trailing string φ expressed y equations presented in Tale A4. Also shown elow each chart are main transformation points of toryces that occur at the orderlines etween quadrants of circular diagrams of the toryx vorticity V (Fig. A5) and the toryx reality R (Fig. A6). 6

17 Tale A4. Toryx relative spacetime parameters as a function of the steepness angle of trailing string φ at the point a (Fig. A3). Parameter Leading string Trailing string Radius cos t Wavelength Length of one winding l cos t (A8-) (A8-3) (A8-5) l cost cos t cost cost cost (A8-) (A8-4) (A8-6) The numer of windings w sin t (A8-8) Translational velocity t (A8-9) t sin t (A8-) Rotational Velocity r sin t r cos t (A8-) Spiral velocity sin t (A8-3) (A8-4) Frequency sin t ( cos t ) (A8-5) cos t Cycle time cos t sin t (A8-7) cos t (A8-6) (A8-8) φ 36/ 9 / / / 8 7 / Figure A7. Relative radii of leading and trailing strings, and, as a function of the steepness angle of trailing string φ. 7

18 φ 36 / / / Figure A8. Relative toryx outer radius as a function of the steepness angle of trailing string φ. φ 36 / η i / / i i Figure A9. Relative wavelength of the trailing strings η as a function of the steepness angle of trailing string φ. 8

19 φ 36/ w i / / i i / i Figure A. The numer of windings of the trailing strings w as a function of the steepness angle of trailing string φ. φ 36 / βt / 9 i 8 / i 7 i Figure A. Relative translational velocity of the trailing strings β t as a function of the steepness angle of trailing string φ. 9

20 φ 36 / βr 9 8 / 7 / Figure A. Relative rotational velocity of the trailing strings β r as a function of the steepness angle of trailing string φ. φ 36 / δ / 9 i 8 / i 7 i Figure A3. Relative frequency of leading strings δ as a function of the steepness angle of trailing string φ.

21 φ 36 / δ / / Figure A4. Relative frequency of trailing strings δ as a function of the steepness angle of trailing string φ. 9 Toryx spacetime topology Spacetime topology of the toryx includes inversions of its leading and trailing strings. To illustrate the inversion of its circular leading string let us envision it in the form of an extremely thin and narrow circular rion with the radius r (Fig. A5). We assume that the leading string is propagating outward when r >. In that case, the outer color of the rion is lack, while its inner color is white. As r decreases while remaining positive, the rion egins to appear smaller, ut still propagating outward. When the rion radius r approaches positive infinility ( ), the rion reduces to the infipoint. After the rion radius ecomes negative (r < ), the rion turns inside out, or ecomes inverted, making its external color white and the internal color lack. To illustrate the inversion of the trailing string let us envision it in the form of an extremely thin and narrow toroidal rion (Fig. A6). We assume that the trailing string propagates outward when the radius of leading string r is greater than the radius of real inversion string r i (r > r i ). In this case, the radius of the trailing string is positive (r > ) and the outer color of the toroidal rion is lack, while its inner color is white. As r decreases while remaining greater than r i, the radius of the trailing string r likewise decreases ut remains positive. Consequently, the toroidal rion will appear slimmer and will have fewer windings w.

22 Fig. A5. Inversion of leading string. When r ecomes infinitesimally close to the radius of real inversion string r i, the radius of the trailing string r approaches positive infinility ( ). So, the toryx reduces to a circle seen in Figure A6 as a colorless edge of a circular rion. When r < r i, the sign of the radius r changes from positive to negative and the rion ecomes inverted. Consequently, its outer color ecomes white while its inner color ecomes lack. Fig. A6. Inversion of the trailing string. Figure A7 shows metamorphoses of the toryx leading and trailing strings as a function of the steepness angle of trailing string φ. Four kinds of inversion toryces are located at the oundaries of the circular diagram etween the four main groups of toryces.

23 Figure A7. Metamorphoses of the toryx leading and trailing strings as a function of the steepness angle of the trailing string φ Negative inversion toryx ( /36 ) - At this point, oth leading and trailing strings ecome inverted. Thus,,. As φ crosses the orderline at /36 the toryx vorticity V remains negative, while the toryx reality R inverts from imaginary to real. The toryx appears as two parallel lines separated y the distance equal to the diameter real inversion string. Real inversion toryx ( 9 ) At this point, only trailing string ecomes inverted. Thus,, and w. As φ crosses the orderline at 9, the toryx reality R remains real while its vorticity V inverts from negative to positive. It appears as circle with the relative radius. Positive inversion toryx ( 8 ) At this point, only spherical memrane ecomes inverted. Thus,,, and w / i. As φ 3

24 crosses the orderline at 8, the toryx vorticity V remains positive while its reality R inverts from real to imaginary. The toryx appears as an extreme case of a spindle torus with inner parts of its windings touching one another. Imaginary inversion toryx ( 7 ) At this point, only the leading string ecomes inverted. Thus,, and w. As φ crosses the orderline at 7, the toryx reality R remains imaginary while its vorticity V inverts from positive to negative. The toryx appears as a circle with the relative radius approaching -. The circle is located at the plane perpendicular to the plane of the real inversion string Located etween the inversion toryces on the circular diagram of Figure A are the toryces that elong to their four main groups. Figures A8 and A9 show the transformations of toryces within each main group. Figure A8. Metamorphoses of real toryces: a) negative and ) positive. Real negative toryces (Fig. A8a) - These toryces elong to the top right quadrant of the circular diagram shown in Figure A7. Trailing strings of these toryces are wound counter-clockwise outside of the real inversion string. As increases,,,, and w decrease, so that the trailing string appears like a conventional torus. 4

25 Real positive toryces (Fig. A8) - These toryces elong to the top left quadrant of the circular diagram shown in Figure A7. Within this range the trailing string is inverted, so that its windings are now wound clockwise inside the real inversion toryx. Asincreases, oth and decrease, ut w and negative values of increase. Consequently, the toryx appears as an inverted toroidal spiral. Figure A9. Metamorphoses of imaginary toryces: a) positive and ) negative. Imaginary positive toryces (Fig. A9a) - These toryces elong to the ottom left quadrant of the circular diagram shown in Figure A7. Here changes its sign from positive to negative and w is expressed y negative imaginary numers. As increases, and negative imaginary values of w decrease, while negative values of and increase. Within this range, the trailing string is still inverted and its windings are wound inside the imaginary inversion toryx. The opposite parts of windings of the trailing string intersect with one another and the toryx appears as a spindle torus. Imaginary negative toryces (Fig. A9) - These toryces elong to the ottom right quadrant of the circular diagram shown in Figure A7. Here the leading string ecomes inverted. As the negative value of increases, the negative values of also increase, while positive imaginary values of w continue to 5

26 increase too. Within this range trailing string propagates outward and its windings are located outside of the imaginary inversion toryx. Toryx Law of Motion According to the 3D-SST, toryces are created y the polarization of quantum vacuum. To maintain created toryces in polarized states, they must propagate in respect to one another according to the derived toryx law of motion. In application to the toryx this law is descried y equation (A-9) presenting the relative velocity of toryx leading β as a function of the relative radius of toryx leading string as shown elow: V (A-) c Notaly, the toryx law of motion applied to toryces is more general than classical law of motion applicale to atomic electrons and celestial odies for which >>. In these cases, equation (A-) reduces to the form: (A-) Figure 3. Toryx law of motion versus classical law of motion as a function of the relative radius of leading string. Figure A3 shows plots of equations (A-) and (A-). When >, the difference etween the calculated values of β from the toryx and classical laws of motion ecomes very small; it continues to decrease as increases. But, as decreases from to, this difference increases progressively. According to the 6

27 classical law of motion, as decreases from to, β sharply increases and approaches positive infinity ( ). The toryx law of motion, however, yields a completely different trend of β within the same range of. Here, as decreases, β initially increases and then, after reaching its maximum value of (corresponding to the velocity of light c) at =, β decreases sharply. Notaly, in application to the atomic electron of the hydrogen atom at the ground level, the magnitude of is close to 4,; for nucleons, however, magnitudes of <. This may explain why the classical law of motion cannot e applied to nucleons. Acknowledgements I am grateful to several scientists for making their valuale and stimulating comments on my ideas. Among them is Professor Carlo Rovelli (originally from Department of Physics and Astronomy of the University of Pittsurgh), who made encouraging comments on my idea of a multiple-level Universe. Thanks to the recommendations made y Dr. Akhlesh Lakhtakia (Department of Engineering and Mechanics, the Pennsylvania State University) I was ale to pulish the first three papers descriing my initial ideas on this suject [9]-[]. I appreciate my continuous communication with Mr. Nouyuki Kanai from Osaka, Japan who followed my research on spirals for the last several years. I am very grateful to my daughter Ellen for stimulating my ideas regarding to the fine toryx structure and to my son Gene, whose spontaneous comments on the sizes of multiple universes had triggered my research on spirals almost two decades ago. He then created computer graphics for the covers of my ooks and provided a valuale technical support for pulications and presentations of this suject. References [] Bostick, W. Mass, Charge, and Current: The Essence of Morphology. Physics Essays, Vol. 4, No., March 99, pp [] J-L. Camier and D.A. Micheletti, Theoretical Analysis of the Electron Spiral Toroidal Concept, NASA/CR--654, Dec.. [3] Chen, C., Pakter, R. and Seward, D.C. Equilirium and staility properties of selforganized electron spiral toroids. Physics of Plasmas Vol 8 (), Octoer, pp [4] Sarg, S. A Physical Model of the Electron According to the Basic Structure of Matter Hypothesis. Physics Essays, Vol. 6, No., 8-95, 3. 7

28 [5] Penrose, R., The Road to Reality - A Complete Guide to the Laws of the Universe, Alfred A. Knopf, New York, 5, pp [6] C.W. Lucas, Jr. A Classical Electromagnetic Theory of Elementary Particles Part, Interwining Charge-Fiers. The Journal of Common Sense Science, Foundation of Science, Vol. 8 No., May 5. [7] Gauthier, R.F. FTL Quantum Models of the Photon and the Electron, AIP Conf. Proc. Vol. 88, January 3, 7, pp [8] Gauthier, R.F. FTL Quantum Models of the Photon and the Electron. [9] Ginzurg, V.B., Toroidal Spiral Field Theory, Speculations in Science and Technology, Vol. 9, 996. [] Ginzurg, V.B., Structure of Atoms and Fields, Speculations in Science and Technology, Vol., 997. [] Ginzurg, V.B., Doule Helical and Doule Toroidal Spiral Fields, Speculations in Science and Technology, Vol., 998. [] Ginzurg, V.B., The Unification of Strong, Gravitational & Electric Forces, Helicola Press, IRMC, Inc., 3. [3] Ginzurg, V.B., The Relativistic Torus and Helix as the Prime Elements of Nature, Proceedings of the Natural Philosophy Alliance, Vol., No., Spring 4. [4] Ginzurg, V.B., Prime Elements of Ordinary Matter, Dark Matter & Dark Energy, Helicola Press, IRMC, Inc., 6. [5] Ginzurg, V.B., Prime Elements of Ordinary Matter, Dark Matter & Dark Energy Beyond Standard Model & String Theory, The second revised edition, Universal Pulishers, Boca Raton, Florida, 7. [6] Ginzurg, V.B., Prime Elements of Ordinary Matter, Dark Matter & Dark Energy Beyond Standard Model & String Theory, The second revised edition, hard cover, 7. [7] Ginzurg, V.B., The Origin of the Universe, Part : Toryces, Proceedings of the Natural Philosophy Alliance, 7 th Annual Conference of the NPA, at California State University Long Beach, Vol. 7, 3-6 June, pp [8] Ginzurg, V.B., Basic Concept of 3-Dimensional Spiral String Theory (3D-SST), Proceedings of the Natural Philosophy Alliance, 8 th Annual Conference of the NPA, at the University of Maryland, College Park, USA, Vol. 8, 6-9 July, pp. 9-. [9] Seife, C., Zero, Viking, Pulished y the Penguin Group, Penguin Putnam, Inc. New York, USA,. 8

29 [] Lewis, H., Geometry A Contemporary Course, Third Edition, McCormick-Mathers Pulishing Company, Cincinnati, Ohio, 973, pp Copyright y Vladimir B. Ginzurg 9