Tetryonics [5] - Quantum Geometrics

1 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 81.00 - Tetryonic Geometry title page 2 Copyright ABRAHAM [2008] - All rights reserved Geometry and the Theory of Everything Plato Euclid 3 5. DOLE Zero c:luntatUllle m E ldhlng Pas: ｾｊｈｰ＠ ｴｅｾ｡ｲ･＠ tC:Li!rl.fatureo l'lc !!:j1ti!IDI"'iil!'fr)" (c.330-275 BC, fl. c.300 BC) (c.428-348 BC) The Socratic tradition was not particularly congenial to mathematics, as may be gathered from Socrates' inability to convince himself that 1 plus 1 equals 2, but it seems that his student Plato gained an appreciation for mathematics after a series of conversations with his friend Archytas in 388 BC. One of the things that most caught Plato's imagination was the existence and uniqueness of what are now called the five "Platonic solids". It's uncertain who first described all five of these shapes- it may have been the early Pythagoreans- but some sources (including Euclid) indicate that Theaetetus (another friend of Plato's) wrote the first complete account of the five regular solids. Presumably this formed the basis of the constructions of the Platonic solids that constitute the concluding Book XIII of Euclid's Elements. In any case, Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of (non-planar) points in space, and late in life he expounded a complete "theory of ･ｶｲｹｴｨｩｮｧＧｾ＠ in the treatise called Timaeus, based explicitly on these five solids. Interestingly, almost 2000 years later, Johannes Kepler was similarly fascinated by these five shapes, and developed his own cosmology from them Tetryonics 81.01 - Geometry and the Theory of Everything 3 Copyright ABRAHAM [2008] - All rights reserved Tetractys The Greek Tetractys is a triangular figure consisting often points arranged in four rows: The tetrad was the name given to the number fourin Pythagorean philosophy there were four seasons and four elements, and the number was also associated with planetary motions and music The Tetractys historically symbolized the four elements [Earth, Air, Fire, and Water] and the relationship between Humanity and the cosmos created by GOD one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. v ·················· ········· ......········ As a mystical symbol, it was very important to the secret worship of the Pythagoreans. As a mystical symbol, it was very important to the secret worship of the Pythagoreans . ...... ....... ........ .... .. .... ｾ＠ ｾﾷ＠ @ ｾ ｾ ｾ Ｂｾ ＨＺｯ＠ ･＠ ｭｯ ｾ＠ male child female @@@@ fire Sacred numbers earth The Cosmos 11 A air The Tetractys can be re-organised to represent the space-time geometries of all EM mass-ENERGY-Matter .... ······· ··········....······· .... c2 ....•·•••••••• ···························· The single triangle in the first row represents zero-dimensions (a point) A vector direction in one-dimension can be represented as a line between any two points The second row represents a Boson (two-dimensions in a plane defined by a rhombus of three triangles) The whole figure folded represents three-dimensions (a tetrahedron defined by four apex points) Photons of ElectroMagnetic mass-Energy quanta are represented by two opposing triangles Tetryonics 81.02 - Greek Tetractys water 4 Copyright ABRAHAM [2008] - All rights reserved The Greek Zodiac 18 February- 20 March ·-. 12 20 March- 19 April .. · 19 April- 20 May Capricom Gemini ·- . .. · 20 May- 20 June 21 December- 19 January 9 3 Sagittarius 21 November- 21 December ........-····· 22 July- 22 August ··········... ...··········· 23 October- 21 November 22 September- 23 October 6 ··········... 22 August- 22 September The Greek Elements Tetryonics 81.03 - The Greek Zodiac ..· 5 Copyright ABRAHAM [2008] - All rights reserved Equilateral Triangles An equilateral triangle is a triangle in which all three sides are equal v ., Equilateral triangles are symmetrical in ./ •••..• many different ways •.. ..- .... Any six equilateral triangles joined can make a hexagon. .. .. "" The tesselation of odd numbered equilateral triangles creates square numbers It is unique in that it is the only polygon that can be tiled [or divided] and produce only identical geometries and squares numbers That is, 1+3 =4 1+3+5 = 9 1+3+5+7=16 1+3+5+7+9 = 25 1+3+5+7+9+ 11 = 36 1 +3+5+7+9+ 11 + 13 = 49 1+3+5+7+7+9+ 11 + 13+ 15 = 64 etc The equilateral triangle is eminently suited for the construction of fractals hv v2 ｾＭ＠ An equilateral triangle is simply a specific case of a regular polygon with 3 sides Tetryonics 81.04 - Equilateral Tessellations 6 Copyright ABRAHAM [2008] - All rights reserved The Pythagorean Theorem Pythagoras of Samos In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle) Though attributed to Pythagoras, it is not certain that he was the first person to prove it. The first clear proof came from Euclid, and it is possible the concept was known 1000 years before Pythoragas by the Babylonians The square of the hypotenuse of a triangle is equal to the sum of the squares of its sides. about (570- 495 BC) .. : .·: ＼ Ｚ Ｚ Ｑｾ ··=..... ｾ＠ Ｚ....Ｍ＠ ..... ﾷ］ｾ＠ In Physics · ··· : SQUARED numbers are ···· ... 'ｾｅｑｕｉｌａｔｒ＠ ........... ...........geometries . ........... .......... . : Since Greek times squared numbers have incorrectly been identified with square geometries : ·.·. : : : . fire air earth water Pythagorean Tetractys : ..................................... . .... : : : . ﾷｾ＠ .. .. .. ··......... .. Equilateral triangles are also squared number geometries Tetryonics 81.05 - Pythagorean Theorem The Pythagorean equation is at the core of much of geometry, its links geometry with algebra, and is the foundation of trigonometry. Without it, accurate surveying, mapmaking, and navigation would be impossible, but its application to the energy-momenta geometries of ElectroMagnetic fields and Matter in motion in Physics is erroneous and must be corrected for science to advance 7 Copyright ABRAHAM [2008] - All rights reserved Hexagons I 3 4 8 A regular hexagon can be subdivided into six equilateral triangles Hexagons are the only regular polygon that can be subdivided into another regular polygon Energy geometries An interesting relationship between circular and hexagonal geometry is that hexagonal patterns often appear spontaneously when natural forces are trying to approximate circles Atomic nuclei geometries Hexagons are the unique regular polygon such that the distance between the center and each vertex is equal to the length of each side Six is a highly composite number, the second-smallest composite number, and the first perfect number. Hexagons can be tiled or tessellated in a regular pattern on a flat two-dimensional plane That is, 1*2*3=1 +2+3=6 Tetryonics 81.06 - Hexagonal geometries Hexagonal tessellation is topologically identical to the close packing of circles on a plane 1 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 81.07 - The Platonic Solids 10 Copyright ABRAHAM [2008] - All rights reserved Tetryonic Solids +tetryon 4faces 6 edges 4 vertices up quark Despite their unique topologies Tetryonic solids are not unlike Platonic solids save that their toplogies are comprised entirely from complex hitherto undescribed 4nx equilateral Planck mass-energy momenta geometries that also match the o o ｾ＠ 2 - tetryon ＠ ｾ 2 8faces 12 edges 6 vertices z down quark 4n V1 QJ ·;:: ..j...J QJ E tetryons 4n regular deltahedrons 0 QJ 0) t"""t' t"""t' V1 (1) QJ """'' 0) t"""t' 0 0 0 -c V1 ｾ＠ s: OJ 12n leptons 12n quarks 12n 0 ..j...J ,._ (1) 0 3 (1) QJ ..j...J ..j...J X ｾ＠ ro 3 OJ V\ V\ lO regular deltahedrons a. 0 0 lO (5" V\ Bn t"""t' 20n Baryons 36n """'' (5" V\ Their equilateral topologies are best described as regular topologic-deltahedrons: 12jaces 18 edges 8 vertices neutrino electron 2 tetra-delta-hedrals octa-delta-hedrals dodeca-delta-hedrals icoso-delta-hed ra Is tetryons quarks leptons Baryons 4n external charge fascia 8n external charge fascia 12n external charge fascia 20n external charge fascia note: Charged mass-energy fascia geometries and edges become "hidden" upon the meshing of delta-hedra to form Matter topologies Tetryonics 81.08 - Tetryonic Solids 2 Neutron 20 faces edges 12 vertices 30 Proton 11 Copyright ABRAHAM [2008] - All rights reserved Euclidean geometry XII Euclid I IX ! III VIII \ .... ｾｉｴＭ (c.330-275 BC, fl. c.300 BC) ﾷＭ . . .___________. .. . ｾ＠ VI Arguably the most influential Mathematics book ever written is Euclid's 'The Elements' In all, it contains 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge. The Elements- Book 1 -Definition 20 Euclid's five general axioms were: /,.... . . .······················;.:.:::.:.::.::::T···········---------------.....................,\ Things which are equal to the same thing are equal to each other. If equals are added to equals, the wholes (sums) are equal. A\.\ .................. Of the trilateral figures, an equilateral triangle is that which has its three EQUAL sides ,B A' _______ ····... >·< _.. J E _____ _ .-- ............... . Euclid's Elements- Book 1 -Proposition 1 Method of constructing an Equialteral triangle Tetryonics 81.09 - Euclidean geometry If equals are subtracted from equals, the remainders (differences) are equal. Things that coincide with one another are equal to one another. The whole is greater than the part 12 Copyright ABRAHAM [2008] - All rights reserved The number rr is a mathematical constant that is the ratio of a circle's circumference to its diameter. Archimedes As its definition relates to the circle, rr is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism Incorrect identification of Pi [c/d] as opposed to Pi radians in Physics has led to the inappropriate association of spherical particles to the physical sciences whereas equilateral triangles & tetrahedra form its true geometry c. (287 BC- c. 212 BC) Pi 1t rr is an irrational number its decimal representation never ends and never repeats. c/d The ratio C/d is constant, regardless of the circle's size .......... 3.141592654........ Proof of the fact that C=2nr and how Archimedes proved it Draw any circle. Make a point anywhere on the circumference of the [green] circle. __ ... -··· Use that point as the center of a second [blue] circle with the same radius as the green circle. / The edge of the blue circle should touch the center of the green circle. .............//········ Draw the line segment connecting the centers of the two circles. That forms the radius of both of the circles. Now draw the line connecting the center of the blue circle to where it crosses the green circle on both sides, and complete the triangles. a r You should have two equilateral triangles whose sides are equal to the radius of the green circle Now extend all of the radius lines so they become diameter lines, all the way across the circle, and finish drawing all of the triangles to connect them. You've got six equilateral triangles now, that make an orange hexagon. So the perimeter of your hexagon is the same as six times the radius of your circle. But your circumference is a little bigger than the perimeter of your hexagon, because the shortest distance between two points is always a straight line. ···· ....... , pH= 6r This shows you that the circumference of the blue circle has to be more than 6r, so if C=2nr then n (pi) has to be a little bigger than 3, which it is. The more sides we draw on our polygon, the closer we will get to the real value of pi (3.14159 etc.). Using a polygon with 96 sides, Archimedes was able to calculate that rr was a little bigger than 3.1408 Tetryonics 81.10 - Archimedes & Pi 13 Copyright ABRAHAM [2008] - All rights reserved The Golden Ratio a velocity vectors Two quantities are in the golden ratio (<p) if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The height of the triangle [ALM] produced by the bisecting line is l/2 the height of the height of [ABC] The figure to the right illustrates the geometric relationship A __ L M a+b is to a as a is to b ... -····· ｸ Ｚｾ ｾ b a+b An intriguing showing of <p in an equilateral triangle was observed by George Odom, a resident of the Hudson River Psychiatric Center, in the early 1980s [1.61803] 1nterior 1ine length [LM] of the bisector 1ine is equal to l/2 the side of the original triangle [BC] [AB] or [CA] 0 ｾＭ ｾ ｾＭＮ｟ ｾ ｾ ｜ ﾷ ｙ＠ M Ｍｾｐｨｬｲ｡ｴｩﾷｯｳ･ The exterior lines [My] and [Lx] are y interior line [LM] Let Land M be the midpoints of the sides AB and AC of an equilateral triangle ABC Golden ratio phi n== h -21f Tetryonic geometry reveals the maximum E-field amplitude of the reduced Planck constant to be an example of the Golden ratio in physics t B Kinetic energies The area of the E-field permittivity diamond [ALDM] produced by the golden ratio bisector is l/2 the area of the original equilateral '•,,,_____ Planck energy triangle [ABC] ... ·· ............ '•, mass energy By measurement and the Intersecting Chords Theorem MX·MY = AM·MC. which is of the form (a + b)·b = a·a. Denoting a/b = x, we see that 1 +X=X 2 , Planck's constant [hJ Tetryonics 81.11 - The Golden Ratio in Planck Energies George Odom 14 Copyright ABRAHAM [2008] - All rights reserved The Golden Rhombus Applying the golden ratio (q>) to quantum scale electrodynamic geometry we can quickly determine that the linear momentum and magnetic moment vectors of photons & EM waves can also be expressed as a golden ratio 1 __ vector velocities vs magnetic vector ... -····· >: < Golden ratio phi [1.61803] '•,, _____ _ magnetic Force vs '•, c2 ----s-ec··-- electric Force Tetryonics 81.12 - The Golden Rhombus photons of EM energy 15 Copyright ABRAHAM [2008] - All rights reserved A v The 3 planes of Cartesian co-ordinates An equilateral [square J triangle divided into 12 equal semitones 2r--.... ................. ......... Musical Notes Planetary Orbits .,.----·····- .....······ ｴｯｰｬｾ ｍ｡ｾｴ･ｲ＠ Consetvation Ｂ ｧｩ･ｳ＠ Lrws hy X B y'2 .............. 'PY,thagoras is credited with the discovery that the intervals between harmonious musical notes always have whole number ratiq.S: \..... ······.... ·------... _ An equilateral [squareJtriangle divided into 6 equal semitones and 6 equal quarter tones ...../ c _..// ,.// With an Equilateral triangle, draw lines ｦｲｾ ﾷ ｴｯｾ＠ center of the circle to each vertex and each midpoint, creating six right ｴｲｩ｡ｮｧｬ･＿Ｌ ﾷ ｡Ｇｳ Ｍ ｾｨｯｷｮ＠ with six different colored triangles. Each right trialigte. has a radius of the circle for one leg, and half of a side of the original ｴｲｩ｟｡ｮｾｬ･＠ for another. Any two right triangles sharing one of the'rad!9llines must therefore be congruent, and that implies that_the-hypotenuses of the triangles are all equal. ﾷ ｴｦｬ･ｹ＠ combine to form is equilateral This in turn shows that the six rigiit'tFiangles are all congruent, and so the large ｴｲｩｾｮｧｬ･ ........ .. ............ Tetryonics 81.13 - Equilateral harmonies 16 Copyright ABRAHAM [2008] - All rights reserved Equilateral Fifths B 5 Fa ·. c So La ［ｾＬＮＧ＠ A .////. . . . /·· Ti D flat D A flat G \.\· · · · · · · · ·. . ../............/..:/ E flat B ............ ,.,.,.,.' E 4 F Musical Notes Tetryonics 81.14- Equilateral Fifths A G 17 Copyright ABRAHAM [2008] - All rights reserved • mass-energ1es 20 radiated SQUARED numbers ODD numbers hv 48, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 1, 4 4, 8 L__ 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 .______ 16 L...-----24 __j ___j ｌＮＭＳＲｾ＠ ｾＭＴＰ＠ ｾＭＴＸ＠ ｾＭｳＶ＠ ＶＴＭｾ＠ L . . . _ __ _ _ _ _ _ _ _ _ p L . . . _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Equilatera 1 Ｍｾ＠ ｾＭＸＰ＠ ｾＭｧＶ＠ Ｍｾ＠ L . . . _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ＸＭｾ＠ Ｍｾ＠ Tetryonics 81.15 - Equilateral Octaves Octaves 18 Copyright ABRAHAM [2008] - All rights reserved Ternary Diagrams Viviani's Theorem implies that lines parallel to the sides of an equilateral triangle provide (homogeneous/barycentric/areal/trilinear) coordinates for ternary diagrams for representing three quantities A,B,C whose sum is a constant (which can be normalized to unity). 0% R 100% 100% Ternary diagrams should NOT be used to model EM field strengths In a ternary plot, of EM energy the Electric field [EJ and the Magnetic dipole [N-5] must renormalise to 1 X A ternary diagram is simply a triangular coordinate system in which the 3 edges correspond to the axes. 100% 100% Trilinear charts are commonly used for finding the result of mixing three components (such as gases, chemical compounds, soil, color, etc.) that add to 100% of a quantity. Tetryonics 81.16 - Ternary Energy Diagrams 19 Copyright ABRAHAM [2008] - All rights reserved Whilst the Pythagorean Theorem boasts a slightly greater economy of terms than the Eutrigon Theorem (Wayne Roberts 2003), the latter contains an important area not included in the former: Eutrigons the area enclosed or swept out by the three points of the triangle in question are an important new class of triangle (mathematically defined by Wayne Roberts), as the analogue of the right-triangle in orthogonal (Cartesian) coordinate geometry ab -- - Kepler's Second Law of planetary motion the area of any eutrigon is equal to the combined areas of the equilateral triangles on legs 'a' and 'b: minus the area of the equilateral triangle on its hypotenuse 'c'. The Pythagorean theorem The square of the hypote nuse of a triangle is equal to the sum of the squares of its sides . .· .· .•:. ·. ·.. · · . .... ·........ﾷｾｲ · ; ·. Ｒ＠ ·· .. ·.·.. i : Eutr·1gon ... . . ....... ·: geometry a b = a2 + b2 - c 2 Since Greek times square numbers have been incorrectly identified with square geometries Equilateral triangles also form square number geometries The orbit of every planet is an ellipse with the Sun at one of the two foci. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. The algebraic form of the Eutrigon Theorem, Oike the algebraic form of Pythagoras' Theorem), is proven to be special case of the Cosine Rule••. Tetryonic theory reveals the equilateral [square] energy geometry that reveals the 'harmonics at play' in physical laws such as the second law of Kepler, and in many other phenomena in physics, chemistry, cosmology, biochemistry and number theory thus providing the foundation for the mathematics of quantum mechanics Tetryonics 81.17 - Law of Cosines 20 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 82.01 - Number theory 21 Copyright ABRAHAM [2008] - All rights reserved """"........ ... ........ ""., " .. "...... .. ...... .," .. .,., TETRYONlCS ....... .. ...... COUNTlNG POLYGONS Tetryonics 82.02 - Counting Polygons 22 Copyright ABRAHAM [2008] - All rights reserved SQUARED energies in quantum mechanics are EQUlLATERAL geometries lOrn ----,----,----,---I E 0 --- I I 1 I I I I I I I I I I I I I I I I I I I I I I I 1 I I I I I I ,- - - - ...,- - - - ,- - - - ----,----,----,---I Equilateral Square area= ＨｾＪ｢＠ )*h 0 \.0 ,.... Circles M ,.... Triangles 15.197 can be created by a number of planar geometries For a long time it has been assumed by scientists (and mathematicians) that circular [and squared] geometries are the geometric foundation of all physics, leading to a serioulsy flawed model of particles and forces in quantum mechanics b 2 =pi *[5.642 ] =100 Tetryonic theory now reveals that quantised equilateral angular momenta creates the foundational geometry of all the mass-Energy-Matter &forces of physics Tetryonics 82.03 - Squared Areas in Physics b h [.5x15.197] x 13.160 =100 23 Copyright ABRAHAM [2008] - All rights reserved 0.987654321 80/81 SQUARE ROOTS ODDS SQUARES 19 ＮＭｅＺｾｖｎ＠ 1/2 1 3/2 2 5/2 3 4 7/2 9/2 5 9/2 4 7/2 ｄＱｓｔｒｂｕＰｎｾＭ＠ Tetryonics 82.04 - Geometric Math 3 5/2 2 3/2 1 1/2 24 Copyright ABRAHAM [2008] - All rights reserved lntegers The integers (from the Latin integer), literally .. ｵｮｴｯ｣ｨ･､Ｇｾ＠ hence .. whole .. in Tetryonics it is the basis for the Planck charge quantum n L 2n-1 [ [n] + [n-1] ] [2n-1] 1 Equilateral geometries form SQUARE number geometries Triangular numbered geometries are NOT equialteral geometries 1,4,9, 16,25,49,64,81, 100,121,144, .... 0, 1, 3,6, 10, 15,21,28, 36,45, 55, .... 1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 Equilateral energy quanta form a normal longitudinal distribution Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component Tetryonics 82.05 - Integers 25 Copyright ABRAHAM [2008] - All rights reserved Bosons are a transverse measure of sca1ar energy momenta ODD numbers An odd number is an integer which is not a multiple of two. 1 Quantum 3 5 Bosons have ODD number 7 quanta 9 11 Quantum Levels 13 15 17 An odd number, when divided by two, will result in a fraction Tetryonics 82.06 - ODD numbers 26 Copyright ABRAHAM [2008] - All rights reserved Tau radians Around the whole outside of a circle, there are about 6.283 radians- or, Tau radians 3.141592654 1/4 .Ｎｲｾ｡＠ ___ _ Ｒｲ｣ ｬ ｟ ｾ Ｚﾷ＠ ,/ 1'2 0 _,.-·· \._ ·------.ＮＬ ｾＯ Ｓ＠ 9P 60 ·-., / ' . ,/ \ /: 6.283185307 ··-.. rc/4 ,..... ,'_.4'5 ··..... : ......... ｾＭ ＭｴｬＧｩｗ ............ __ ,,' :. rc/6 ...... , ....··' ... ········ '. ••• TC,;j.oj : ,.- ,, Ｎ｟ｾ .. 3 '() \ \ .... Ｍｾ ［ Ｒｲ｣＠ ,.'' ·. 210 - __ .... ·· --- 3iri2 ___ _ 3/4 1' is a more 'natural' radian system for geometric physics than 1t Tau = 27t =360 degree rotation about a point 31 ful1 (J' 2 4 r· 1t 1.0 1 historically defined as the ratio of a circle's circumference to its DIAMETER should be redefined in physics to I 8 .2 I I -1.0 T T 3T 2 4 T co (} 12 T' 1.0 0 .5 2 1' the ratio of its circumference to its RADIUS -0 .5 in doing so many of the n/2 terms common to physics will re automatically rationalised and will better reflect the Tetryonic geometry of mass-ENERGY-Matter in motion - 1.0 Tetryonics 82.07 - Tau geometry ll gI 4 5 ()' 6 I I I I I I I I T T 3T 2 4 T 27 Copyright ABRAHAM [2008] - All rights reserved Photons are a 1ongitudina1 measure of sca1ar energy momenta EVEN numbers An integer that is not an odd number is an even number EM waves are Photons are comprised of EVEN EVEN number numbered quanta quantums If an even number is divided by two, the result is another whole number 1/2 1 3/2 2 5/2 3 7/2 4 9/2 4 7/2 3 5/2 2 3/2 1 If an odd number is divided by two, the result is a fractional number Tetryonics 82.08 - EVEN numbers 1/2 28 Copyright ABRAHAM [2008] - All rights reserved Triangular numbers Historically, a triangular number counts quanta that can pack together to form an equilateral triangle 1,3,6, 10, 15,21,28,36,45,55, .... this form of geometric counting of same charges over-complicated the simpler physical reality n 2v-1 L [ [v] + [v- 1] ] 1 Equilateral chords or quantum levels are ODD numbers 5 ]] ]2 ]] 6 1,3,5,7,9, 11, 13, 15, 17, 19,21,.... 7 ]1 ｝ｾ＠ 17 ]8 23 24 ＳＵｾ＠ 1 3/2 2 1,4,9, 16,25,49,64,81, 100,121, 144, .... ｾＵ＠ 35] 3] 30 1/2 ｝ｾ＠ ｾＴＩＮ＠ 8 Equilateral geometries form SQUARE numbered geometries ]35 ]3 ｝ｾ＠ ＳＵｾ＠ 4J.(Q) 5/2 3 7/2 4 9/2 [2n-1] 4 7/2 3 5/2 2 3/2 1 Triangular energy quanta form normal distributions Tetryonics 82.09 - Triangular Numbers 1/2 29 Copyright ABRAHAM [2008] - All rights reserved Squared numbers ODD A square number, sometimes also called a perfect square, is the result of an integer multiplied by itself Quantum Energies L ooo ｾＲ＠ ｾＭ＠ + 3 + 5 Square numbers result from the summing of consecutive ODD numbers + 7 ODDS + 9 2n-1 + [ [n] n L [2n-1] 1 + [n-1]] Energy levels have SQUARE number quanta 11 + 13 + 15 + 17 Compton Frequency In Tetryonics Square numbers produce equilateral geometries Tetryonics 82.10 - SQUARE Numbers 30 Copyright ABRAHAM [2008] - All rights reserved Square roots A square root of a number is a number that, when it is multiplied by itself (squared), gives the first number again. 1 [ 1- 0 J 1 [ 1- 0 J -i and +i Root of positive one Rootofnegativeone They reflect the real non-negative linear momentum of a system Square roots of negative numbers have a basis in physical reality A whole number with a square root that is also a whole number is called a perfect square in Tetryonic theory they are actually equilateral geometries Tetryonics 82.11 - Square Roots 31 Copyright ABRAHAM [2008] - All rights reserved Rea 1Num hers In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, Quantum levels +n -n Wave probabilities Tetryonics 82.12 - Real Numbers 32 Copyright ABRAHAM [2008] - All rights reserved to ONE v Quantum levels SQUARE numbers ODD numbers m [2n-1] ::J the sum of consecutive ODD numbers 1, 3, 5, 7, 9 ..... . 1+3+5+7+9+ ..... . Ｍｅ｣ｾ l 2 3 4 6 ｛ｾｒｬ｝ 5 Ｍ 7 8 Ｍ 9 8 ｎ Ｍ Ｍｾ 7 6 4 5 ｛ｒｾｬｲ｝ Linear energy momentum NORMAL DISTRIBUTION Tetryonics 82.13 - One to Infinity squared ＭｾＳ 3 2 l ('[) ...... lO ('[) Vl 33 Copyright ABRAHAM [2008] - All rights reserved Basic Properties of nested scribed Equilateral Triangles Given an equilateral triangle of sides altitude vms 3S V374s2 area perimeter / '' / / / / / '' / ' \ \ I \ I \ I I V3/6S in-radius \ s I I I I I I I I s \ \ I I I I I I I \ VifJi2 s2 in-circle area \ \ circum-radius V3/3S ' s / ------- Tetryonics 83.01 - Basic nested equilaterals circum-circle area VITJ3s2 34 Copyright ABRAHAM [2008] - All rights reserved Tetryonic [equilateral] geometry v A The equilateral triangle exhibits 'square symmetry' it can always be divided into n 2 [number] of smaller self-similar parts All triangles are flat euclidean 1t radian geometries a ........ /.. :c ------· [1801 An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center Tetryonics 83.02 - Tetryonic geometry 35 Copyright ABRAHAM [2008] - All rights reserved CHARGE Energy QAM Scribed equilatera1 geometries /. / ...........······ ···········...\ ......\ reflect space-time's geometric relationship with charged mass-energy h . . . . . . . . . /..... ｨ ｾ＠ 1,____ _ _ _ _ _ ｾ Ｏ Ｒ＠ :2 ｾ ........ ﾷＭ · · · · · · · · · . . . .. ｾ Ｍ ｾ Ｍﾷ＠ ﾷＭ ｾ Ｍ ｾ Ｍﾷ＠ Ａ＠ ,' QAM per Energy per spatia1 unit spatia1 unit mass 11' c ····... The triangle of largest area of all those inscribed __./ ···... in any given circle is equilateral __ ...- seconds ﾷＭｾ＠ m s Tetryonics 83.03 - Inscribed Triangles 36 Copyright ABRAHAM [2008] - All rights reserved Circumscribed Triangles reflect Energy's relationship with Time v Positive Planck Charge Negative Planck Charge ············... ········...... The peri miter of an equilateral triangle is The radius of the circumscribed circle is lnscribed circles hv · -. . · · ·- · ·-· -· - -· ______________________________ __y2 ......___ The equilateral triangle has the smallest area of all those circumscribed around a given circle Tetryonics 83.04 - Circumscribed Triangles v'3 R==- - 37 Copyright ABRAHAM [2008] - All rights reserved Circumscribed circles By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2 v ........ The ratio of the area to the square of the perimeter of an equilateral triangle The ratio of the area of the incircle to the area of an equilateral triangle 1 ....········ / 12v'3 ｾ Ｍ Ｍｾ Bosons /.../··········· ........ ·y'3 ｾ ｾＭ ｾ＠ circumscribed circle .............. ___ .......... .......... 1f ·······-.............. _______ｃ ｟ ｾ Ｍ -- ｾ＠ Photons ,.//_./······· ... ······· Equilateral triangles and Tetrahedrons will scale at exactly the same proportion as Circles and Spheres scribing them Tetryonics 83.05 - Circumscribed Circles -- ···········............ .,..,. ... ' 38 Copyright ABRAHAM [2008] - All rights reserved Charged mass-ENERGlES Charge is the result of quantised angular momenta Time is a measure of changing quantised angular momenta [the inscribed circular fiux of energy in equilateral geometries] [the circumscribed spatial co-ordinate of equilateral energy geometries] circumscribed inscribed charge ···... time ······ ........ __Ｑ［ｾ Ｍ Ｍ Ｍ It is the equilateral geometry of energy not a classical vector rotation that creates QAM Ｍﾷ ﾷ ﾷ ﾷ＠ Scalar EM mass is a measure of equilateral Planck energy per spatial co-ordinate system ｃｨｾ a vee or ｭ･｡ｳｵｲｾ＠ ｟ ｲｧ･､＠ ass-energies can be ､･ｳ｣ｲＢ｢ｾ Ｍ ｡ｾ＠ of scala Energy momenta-p·er unit of i e [inscribed circular fluxes·Of·eqJ.JJiateral energy per ｣ｩｾｵｭｳｴｔ｢･､＠ ·------- -----·· temporal geometry] Tetryonics 83.06 - Charged mass-ENERGIES 39 Copyright ABRAHAM [2008] - All rights reserved Trigometric functions Standard trigometric functions must be carefully applied to measurements of equilateral Planck mass-energy geometries in scribed circular space-time co-ordinate systems . s1n 8 opposite hypotenuse cos 8 adjacent hypotenuse tan 8 opposite adjacent ｾａ＠ Q) o!-1 "(ii 0 ore '!)lise 00- ···- .... _ 0 D 8 adjacent ......... ___________ ···- ... ____ _ ｾ＠ ,......, w .....,. _________________________________________ C"' .rJ'J,....ｾ＠ Ｍ r cos _____________ ...................................................................... ....., a The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the SINE of the angle gives the length of they-component (rise) of the triangle, the COSINE gives the length of the x-component (run), and the TANGENT function gives the slope (y-component divided by the x-component) .......................................................................... E .3c ····- ... ____ ----------- Q) E 0 E ........................................................ ;u Q) c :.:::i ＮｩＡＺ］ｾ｡ＭＦ＠ Tetryonics 83.07 - Trigometric functions ..... ｾ｟＠ ....._ _ _ _....... ...__ _.....,.____....... ｾ＠ 40 Copyright ABRAHAM [2008] - All rights reserved The roots of scribed equilatera1 triangles Scalar equilateral energies map directly onto circular space-time co-ordinates through their square root linear momentum 0\ 0 0 0.866 r The ratio of the side of an equilateral triangle to the radius of its incribed circle is sqr [3]/2 0"' r The ratio of the circumscribed circle of an equilateral triangle to its inscribed circle is 2:1 a Q.J The ratio of the side of an equilateral triangle to the radius of its circumscribed circle is sqr [3] .. d 0"' .. 1.732 ie!-. w ·· ... ｾ､ｲｯ＠ 2:1 d r--------------------------------------radius Ｍ equilateral b1angle ﾧｾｦ diameter ｾＭＺｮ a ----e-qu-ila-te-ra-1s-id-e- - - - 2-:,----r-ig-ht-an-g-le-ba-se_ _ _ _ b Tetryonics 83.08 - Linear mapping of energy momentum linear momentum d 41 Copyright ABRAHAM [2008] - All rights reserved Equilateral triangles and scribed circles ·········· ... Ek == "t.i.J II ｲＮ ｬ ｾ＠ .. ..;;_·;;····/ 2 me E==-- R .Y ｾ Ｍ ｬ［Ａ ................................................. --- ＮＭ All the relativistic relationships historically attributed to circularised energies and modelled through the use of the math of right angled triangles are in fact the result of equilateral, scalar geometries Tetryonics 83.09 - Lorentz Correction geometry 7112 Jl- (vjc) 2 - 2 1TIC 42 Copyright ABRAHAM [2008] - All rights reserved Tetryonic lnfinite Series Finite sequences and series have defined first and last terms, is a infinite sequence of square numbers, the result of adding all those terms together [or their geometric inverse] ..................... ···············.............\ ········· ·······........ ........ ...... ··..... ·••·•... \,\\ .2 ', cc ｾ＠ ｾ＠ n= l ............ 1. == lim ( -----::1 + -.1.· + · · · + -1. ')· . -n2 n--++cc ·. 12 22 n2 ' whereas infinite sequences and series continue indefinitely ,?r ····... The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. ··················· \ ｾ＠ ｾ＠ .... ｾ＠ 1 == 1 + -. 1(.2)·. ｾ＠ 22. 1 w2 + -.. 32 + ... == -.6. ················ 1 ｾ＠ ｾ＠ ｾ＠ ｾ＠ ··.... .. Tetryonics now provides a geometric solution to visualising and solving the Basel problem ...·· ········ >1 n- ｾ＠ n2 ｾ＠ ｾ＠ ｾ＠ ｾ＠ .. .... ｾ＠ ! ! ..... .. ········ ·········· ·········· ··············································· The entire sum of the series is equal to ｾ＠ 1 .645· twice the size of the radius of the largest inscribed circle which is equal to the largest circle circumscribing the triangular series. Tetryonics 83.10 - The Infinite Series 43 Copyright ABRAHAM [2008] - All rights reserved lnverting the Circle Electric Permittivity Magnetic Permeability Jlo ElectroMagnetic fields 2D spatial co-ordinates Tetryonics 83.11 - Inverting the Circle 44 Copyright ABRAHAM [2008] - All rights reserved lrrational numbers an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals sin(x)=-square root of (3)/2 2 0.866025403 ...... . V3 E ;::::l rJJ I= u +-1 Q) Q) f.- E t..E f.E +-10 0 f.- u Q) ;> rc 1 .E 1 1 Pythag<pras' theoreom and irrati<pnal numbers expressed in terms of right-angled triangles in Physics offer a 'half truth' regarding the equilateral geometry of Energy Tetryonics 84.01 - Irrational numbers Q) 45 Copyright ABRAHAM [2008] - All rights reserved Leibniz linear momentum v ..··················· Newton Newton focused his work on linear momentum which he developed into his famous laws of motion Newtto1n1 a1n1d lenb1n1nz dnsagreed ab<O>IUit Wlhlalt tlhle ｗ＼ｏ＾ｉｦＧｾ､＠ US made <O>f alln1d lhi<O>W DtS plhlysncs slhlaped <O>IUI T' scne1n1tnflilc <C<O>In1<eepts of f<O>If'<ee, eln1elf'9)y, a11n1d mome1n1t1U1m 1retlfY<O>In1ncs ｬｦＧ･ｶ｡ｾｳ＠ tlhle ｰｬｨｹｳｮ｣｡ｾ＠ ｲ･ｾ｡ｴｮ＼ｏ＾ｉＱｳｬｨｰ＠ tlhley botlhl des<elf'nbed ｭ｡ｴｬｨ･ｮ｣ｾｙ＠ geometlf'n<e propertnes of ･ｱｩｕｮｾ｡ｴｬｦＧ＠ ｉｐＧｾ｡Ｑｮ＼･ｬｫ＠ eln1telf'9JY m(Q)me1n1ta Sca1ar Energy ', .............. p=mv ·-----------., Tetryonics 84.02 - Newton vs Liebniz as 46 Copyright ABRAHAM [2008] - All rights reserved Geometric Square Roots In geometrical terms, the square root function maps the area of a square to its side length. .5 Square root of 1 .5 "In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations" Square root of 1 Square root of 2 Square root of 3 ''As to geometry, in particular, the abstract tendency has here led to the magnificent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstract reasoning and symbolic calculation in the sense of algebra. Notwithstanding this, it is still as true today as it ever was that intuitive understanding plays a major role in geometry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry" David Hilbert [Geometry and the Imagination] Tetryonics 84.03 - Square root geometry 47 Copyright ABRAHAM [2008] - All rights reserved Square Roots in Physics In mathematics, a square root of a number a is a number [n] such that [n]2 = x, or, in other words, a number [n] whose square (the result of multiplying the number by itself, or [n x n]) is x. v Modern calculators use the Tetryonics uses the Square Root Algorithm geometric Square Root to calculate the value to calculate the value lt is an approximate numerical value lt is an exact geometric value v v E E= 0 In classical geometry, the square root function maps the area of a square to its side length. 5 10 15 20 25 Geometry can easily map irrational numbers Tetryonics 84.04 - Square Roots in Physics In physics, the square root function maps ENERGY [E) to momentum [mv] 48 Copyright ABRAHAM [2008] - All rights reserved The Square roots of n Historically, any number raised to the power of 2 has been modeled using a polygon--the square That's why we call raising a number to the second power "squaring the number." v 25 1 2 3 4 5 4 3 2 1 is distinct from vector [v ]elocity note: scalar linear momentum [mv] 36 12 3456543 21 3 5 9 49 1234567654321 [In physics square numbers are in fact equilateral geometries] The perfect squares are squares of whole numbers. Here are the first eight perfect squares 15 Tetryonics 84.05 - Square roots of n 64 49 Copyright ABRAHAM [2008] - All rights reserved The Square root of Negative 1 v Magnetic fields are out of phase with Electric fields Positive fields are out of phase with Negative fields Leonhard Euler Euler's Formula Euler's formula is often considered to be the basis of the complex number system. In deriving this formula, Euler established a relationship between the trigonometric functions, sine and cosine, and e raised to a power eix =cos (x) + isin(x) a mathematical description of EM-Energy waveforms (15 April 1707- 18 September 1783) sinx =x x3 xs 3! 5! - - +- .X2 - ... 1/2 x2 1 3/2 2 5/2 3 7/2 4 9/2 [Ji) 9/2 4 7/2 3 5/2 2 3/2 x xs ,ex= 1 + x +·- +·- + ·- + ·- + ___ ﾷ Ｍ ｾ＠ Ｓ ｾ＠ ·! Ｕ ｾ＠ x3 -1 x in declllna.l radians 1 1/2 x2 x 3 x x5 sin x +cos x = 1 + x - - - - + - + - + . __ 2t 3 r 4·!. :P 1 -l X oos .x = l - -2r + -4-r - ·-- 1 Tetryonics 84.06 - Square root of Negative One 0 50 Copyright ABRAHAM [2008] - All rights reserved Geometric means geometric square root The geometric mean of two numbers, is the square root of their product of positive one geometric square root of negative one ;:::s- .· ｾ :n \ Ｍﾷ＠ \ ·····--...____·...... a _. . . _ _._ . . . . . . .)< ............. .. ...... " ··--------········->" ｾ＠ In physics, the geometric mean of two superpositioned fields produces a vector square root Force It is generally stated that the geometric mean applies only to positive numbers. In Tetryonic geometry the geometric mean applies to positive & negative fields. F = -G 11111112 r2 Tetryonics 84.07 - Geometric Means 51 Copyright ABRAHAM [2008] - All rights reserved Superpositioning When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. constructive interference ln phase Additive N -- -- IV '• N w ('V") N ｾ＠ "¢ Ll') N w \0 ...... r-.... N ... co ｾ＠ w \0 N Ll') ｾ＠ "¢ N ('V") N ............................................... N Subtractive Out of ohase destructive interference When two or more waves traverse the same space, if the summed variation has a smaller amplitude than the constiuent component variations. The lines of force Tetryonics 85.01 - Addition [Phase superpositioning] 52 Copyright ABRAHAM [2008] - All rights reserved Tetryonic Multiplication table A multiplication table is a mathematical grid used to define a multiplication operation and its results Multiplication Table 1 2 3 1 1 2 3 2 2 4 6 6 6 n w 7 1 M ｾ＠ 8 8 W ｾ＠ 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12 8 ro n M w w 1 I ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ ｾ＠ a G G ｾ＠ ｾ＠ ｾ＠ M ｾ＠ ｾ＠ ｾ＠ In Tetryonics multiplication tables can also be based on EQUILATERAL geometries ｾ＠ ｾ＠ n ｾ＠ m n n 1 ｾ＠ M ｾ＠ 00 Historically Multiplication tables have been based on Square geometries 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 The integer multiplicitors are colour coded Tetryonics 85.02 - Tetryonic Multiplication 53 Copyright ABRAHAM [2008] - All rights reserved Rhombic Multiplication Tables 2 2 3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 :?4 32 40 48 56 64 72 80 9 18 21 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 10 1 2 2 4 5 6 7 8 4 5 9 10 6 2 3 4 7 8 7 8 9 10 J 9 10 11 12 13 14 15 16 17 18 19 2 4 5 6 7 2 Tetryonics 85.03 - Rhombic multiplication tables 8 9 10 4 5 6 7 8 9 10 54 Copyright ABRAHAM [2008] - All rights reserved Photonic Root Tables Tetryonic multiplication table can take a number of geometric forms lnteger median Square root median Table read diagonally 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 Tetryonics 85.04 - Photonic Root Tables Table read from centre to outside edge then down 55 Copyright ABRAHAM [2008] - All rights reserved Divide [division] tables Minus [subtraction] tables X Times [multiplication] tables Plus [addition] tables Tetryonics 85.05 - Tetryonic Flash Math 56 Copyright ABRAHAM [2008] - All rights reserved ｾ＠ ｾａ＠ A A A A A A A A A A ｾ＠ A AA A A A A A A A A A A A A A AA A 3 4 5 6 2 3 4 5 6 2 3 5 6 2 3 4 5 6 2 3 4 /:).. D.. IE&. ｾ＠ A A A A A A A A A A A A 7 8 9 9 7 ｾ＠ A A A 0 A A A A A 7 8 7 D.. ｾ＠ D.. ｾ＠ it:& IE& u;& Am Tetryonics 85.06 - Flash Math Cards 9 0 9 0 57 Copyright ABRAHAM [2008] - All rights reserved tetryons anti-Matter Tetryonics 85.07 - Tetrad Math Tables 58 Copyright ABRAHAM [2008] - All rights reserved lrrational Numbers An irrational number is defined to be any number that is the part of the real number system that cannot be written as a complete ratio of two integers Sin rr/3 __ ... ---------· ·-. /// ...................······ One well known ｩｲ｡ｾｊＶｮｬ＠ number is n_...-...- lrratfq.nal numbers can be easily ｲ･ｰｾｳｮｴ､＠ geometrically Tetryonics 86.01 - Irrational numbers 59 Copyright ABRAHAM [2008] - All rights reserved 1 0 1 2 Exponentials & Logarithms 10 e and the Natural Log are inverse functions of each other: 100 ex is the amount of continuous GROWTH after a certain amount of time. Natural Log (In) is the amount of TIME of continuous growth to reach a certain level 1,000 3 ...······································"""" 10,000 = 10 4 log (1 0,000) .. 4 n How much growth after x units of time (and 100% continuous growth) ...··.· ············· ···... ······....... ｾ＠ GROWTH ·•···... ln(x) lets us plug in continuous growth and get the time it would take. ln(x) == lim n(x l fn - 1) . n--')oo ｾ＠ ··... ｾ＠ ....····· · · · · · · · · · · · · · seconds . . . . · · · · · · · · · ············································ time Tetryonics 86.02 - Exponentials & Logarithms ...·· PERIOD 60 Copyright ABRAHAM [2008] - All rights reserved Exponential growth GEOMETRIC growth = e = li!!l( 1 + ｾ＠ y GROWTH e represents the idea that all continually growing systems are scaled versions of a common rate 2.718281828 v 1t .......·········•········································ ············································•·········... ; ..····················· ····················... / .... Pi is the ratio between circumference and diameter shared by all circles. e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all the change happens all at once at the end of a period of time- ie quanti sed growth) It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface areas Pi radians are equally important and show all quantised equilateral energy geometries are related to their scribed circles e \ ··.. .. ······· \.............. ....·· ······· changing quantum energy per second 'N.2 .....•../ / e shows up whenever systems grow exponentially and continuously..... radioactive decay, interest calculations and populations ··································sec ......··························· ······················ tetryons EM fields quarks e can he app1ied to the equi1atera1 energy geometries of physica1 systems leptons on1y where the rate of increase is a integer factor of a squared number Tetryonics 86.03 - Exponential growth Baryons 61 Copyright ABRAHAM [2008] - All rights reserved Exponential energy levels Nuclear Energy levels Radioactive decays follow exponential curves determined by the Tetryonic topology of the sub-atomic particle families Quantum Levels 400000 ｾ＠ 6000 350000 300000 ｾ＠ Quantum levels en 250000 ｾ＠ Q) c 200000 UJ Baryons 150000 Proton Neutron 100000 12 [24-12] 0 50000 [18-18] antiNeutron antiProton 0 Particle families EM Field The emission and absorption of bosons and Photons within sub-atomic nuclei ｾＮｔｳｴ＠ Planck quanta [[｣ｯ ｾｯ ｝Ｎ ｛ ｭ ｮｶ ElectroMagnetic mass Ｒ Quantum levels ｝ｊ＠ leptons ElectroMagnetic mass ＱＬｾ｛＠ EM Field [ｅｯ ｾｯ 12 velocity Increase and decrease in integer amounts according to the charged Tetryonic topologies of the particles involved 121r ｛ ｛ ｾ Ｚ Ｉ ｾ＠ Ｎ ｛ ［ ｭ［ 12 [12-24] [12-0] Electron Positron Quarks and Leptons ｛ ｾ ｾ 0 [6-6] Neutrino Ｒ｝ ｝＠ velocity 8 Up [10-2] Down Strange Charmed Top Bottom 4 [4-8] Quantum levels Tetryons 4 [4-0] Planck quanta ｝Ｎ ｛ ｭ ｮｶ ElectroMagnetic mass Ｒ Ｑ＠ ｝ｊ＠ 0 [2-2] 4 [o-4] velocity Quantum levels Tetryonics 86.04 - Sub-atomc exponential growth 62 Copyright ABRAHAM [2008] - All rights reserved Series addition & the Riemann Zeta Function The second series addition of the Reimann Zeta function is where x=2: (piA2)/6=1 + 1/2A2+ 1/3A2+ 1/4A2+ ... (the sum of the reciprocals of the squares) ｾＨＲＩ＠ = 1 12 1 22 + 32 1 42 1t2 6 f(n) 1 1 n=l n X 1/4 In mathematics, the Riemann zeta function, is a prominent function of great significance in number theory. It is a named after German mathematician Bernhard Riemann. It is so important because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics 1/4 1/9 1/16 l/25 1/36 1/49 1/64 IS\& The mystery of prime numbers Question: which natural numbers are prime? how are they distributed among natural numbers? Primes are basic building blocks for natural numbers: We donlt know how to predict where the prime numbers are: -any natural number is a product of prime numbers -a prime number is only divisible by itself and by 1: (it cannot be further simplified) "Prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance but also exhibit stunning regularityll (Don Zagier, number theorist) Tetryonics 86.05 - Riemann Zeta Function 63 Copyright ABRAHAM [2008] - All rights reserved Adding the odd numbers in order produces the square numbers 0 1 ,.;\ [1+0] [2+1] [3+21 [3+4] SPECTRAL LINES 3 5 4 /'\iF]\ 7 Ｏ｟｜ＧＺ [12-02] [22-12] [32-22] 1 /'i/\ ｾ Ｍｾ｟｜Ｏ＠ 9 16 Apart from 2, all primes are odd numbers; the difference between two consecutive squares being odd, every prime can be expressed as the difference between two squares [42-32] KEM ENERGIES [6+5] [7+6] Primes can be modelled as Bosons [ODD number energy momenta geometries] Primes can be expressed as Bosons [the difference of two squared energies] [9+8] [10+9] [12+11] [2n-1] [15 2-14 2 ] [16 2-15 2] [15+14] [16+15] 37 ＮＨＺＩｾ＿Ｌ [18+19] ｪ ﾷ ｽ ｾＭ＿ＺＨＯＧﾷＡ＠ 361 20 [21+20] 41 43 Ｏ｟｜ＬＨｩＮ [22+21] [24+23] [21 2-20 2] [22 2-21 2] /'<.-7'\)"\)"\)\,)"\)'\)'\)'\)"\-lr:.)"<.)\,)\,)"·:,)"·:.)"·:.)"·:.)"\)"·:.)\. 441 ｾ ｬＬ ｲ ﾷ｜ＨＧ＠ 484 23 47 Ｎﾷｾ｜ＨＺＩ＿Ｌ ｾｊ ｾＺＬ＿ＩＨＯＧＲ｜ﾷＡ＠ 576 25 26 [27+26] [30 2 -29 2 ] [31 2-30 2] [30+29] [31+30] [34+33] [36 2-35 2] [37 2-36 2] [36+35] [37+36] 89 [45+44] Ｎﾷｾ｜ＨＺＩ＿ＬＧｴ ｦｾ ｾｬＺｴ［ＨＬ＿Ｏ｜ＩＧﾷＡ＠ 2 2 [45 -44 ] 2025 46 The difference of two squares is (n+1) 2 - n2 = (n 2 + 2n + 1)- n2 = 2n + 1 93 22o9 48 [49+48] 97 Ｎﾷｾ｜ＨＺ＿ＩｴＬＯＧ Ｉ［ｾ ［ﾷ｜Ｌ＿ｾＺＩＨｴＯＧＡ＠ 2401 1 2 3 4 56 7 8 9 10111 213 141516171819202122232425 2627282930313233 3435363738394041424344 454647484950 494847 4645444342414039383736353433 32313029282726252423222120 19181716151413 121110 9 8 7 6 54 3 2 1 [n-1] all energy quanta create normal distributions Tetryonics 86.06 - Prime numbers [n-1] 64 Vl Q) Q) ｾ＠ o...c+-..._ Q) ｖｬｾ＠ 0 Q) :::l ｾ＠ ·ro Q. $ Q) f'-• • T ..0 c ｾ＠ ｾ＠ iU ｾ＠ ｖｬｾ＠ c ｾ＠ . :::l 0 u Q) ·---. ｾ＠ Q) Q) Q) 0 ﾷｾ＠ .1-J ::J ,...0 ﾷｾ＠ ;.... Ｍｾ＠ ｾ＠ ｧｾ＠ § ｾ＠ Ｚ［ｾ＠ _g .p ＫＭＧｉＮｾ＠ ｾ＠ 0 +-' ..c 0) Ｍｾ＠ "0 ｾＲＮＸ＠ :.c +-' ｾ＠ Q) u ｾ＠ E ..0..... ., 0 ｾｩ＠ 0) ｾ＠ 1.1= n:l Eli ｾ＠ :::;, E ＧＰｾ＠ ｾｌＡ＠ ..0 ｾ＠ Q) ..... Q) Vl ..c Q) +-' 0) a_ ·v; ｾ＠ u ｾｪ＠ 2 ｾ＠ ＺＭＧＮｾ＠ ｾＧＮＬ＠ ca ｾＧＮＬ＠ :-'',; ｾＧＮｩ＠ ｾ＠ ｾｪ＠ ::-...,; ｾｪ＠ ｾ＠ ｾ＠ ; ...... ｾｪ＠ ｾ＠ ｾ＠ .....; ｾｪ＠ ｾｪ＠ ﾷｾ＠ 0 ｾ＠ ｾ＠ ::-.....; :'...; cu E ｾ＠ ｾｪ＠ ｾ＠ ｾｪ＠ ::-',; ::-..,; ｾｪ＠ :s- .1-J r;n ｾＧＬ［＠ ; ...... ｾ＠ $ ] ] "§_ ｾｪ＠ ｾＧＬ［＠ ｾ＠ n:l ..c 5=:oE ::-',; ｾＧＮＬ＠ cＢＧｾ＠ u :::;, ｾ･ｊ＠ ｾｪ＠ :-'',; -a;i E c n:l ·- c ｾｪ＠ z2 +-' E 0) 0 Vl Q) ..CQ) ｾ＠ E .!o ..., 0 I!! u .!"C n:l ［ｾ＠｣ 0 c IKe' c E ｾ＠ n:l +-' Q) ..cn:l..C N >- +-' 0) "0 c ｮＺｬｾ＠ Q. ·- c ｾ＠ x "§ Ｍｾ＠ Q) ｾ＠ :5--:..0-o 0 "0 ·- 0 "0 -oO"OC Vl ｾ＠ n:l '+- E ｾ＠ +-' 0 I'-. n:l roc>-0) r;n ｾ＠ 1a .c ..., cu ｾ＠ "0 .E 8 ＠ｾ ｾ＠ ｧｾ＠ 0 £:560.. Q) +-' Q) Vl 0 n:l c +-' n:l n:l :::l ..... ..c u Q) +-' ｾ＠ ..c Vl - .3 ·- ·- "0 0 Q) ..... 0 ..... 0 0 Q. 0 Q) ..c ..c .._ u -- Q) >-..c+-' Q) ｾＺＵ＠ ..c Ｍｾ＠ ..... +-' Q) n:l 0) E n:l 0 ..... c Q) >-O...QJ c n:l +-' n:l 0) '+- E Qj c 0 c Q) ..c I! N + c. ｾｪ＠ :-''..; ｾｪ＠ .!!! Tetryonics 86.07 - Prime number distributions .§ § ｾｪ＠ :...,.; ｾｪ＠ :...,; ｾ＠ ｾ＠ ｾ＠ ;.... (JJ :',.; ,...0 ｾｪ＠ E ｾ＠ (JJ Copyright ABRAHAM [2008] - All rights reserved ﾷｾ＠ ｾ＠ E ｾ＠ , :t',; ::J ｾ＠ Vl Q) ｾ＠ E ..... Q) Q. ·-Q_E ..0 Q) +-' :::l "(0 +-' :::l >- Vl c ..o ·- E ..... Q) Q) ·- :::l c c c ｾ＠ "3: "3: ;.... ｾＲ＠ Q) ｾ＠ c n:l Q) Vl ｾ＠ ｾ＠ ｾ＠ "0 Vl Q) c Q) Q) ·- o o- E >- Q) ·ｾＰＩＮ＠ '+Q) o E Qj Vl n:l tt::.--:.C ·- M Q) c 2: ｾｅｏｊ＠ Q) "0-.:;t> +-" ＺＵｾｅ＠ ....- n:l Qj ·= 2 ,..... ....... n:lQ. n:l E Q) c ｾ＠ Ｍｾ＠ ｾ＠ E a_ E ·- c ·a_ "3: a_ n:l +-' ｾ＠ ｾ＠ ｾ＠ +-' Q) ｾ＠ E- Ｍｾ＠ Q. c n:l Q) n:l .......... X <x::-2-2 c ro ..... ｾ＠ ｾ｣ｵｅ＠ c $ +-' ｾ＠ E :::l ｾ＠ =n:l Q) +-' Q) "OVl..C ..... n:l +-' ﾧＺｾ＠ c Q.Q) c..o :'..; c II ,.... a. N a. Q) Q) Q) ＡＧＭＮｾ＠ ..c ..c ｾ＠ +-'+-'Vl '+- ..c ｾ＠ 0+-'M "3: Ｍｾ＠ "0 ｾ＠ ｾｪ＠ ｾ＠ ｎｾ＠ "'"' ::! ｾ＠ s j"' ｾｪ＠ ｾ＠ｾｪ＠ ｩｾ＠ ｩｾ＠ ｾｪ＠ ｾ＠ :'..; ｩｾ＠ :...,.; ｾｪ＠ :...,; E ｾ＠ ｾ＠ 0 -2 N ｾｪ＠ !-'...,; 0 \O+J -o... ｩｾ＠ ｾｪ＠ :',; :',; :-''..; ｾｪ＠ ｾ＠ Vlco ｍｾ＠ n:l C :;:; o... Q) Ｍｾ＠ ｾＭｅ＠ ｣ｾＺ［＠ ..... Q) c Q) ｾ＠ E 0...Q) ｾ＠ E ·.: c 0 Q) E E :5 ·.: 8 ｾＮＸ＠ I :...,.; ｾｪ＠ "'C :...,.; !"'...,; :r...,; 0 ＮＭｾ＠ .3 n:l Q.C n:l ＠ｾ Vl Q) E -Q) '+0 "§_ $ Q) QJ_c= c +-' Q. Q) Vl n:l 0)'-0) Ｍｾ＠ Q)+-'+-' ｾ＠ "3: ·- Q) Q) ..0 ｾ＠ .... :.t:i n:l ｾ＠ c II \() c c +-' +-' n:l Q) Q) ..c ..Q:::l+-' Vl "0 Vl Q. ..... Vl n:l ｑＩｾ＠ E-s E +J 0 Q) .;: E Vl Q. :5 Ｍｾ＠ $ 0 ｾﾧ＠ +.- Q) Vl Q) ..c · Q) +-' c E ｾ＠ ｾｪ＠ ..... ｾ＠ . )6ES ｾＭﾧ｟＠ E+-'+-- E ·.: c cu ｾＧａ＠ ｾ＠ QJQ);::::: ..0 ·-E a_ E 0. o ·- E :...,; ｾ＠ "' .!!! Q) "3: § +-' Vl >-.._ ｾＭＲ＠ UJ c Ｂｾ＠ ﾧｾ＠ "'"' 65 Copyright ABRAHAM [2008] - All rights reserved Titu Digital roots of Primo JtUIJlbBrs The digital root (also repeated digital sum) of a number is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. 3 11 12 13 4 5 6 14 15 16 8 11 18 9 10 19 20 21 22 23 24 25 26 27 28 29 30 33 37 .38 39 40 31 32 41 42 43 44 45 46 lt7 48 49 50 51 54 $4 35 36 1 ss sa 69 60 52 53 61 62 63 64 65 66 71 72 73 74 75- 7'6 77 78 79 80 56 57 67 68 69' iO 81 82 ·83 84 85 86 87 88 89 90 91 Tetryonics 86.08 - Digital roots of Primes 92 93 84 95 96 97 98 gg, 100 66 Copyright ABRAHAM [2008] - All rights reserved Archimedes is given credit for the first calculus. Archimedes infinite series Today's calculus was published by Newton. 1 +.75+.1875+.046875 + .01171875+ ..... . 1 + 3/4 + 3/16 + 3/64 + 3/256 + 3/1 024 + ..... . • • Nested convergent infinite series Tetryonics 87.01 - Archimedes infintie series 67 Copyright ABRAHAM [2008] - All rights reserved 0 Summing a convergent infinite series 0 1/2 + 1/2 + 1/2 3 + 1/2 4 + 1/2 5 + 2 2 3 1/4 + 1/4 + 1/4 + 1/4 4 + 1/4 5 + 00000000000 00 L u.n == ao + a1 + ｾ＠ + ··· n=O ODDS= 2n-l Summing the dissimilar coloured equilateral triangles gives unity 1/3 Tetryonics 87.02 - Summing a convergent Infinite series 00000000000 68 Copyright ABRAHAM [2008] - All rights reserved 1/3 Summing the dissimilar coloured equilateral triangles gives unity 2 3 4 5 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 0 0 Summing a divergent infinite series Tetryonics 87.03 - Summing a divergent Infinite series 0 00000000000 69 Copyright ABRAHAM [2008] - All rights reserved Pietro Mengoli The Basel Problem Leonhard Euler The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. 1.644934 (1626- June 7, 1686) (15 April1707- 18 Sept 1783) The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series f(n) = 1/n 2 Tetryonics 87.04 - The Basel Problem 70 Copyright ABRAHAM [2008] - All rights reserved lntegrals of mass-energy is a means of finding scalar areas using summation and limits. Integration is a micro adding of CONTINUOUS quantities. The summation of equilateral energy momenta quanta with respect to their linear vector components 00 dv ..············································ ·········································.... f(x) ··•··... ...···· n The Integral of the continuous area under the curve is the summation of an infinte number of disctrete rectangular measurements made to a specified limit Integration is a special case of summation. An integration isn't a simple summation, but the limit of a sequence of summations Integration is defined as the limit of a summation as the number of elements approches infinity while a part of their respective value approaches zero. All Planck energies are discrete equilateral geometries Summation is the finite sum of multiple, fixed values. 11 00 X n · · ·. ············... ﾷｾＭ＠ F . .· · · ....··· ｾＮｴｩｭ･＠ lntegrating the energy quanta contained within equilateral charge geometry gives the variable Force required to acheive changes in motion [Energy, work, acceleration J Summation is a macro adding of DISCRETE quantities. Tetryonics 87.05 - Integration '·,, · . .. - sedond ' 2 •••• ••••• ...... mass is the surface integral of EM energy geometries per unit of time 71 Copyright ABRAHAM [2008] - All rights reserved "The calculus of infinitesimals" Scalar Energy vector Force The fundamental theorem of calculus simply states that the sum of infinitesimal changes in a quantity over time adds up to the net change in the quantity. Lelbnl:ts vis viva (latin for IMng force) lsmv2, v Much of Newton's work centred around momentum and changes to It [mass.velocity] F =m E mv 2 [dv/dtJ Sir Isaac Newton Gottfried Wilhelm von Leibniz (July 1, 1646- November 14, 1716) (1643-1727) ｾ ｾ ｾ Ｍ //./ mi - - -. .\ .. ﾷ ..... integral calculus ............. ｾ ﾷ ｾＭＮ ｾ •, •· .......ｳＮ＼Ｚ｟ｾＹ＠ Ｎ＠ differential calculus Ｒ＠ .. ···• The founders of calculus thought of the integral as an infinite sum of rectangles of infinitesimal width The fundamental theorem of calculus is a theorem that links the concept of the integral with the derivative of a function. Tetryonics 87.06 - Calculus .. .... ... .. .... '' '' ｾＮ＠ ····-..ｾ＿ＮＭﾷ＠ v ｾ ﾷ＠ ... ," .. .. .. . .. . . .... In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes 72 Copyright ABRAHAM [2008] - All rights reserved [L@O @mJ ｯｾ＠ scalar energies linear momentum geo111etries linear momentum is the square root vector force of the scalar energy required to do a set amount of work d[mv]!dt = dp!dt = ma 8 64 52 52 7 49 Energy momentum 36 6 25 5 16 4 9 E t 1 2 5.9 3 4 5 6 ｾ｀｜ｙａＷＱｴﾩｭｊ＠ Energy T110111entu111 7 8 9 10 print out this page 3 then cut out the 10 Planck energy momenta triangles shown p rna] t 2 3 slice fine lines along the pink linear momentum arrows with a razor blade or similar as highlighted under the Newtonian acceleration curve 5 4 6 7 8 9 10 5.9 4.6 3.1 10 21 35 7.2 7.2 mass is energy per second ·······..... ...Ｕ［ＬｾＮＭＯ＠ slide the cut-out triangles into the slots created ......__-----t-------45 rotate the assembled model to show the real force momenta geometry at work Tetryonics 87.07 - Physically modeling the geometric forces of acceleration in calculus 73 Copyright ABRAHAM [2008] - All rights reserved d[mv]!dt = dp!dt = ma linear momentum is the square root vector force of the scalar energy required to do a set amount of work 8 64 I I 7 49 ＳＶ p Ｍｾ ＭＫ＠ 6 linear momentum 25 16 kg m s 4 print out this page p 3 9 E t 1 2 3 4 5 6 7 8 9 5 t 10 2 3 4 5 6 7 8 9 10 v [L@O @[Y1) ｯｾ＠ scalar energies ｳｾｵＺ ﾷ＠ . . . . . , # . . . . . . . . . . . . . .. .//_./······· / ODDnumber E mass-energy momenta relationship square root equilateral energy is linear momentum (M@\Y;\Yl[t@[J1) linear momentum energies\\ / bosons F = ma mass is a scalar constant relating Force to acceleration v. : 2 ..........// . v normal distribution of energies --------- F .....··· Ｍ ｾ Ｍﾭ seconds Planck quanta and their vector linear momentum lie at go degrees to the angle use in the graphs of motion in calculus Tetryonics 87.08 - Tetryonic Planck geometries in calculus F = m !; 74 Copyright ABRAHAM [2008] - All rights reserved Differentiation Differentiation is concerned with things like speeds and accelerations, slopes and curves etc. These are Rates of Change, they are things that are defined locally. d.v d.v d.v ..··•···································· .. F F An increase in a force opposing an object's vector velocity results in DECCELERATION An increase in a force in line with an object's vector velocity results in ACCELERATION !A ｾ ｬｩ｝｛ｊＩｾﾮ A linear measure of Forces acting on physical systems resulting in changes to distance covered per unit of time Ｎ ｾｬｦｩ｜ｊｵＡＧ｀ｴ＠ Ｎ ｾﾮ｣ｴｬｩ｛ｰＩ｀ｊ＠ F' [b)w ｾ＠ ｬｦ｀ｉｔｴＩＨｄｏ｛ｊｩｾ＠ (ID[Ji) ｾ＠ •·•··•·... ·•••··•······ ...... ｀ｬｩｾｯ｣ｴﾮ＠ O[Ji) ·; t1me ............•··•••··•••• ........... : .......... . ｉｔｦｴＩ｀ｾｄｵｩ＠ / •••··· ｬｦＮﾩ ｾ ｾﾮ＠ A scalar measure of Forces acting on physical systems resulting in changes to their rate of motion f = ma v In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. m/s Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. Tetryonics 87.09 - Differentiation a ｾ ＠ｶ ｾ ｴ＠ 75 Copyright ABRAHAM [2008] - All rights reserved Visualising the geometric half-truths of relativistic physics A The source of all the physical relationships of mass-energy momenta and the constants in Physics is the Equilateral Triangle (and all texts must be corrected) v Energy geometries within Physics including Special Relativity with its Lorentz corrections have historically been incorrectly illustrated through the geometry of right angled triangles Physics is geometry, one cannot be separated from the other Equilateral geomtries lead to a intuitive understandings of Physics, Chemistry, Electrodynamics and Gravitation along with all the other apsects of Nature. =E = m V2 v2 ENERGY F ｾ＠ moc2 2 2 4 Generalizing, we see that the square of the total energy, mass, or distance in spacetime is the sum of the components squared. We can see an origin of distance in spacetime relating to velocity in pc in which Energy is subject to Lorentz corrections [v/c] ､ ｾ ｐ＠ dv L...JF == dt == mdt == m a :F == m a. 2 2 E =pc+m0c 6.629432673 e-34 J E == pc. momentum p2 = m v2 Tetryonics 88.01 - Tetryonic vs. Pythagorian geometry Additionally, EM mass can be directly related tot the Energy content of a body by the velocity of Energy E == me? 76 Copyright ABRAHAM [2008] - All rights reserved Velocity In physics, velocity is the measurement of the rate and direction of change in the position of an object. v Velocity (..·: ::·· · ...) m s ........................./ It is a vector physical quantity; both magnitude and direction are required to define it. v ••.• The scalar absolute value (magnitude) of velocity is speed, a quantity that is measured in or ms-1) when using the 51 (metric) system. metres per second: s ••••••••••••••••••·•••••••••••••••••••••···•········ is a 2D radial Space-time ········································•··············... MEASUREMENT .. ..···· ······... ····... Speed is the scalar value of the Distance traveled per unit ofTime v== Velocity is the vector value of the Distance traveled per unit ofTime ｾ＠ m s ...2 (i Velocity squared is the scalar value of the Distance traveled per unit of Time squared (Energy of a given spatial volume) All divergent Energy possesses a vector direction and an associated scalar area whose energy content is quantised ········... ··.. ····················· ·······... ·································· c2 ······················ se·c·on·a s Tetryonics 88.02 - Velocity .·· ...../ ····· Velocity squared m2 s2 77 Copyright ABRAHAM [2008] - All rights reserved Acceleration ,.....················: ·····.., t"see··. .·"j In physics, acceleration is the rate of change of velocity over time [dt] ··......................·•·· a is a 30 Spherical Space-time Acceleration In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. m MEASUREMENT ...············································· Acceleration has the dimensions [Length]/[Time Squared] In 51 units, acceleration is measured in meters per second squared (m/s''2). ··········································•···············... a= &x Ｆｾ＠ &t" In classical mechanics, for a body with constant mass, the acceleration of the body is proportional to the net force acting on it (Newton's second law) [ 2n ]+1 Deceleration &y Acceleration [ 2n ]-1 f =111a hv 2 Fofce ·.. ········... ······... acceleration ｩ ｾ＠ a measure of a vector Force ｡ ｾ ｴｩｮｧ＠ along a line in a spatipl volume · · · · · · · ·· · · s·ec6nt:rs· c4 i · · · · · · · · ···... ..../ .....······· ..·· Tetryonics 88.03 - Acceleration Additionally, for a mass with constant velocity, (ie in an inertial frame) the energy of motion is expressed as its momentum (acceleration causes changes in Energy-momentum) p kg m s 78 Copyright ABRAHAM [2008] - All rights reserved Momentum linear kg m s momentum p In classical mechanics, momentum (pl. momenta; 51 unit kg·m/s, or, equivalently, N·s) is the product of the mass and velocity of an object (p). v Like velocity, momentum is a vector quantity, possessing a direction as well as a magnitude. ..··················································· ·················································.... .. Momentum is a conserved quantity (law of conservation of linear momentum), meaning that if a closed system is not affected by external forces, its total momentum cannot change . .. ..··········· ..·· ··········... Momentum should be referred to in its specific forms to distinguish it in its various forms [Quantised Angular, Linear, Rotational and quantum/nuclear momentum] Although originally expressed in Newton's Second Law, the conservation of momentum also holds in special relativity and, with appropriate definitions, a (generalized) momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, and general relativity. In relativistic mechanics, non-relativistic momentum is further multiplied by the Lorentz factor. ·.h 2 ﾷ ｾＭ ｾ＠ .. Linear momentum is the vector square root ·. ··········... of the mass-energy in any {K}EM field geometry ........... ··... and produces a vector velocity ···· ························........... 1 second.... ································· ································· Energy can be expressed as the square of linear momentum equilateral Planck energy momenta 111Q V 2 Tetryonics 88.04 - Momentum 79 Copyright ABRAHAM [2008] - All rights reserved Energy-momentum relationship Quantum Mechanics The total number of equilateral Planck quanta [quantised mass-energy momenta} is directly related to the square of its linear momentum [mass-velocityJ Newtonian physics v m h v Quantised energy equilateral momenta QMQIIYilttff.§tedllEIYiltergy mo;mtelYilttQJ. &QJ.HQJ.r ｭｾﾧ＠ ｾｉｙｩｬｲｧｹ＠ ｭ｀ｾｉｙｩｬｴｑｊＮ＠ 11.§ rteHQJ.tttedl U» ttlhLr@MglhL ttlhlte teqJ_MiHQJ.ttterQJ.H gteo;mtettry «»f PHQJ.IYilclk g§ c«»mttQJ.IYiltt mnv 2 E Scalar energy linear momentum mv 2 Tetryonics 88.05 - Energy momenta pv 80 Copyright ABRAHAM [2008] - All rights reserved v lnertia1resistance to Force v Inertia is the resistance of any physical object to a change in its state of motion. v Changes to mass-velocity require a corresponding Changes to mass-velocity produces a change in an object's Kinetic Energies .·········· ,.- F F .............. The total intrinsic momenta of all energy waveforms is the sum of their constituent Quantised Angular momenta (mass-energy momenta) Ｍｾ･｣ｯｮ､＠ __ _ Any change in motion results in changes to the Charge geometries creating in turn proportional changes to KEM mass and momenta components F = ma The 'inductive resistance' of Charge quanta fields to changes in their mass-energy momenta content is what we term Inertia q Matter in motion has Kinetic Energies in addition to invariant rest mass-Energy KE = RE - rest Matter v v [v-v] KEM = Mv2 Any change to an object's velocity results in a corresponding change to its mass-Energy momenta which is reflected by its inertia E quanti sed moment of inertia Tetryonics 88.06 - Inertia mass q [v-v] 81 Copyright ABRAHAM [2008] - All rights reserved Quantised Angular momentum As it is a physical [equilateral] geometry QAM is conservative in any system where there are no external Forces and serves as the foundational geometric source for all the conservation laws of physics A major re-<lefinition of quantised angular momentum in physics is revealed [v-v] m ..·······················""............... .. ···········································... ..·· ··.. ..······ ····•... 64 15 classical rotational angular momentum Quantlsed Angular momentum In quantum mechanics, angular momentum is quantised- that is, it cannot vary continuously, but only in ODD number "quantum steps" between the allowed SQUARE nuclear Energy levels ---------- --·------- ｾＭﾷＺＮ｟ \._ ﾷ｟ＺＮｾ Ｚ＠ ... ·.:.:_:::::-_ ..:.. ··········... ｑｬＡ＼､Ｑｔｩｴｾ＠ . ..... ｾ＠ ........._.·._\_ ........_ _ - - -· ----------- Ｎ｟ ｾ ﾷＺＮ｟ ］ｾＮ｟Ｚ ］ＺＮＬｯ ］ＺＮ｟Ｍ ＺＡ［ＧＮＱｾ｟＾ﾷ＠ ｟ｾ＠ A1TilgruJTI<dlw ｍ｜ＨｑＩｭｾＱｔｩｬＺ｡＠ In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system. /\ / ＺＮＭ｟ｾ＠ .. .... \_:.:::_-::::_:::-::_:::::_:::::.... ::.:_.:.:.·_ .. m .. When applied to specific mass-Energy-Matter systems QAM reveals the true quantum geometry and nature of Energy in our universe 2 ........... ·······················.......... 1 second......··························· ····························· ..········ ·······... ··...\ h kg m s /..... 2 mO mass x QAM ···.....................·· Planck's Constant Angular momentum is sometimes described as the rotational analog of linear momentum, in Tetryonics it is revealed to be the equilateral geometry of quantised mass-energy momenta within any defined space-time co-ordinate system Normally viewed as an expression of rotational momentum Quantised Angular Momentum [QAM] is in fact a result of the equilateral geometric quantization of mass-energy Tetryonics 89.01 - Quantised Angular Momentum 82 Copyright ABRAHAM [2008] - All rights reserved Charged geometries All charge geometries are nett divergent Q Divergent energy momenta [v-v] Convergent energy momenta 1 0 3 1 6 3 10 6 15 10 21 15 28 21 36 28 All charge geometries are comprised of finite equilateral energy momenta quanta Tetryonics 89.02 - Charge geometries 83 Copyright ABRAHAM [2008] - All rights reserved Renorma Hsation Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. The problem of infinities first arose in the classical electrodynamics of point particles in the 19th and later in the calculation of Gravitational fields in General Relativity in the early 20th century. ln Tetryonics lnfinities do not exist lnQED lnfinities t11ust be cancelled 1 /,/.......········ ......... \ ｨ ﾷ ｾ＠ .. .....-. ·····........... _.llliiiiiiii.....Y·2 ﾷＭｾ＠ The mass of a charged particle should include the mass-energy in its electrostatic field (Electromagnetic mass) which in turn would approach infinity as the electron radius decreases. Tetryonics solves the problem by clearly differentiating between EM mass and Matter and using finite equilateral geometries for all Matter in motion Initially viewed as a suspicious provisional procedure by some of its originators, renormalization was eventually embraced as an important and self-consistent tool in several fields of physics and mathematics. Tetryonics 89.03 - Renormalisation 84 Copyright ABRAHAM [2008] - All rights reserved Mapping 3D spaces using Recti-linear co-ordinates The adjective Cartesian refers to the French mathematician and philosopher Rene Descartes who developed the coordinate system in 1637 Cartesian coordinates can be defined as the positions of the perpendicular projections of a point onto the two or more axes, expressed as signed distances from the origin. y Since then many other coordinate systems have been developed such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space. 10 X 0 : : 5 : : : : 10 ' : : : : 15 : : ｾ＠ ｾ＠ .. ! ! ! ! ! ! ! ! ! ! y Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more 3D Cartesian co-ordinate [c 31 systems are distinct from spherical co-ordinate [c 41systems Tetryonics 90.01 - Rectilinear co-ordinates 85 Copyright ABRAHAM [2008] - All rights reserved Polar co-ordinates In mathematics, the polar coordinate system is a two-dimensional co-ordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. Action Dynalllics Curvilinear co-ordinates may be derived from a set of rectilinear Cartesian coordinates by using a locally invertible transformation that maps one point to another in both systems Metric Tensors 360 ｾ＠ ........... Ｍｾ ﾷＺ＼＾Ｎ｟ｾ［Ｌ ..... ......... -.. - .......... Ｚ｟ﾷ｜＠ .. ·· 5·· ······ Gravitational acceleration Polar or curvilinear co-ordinate systems are used extensively by Einstein in his theory of General Relavtivity 180 In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the co-ordinate lines may be curved. Reimannian curved space-time Tetryonics 90.02 - Polar co-ordinates 86 Copyright ABRAHAM [2008] - All rights reserved Co-ordinate transformations There are many different possible coordinate systems for describing geometrical figures and they can all be related to one another. Such relations are described by coordinate transformations which give formulas for the coordinates in one system in terms of the coordinates in another system . I I . I I . I I . I I . . • • I I . I I ....... ---'-----'-----'-----'----A ........•...... .,_,.,.,., . I I I 1o ﾷｲＭｾｴｩｪ＠ I ·--·--- ＭＧﾷｊｬＮｾＢＱ＠ ＭＧﾷｊｉＮｾＢＱ＠ I I I I I I I I I I I I • • • I I I • I I I I I . I I . I I I I I I I I . . I I • • J ........ .J ........ ｾ＠ • • • . I I I I . I I . I I . • • . . I I I I ........ .J ........ .J ........ .J ........ J .................. J ........ • I I I I I I I I I I I I I I I I • • • • I I I I I I I I I I I I I • I I I I • I I I I • I I I I I I I I I I I I I I • • • I I I I I I I I I I I I • • • I I I I I I I I I I I I • • • I I I I I I I I I I I I I I • • I I I I I I I I • • I I I I I I I • • I I I I •••• ｦｲｌｴ ｊ ｾ ｪｲｬｴ J ｴｾ ﾷＱＺＭｾ＠ ｳｽ ＡＱＺ＠ s·· : : : : ｾ＠ : : : : ..ｾ＠ : ·--- ___ ._ ____ ._ ____ ,_____ ,____ ..._ ___ _. ____ _. ____ _. ____ _. ____ ____ _. ____ ............................................... .. I I I I I I I I • • I I I I I I I I • • I I I I I I • • I I I I ·--- ---r----r----1-----1----t---1----1----1----1----1----1---- Ｍ ｳ ＭＱｾ＠ ·--- ..........................1·----l·-- -----------..l·--- . . ----..l---- .. ----..l---- ............................................... . X I I I I I I I I I I I I I I I I I I I I I I I I I I • • • • I I I I I I I I I I I 1- I I I I • • • • I I I I I I I I I I I I • • • • I I I I Converting between Polar and Cartesian coordinates I I I I ....... ---t-·---.. ·-··l-----l---------1---- ...................................................... ................................................ . ........... I 0 i i i i y 5 i i i i 10 i .... . I I I I i i i 15 i i opposite adjacent tan 8 rz=12z+Sz y ｴ｡ｮＭｾＨＵＯＱＲＩ＠ 22.619 X solving for the triangle using trigometric functions Tetryonics 90.03 - Co-ordinate transforms ｾ 13 ＶＹ＠ 87 Copyright ABRAHAM [2008] - All rights reserved Spatial co-ordinate systems Cubic "" "" " z ----------------" {----- " "" " "" Spherical Ｍｾ＠ " Spacetime is any mathematical co-ordinate system or model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional with time playing the role of a fourth dimension that is different from the spatial dimensions. ,. TJ.Lv 3 I ,_" "" " "" "y Tetryonics maps spatial co-ordinates through the momenta vectors of equilateral Energy ------------------i seconds 2 Tetryonic Space-Ti111e Cartesian Space-Ti111e v ./ / / / / I I I ,..__ ｾＭ v v ｾＭ ｾＭ ....... v Vv v V / VI / v I I I I I I I I I I / ,t..L 1/ t'l ,..,. L.! ｾ Ｂ ｉ＠ 1\ \ '\ ｾ I"'\ '\ I'., "'-.. ＮＭｾＧＢ＠ r-......._ "' r--... """ ""- " ｾＧＭＮＬ＠ "-., ｾ＠ ｾ＠ r--. r--.. ........ "-., '\ "' 1\ \ \ 1\_ \ \ 1\ \ \ \ \ 1\ \ \ \ \ \ \ • ｾ＠ r-- "r-- ln 'M ..,. , ._ 1-. / vV / v v ........... _ .......... .......... ........ X L EM mass-ENERGY momenta are equilateral geometries v II I--' 1---1---- I"- r-_ I'- I'I'- \ 1'--- "-.._ "' " Ｂ＠ -/ \ \ \ \ \ \ \ \ \ \ \ \ 1\. : \\\ -- ,..__ vv- r--.._ --- 1---- I I I I I I 1/ I I II V /1 / / I I I / / J / v v v / ./ ·'' / ......... . 1,- X ＺｾＯ＠ From a Euclidean space perspective, the universe has three spatial dimensions and one dimension of time [reflected by quanti sed angular momentum]. v / Mapping equilateral Energy geometries onto recti or curvi-linear spatial co-ordinate systems introduces mathematical complexity to a otherwise simplistic geometry for all EM mass-Energy-Matter interactions +y In physics spatial co-ordinates to date have been based on Cartesian co-ordinates when in fact Energy momenta follow a Tri-Linear co-ordinate geometry Tetryonics 90.04 - Spatial co-ordinates 88 Copyright ABRAHAM [2008] - All rights reserved vectors TJ.Lv mass-Matter m =M mass-energy equivalence Euclidean c2 equivalence .................. 1 Dimensional / . . . . . . · · · · · ---- energy-Matter equivalence ········................... velocity Cartesian co-ordinates z GR fails differentiate betweem EM mass-ENERGY-Matter unlike Tetryonic theory ［ｾＭ＠ I ;; "" ," " "" " "" /; {---------- "" " " "" "" " Ｍｾ＠ Spherical ...... Q. ___ _ ............ __ ... -····· Dimensions velocity squared 2 I" ) y Ｍｾ＠ " "" "" " " "" "" 3 Dimensions velocity cubed Space-time co-ordinates The propagation of Energy momenta forms distinct spatial co-ordinate systems 180 3 Dimensions Where angles are typically measured in degrees C) or radians (rad), where 360° = 2n rad = 't radians. quadrature velocity Tetryonics 90.05 - Spatial geometries 89 Copyright ABRAHAM [2008] - All rights reserved Energy per second Tetryonic co-ordinate systems z Energy per second squared M Proton 2D mass-energy • geometrzes X ,. . . .......... \ - ..... ',',, ', ···...... , . · · · · · · · · · · ·. . - Ｚ .....···· --- _,,' ..... - /,,,,,,,,,,, ﾷ ｾ ﾷ ｾ Ｚ ｾ ｲ Ｚ ﾷＺ＠ -..· · · · · · · · · · · · // - _···... pl.a11.a r······ . -/ , __ Ｍ ｾﾷ y Ｍ Ｍﾷ Ｍ ﾷ ｟ ﾷ＠ _______________ _ Planar mass-energy geometries have no z-components Differentiation between 2D mass-energy & 3D Matter is key to extending our understanding of physics Matter topologies have z-components Tetryonics 90.06 - Tetryonic co-ordinate systems 3D Matter topologies Neutron 90 Copyright ABRAHAM [2008] - All rights reserved v ODD SQUARE 9 1 numbers 2 3 4 5 4 3 2 1 hv 1 2 3 4 5 6 ... n ···6 54 3 21 11 normal distributions 12345654321 2 Quantum probability distributions The equilateral geometry and distribution of quantlsed Energy momenta provides the basis for all statistical probabilities In Quantum mechanics, thermodynamic & Information entropy -Including a solution to Heisenberg's Uncertainty Principle thus paving the way forward for a new understanding, and manipulation of physical phenomena at the quantum level 2 3 4 3 2 Normal distributions are extremely important in statistics, and are often used in the natural and social sciences for real-valued random variables whose distributions are not known Tetryonics 91.01 - Quantum Energy Distributions 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 91 Copyright ABRAHAM [2008] - All rights reserved Quantum Probability Distributions The normal distribution is a probability distribution. It is also called Gaussian distribution because it was discovered by Carl Friedrich Gauss. square root momenta v Probabilities are A11 probabilities the square of the are re-nortllalisable Amplitude and sum to Unity n 1 EVEN Longitudina1 Photons ODD transverse Bosons Planck quanta hV. . . .__.. . . ._________ ｾ ﾷＮ Ｒ ﾷ＠ ... ｾ Ｍ Ｑ ｝＠ ............ .2 4 5 6 7 8 9 8 7 6 5 4 ·. . ·····.... Nonnal Distribution c2 ｛ ｾ ｝＠ ........ Tetryonics 91.02 - Normal Distributions and Probabilities 2 1 Square energ1es 92 Copyright ABRAHAM [2008] - All rights reserved Quantum Levels Quanta Distributions transverse bosons 2 v =E = f Quantum Probabilities 1ongitudina1 photons 4 W- Tetryonics 91.03 - Quantum Wavefunction Distributions 93 Copyright ABRAHAM [2008] - All rights reserved Wavefunctions h Qua i]Tll Leve1s Transverse Bosons Longitudinal Photons ElectroMagnetic Waves Quantised Angular Momenta > < Velocity Amplitude Probabilities Linear momentum Squared energies ｅｮ･ｾ Tetryonics 91.04 - Wavefunctions ｟＠ . nlomenta 94 EM waveforms t/) ta ...... z c: r----1 ｾ＠ 0 E- ｾ＠ ta ＠ｾ ::l ta cc: c: ta 0 ·..... ""C 0 ::l ·f "' - £ o L......l ｾ ｾ＠ ｾ ｎ＠ u :I: ..c:...., a..·e;, c: ｾ＠ 0 ｾ＠ All EM waveforms can be measured by either their Transverse EM masses {BosonsJ or their Longitudinal EM masses {Photons] I BOSONS ｯ ｾ Ｌ ｮ＠ > ｾ＠ I I I I I 64 I I I I I I I I I Frequency:(fJ J,. I ＳｦｳｾｴＬ＿Ｖ＠ I 1 ｾﾷ｡＠ I I Planck quanta [[mnv2]] velocity c I rwt- - -· - - ...! - I - I J. Ln- --Ｍ Ｍ ｾ Ｍ ｾＭＺ - - ｾ＠ I I I I I I I I - I 1-- - I I I I I Ｍ ｾ - 1 L - 1 ·- .I_ 1 I Ｍ Ｍ - - :--, - - I ｣ Ｍ ｾ Ｍ I s I CU I .-r --- ｾ＠ c Q) N ... ｜Ｐｾ＠ .. u.. u,--c 0 >u :::::J - - -cu-"" I!!-- -cu -- s _m. ｾ＠ ·o q-5c c.. c. 0 0 ..!: 0... - ::::::J e Copyright ABRAHAM [2008] - All rights reserved - _Fl! --- - - - Ｍ ｾ - - - --- \.!) Ｍ _....,. ·Ltt- - - - - - - - - - - ----------- - - - -cwt- - - - - - - - - - - - - - Vl Bosonsare transverse quanta £Ol c Q) Qj > ｾ＠ ｾ＠ UJ cu "'RS .t::. LL 2hv hf :J Tetryonics 91.05 - EM Waveforms Bosons • · ｌｯｾｧｩｴｵｃＺｮ｡ｬ＠ v) 95 Copyright ABRAHAM [2008] - All rights reserved Quantum Energy distributions f E= 8 Quantum Quanta v 64 f c Longitudinal Photon Frequency Transverse Boson quanta Wavefunction Quantum Energy Levels [n 2] S N S N S N S N s s s Levels Waveforms Bosons Planck quanta ｯ ｾ ＿ ｴ＠ [[mnv2]] EM waves ｅｖ ｾ ｔｃ＠ velocity Tetryonics 91.06 - EM wave Distributions Planck quanta [[mnv2]] velocity H'l' N S S N N S S N N S S N N S S N 96 Copyright ABRAHAM [2008] - All rights reserved Normal Distributions Pierre de Fermat Leonhard Euler The Gaussian distribution sometimes informally called the bell curve. - J1. Ｉ ｾ ｝＠ !2 [1+ erf{·.. xy'2;;2 ｾＭ Pierre de Fermat is given credit for early developments that led to infinitesimal calculus. all ODD numbers are a 2 b 2 = [a-b].[a+b] the difference of two squares 9'1% Ｍｾ ｾ ｌ＠ Th . Normall Ois.tributim1 ｾ＠ 36 Random Events e -x.J-2 {1F11/[1n1 t.ll kl p(l - Ｑ ﾷ｜Ｚｾ＠ 1 } k p=q= Yil,n = B -1 Cent rail Umfrt Theorem In ﾷ ｾ ｾ ｨ･＠ li ｨ ･＠ 11mit of large numbeii'S , s.um .of ｨＱ ､ｾｩＡｮ･ｬＱｴ＠ nd!om v.aFil!lbl:e II ｢ｾ＠ nomnallo/ dl:m buted Scalar Energy EM waves [SQUARE numbers] Transverse Bosons [ODD numbers] n -1 The Bl o:mia I Theorem IPr<[lk] n-1 Leonhard Euler developed a formula which links complex exponentiation with trigonometric functions ot 'thl! ｯＮｾ＠ A bell shaped curve defines the standard normal distribution, in which the probability of observing a point is greatest near the average, and declines rapidly as one moves away from the mean. Pr(x ...: v ｾ Ｍ ｲＮｾ ﾷｾ ｾ ｯｦ ｟ ｾ ﾷ Ｍ ｾ ｟ Ｎ ﾷ ｾ Ｍ n-1 l 2 3 4 5 6 5 4 3 2 longitudinal photon distributions [normal distribution] In probability theory, the nonnal (or Gaussian) distribution Is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function Tetryonics 91.07 - Bell Curves l 1 Copyright ABRAHAM [2008] - All rights reserved Fundamental theorem of Energy momenta A nth level scalar energy momenta waveform has exactly n linear momentum in unit circle co-ordinate systems (with Longitudinal and Transverse equilateral Planck waveforms being orthogonal to each other) +1 scalar energies / / / / E Longitudinal EM waveforms are the waves produced by spark gap discharges 3n/4 '' '' / I I '\ Their energy momenta are co-linear with the wave's direction of propagation '' I I n/4 ', \ \ \ \ \ \ '' / Transverse EM wavefonns are the waves produced by accelerating charges 7n/4 "'"' ' ' Ｑ Ｏ ＺＡｾ Ｍ Their energy momenta are orthagonal to the direction of wave propagation p 2700 -------- mv -1 linear momentum • {Mathematics] l 0 • rJ {Electrical theory) square root of negative one Tetryonics 91.08 - Fundamental Theorem of Energy momenta 98 Copyright ABRAHAM [2008] - All rights reserved Velocity and Time dependent EM fields v ··. SQUARE ROOT momenta Quantum levels SQUARE number ENERGIES h'. . ｾＭ All EM geometries are ｣ｯｮｩ • Ｎ ｰｲ ｟ ｾｳ･､＠ of Transverse quantum levels and L9ngitudinal wave probabilities ··············..... ..-········// c Tetryonics 92.01 - Geometrics 99 Copyright ABRAHAM [2008] - All rights reserved Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear Junctions of the form f( x) = ex, where c is a constant; matrix multiplication corresponds to composition of linear transformations. Matrices v Further developing equilateral Matrices and tensor mathematics to reflect the 2D geometry of EM , KEM and GEM quantum fields, along with the geometric quantisation of mass-energiy momenta and their energy distributions allows for field interactions to be accurately visualised and modelled v v ·······...\\ //_......... ,• ·······...\ ...., 1 nl nS 3 4 3 1x1 n2 ｾｨ ｾ＠ . ..］ ］Ｍ Ｍｾ ｾ Ｒ＠ ｨｾ＠ .....- ......ｾ］ ···... _________｣ ｟ ｾ Ｍ Ｍ ﾷ ﾷ ﾷ ﾷ ﾷ ﾷ ﾷ＠ ··-...________｣ ｟ ｾ ＮＭ Ｍ ﾷ ﾷ ﾷ ﾷ＠ Positive Fields Negative Fields 3 ｹ Ｒ＠ 2 4 4 2 n7 2 2 3 4 2 1 3 4 2 3x3 2 1 1 1 3 3 4 5 1 1 n8 2 1 3 2 2 2 4 3 2 2 1 3 1 l 2 4x4 1 Tensors are geometric entitles Introduced Into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices to Increasingly higher orders. Modifying Square matrices to reflect the equlalteral geometries of Tetryonlc fields allows for the accurate geometric modelling of all Scalar & Vector fields along with their varied Intrinsic quantum energies and physical properties Tetryonics 92.02 - Tetryonic Energy tensors 3 3 4 3 4 5 6 3 4 5 2 3 4 5 6 1 1 3 4 5 3 4 5 6 3 4 5 2 3 4 5 6 1 2 1 2 3 4 1 2 7x7 1 3 1 1 3 4 5 6 7 8 7 2 6 7 1 2 2 6 5 2 n4 SxS 6x6 1 3 1 6 5 1 2 3 n6 2x2 n3 3 4 7 2 3 4 5 6 1 2 1 2 3 4 5 1 2 3 4 1 2 3 1 8x8 100 Copyright ABRAHAM [2008] - All rights reserved Energy momenta Tensors momenta (a property of Energy) is converative PM ------- (E, p 1, p2, p3, p4). Matter (a geometric property) is NOT conservative All standing-wave Matter topologies can be modelled using its Tetryonic charge energy momenta Tensors with an additional Kinetic EM energy-momenta tensor required for Matter in motion PKEM ｾ＠ (E, pS). 20+ 1 [SR] mass-energy momenta can be folded into 3D+ 1 [GR] Matter that can be modelled using 4 Energy-momenta tensors P=L P E =L E all fascia all fascia Positive standing-wave Matter Relativistic Matter in motion v Total Momentum is the total of all quanta linear Momenta in a 3D particle Energy of a massive particle is the total of all Planck quanta (compton frequency) in a 3D geometry : I=L I Note that both 2D mass-energies {Special Relativity1 u and 3D Matter [General Relativity1 have distinct Energy momenta all fascia Inertial mass is the total of all inertia in a 3D particle Negative standing-wave Matter PhotoVts have V\.IA{( eVtergy -W\OW\eVttuW\ Relativistic Matter in motion v It is the 3D Tetrahedral topologies that provides a definative basis for Matter teV\.SOV'S Tetryonic Energy momenta tensors should not be confused with Four vector tensors which map energy-momenta vectors in 3D spatial[cartesian] co-ordinate systems c Tetryonics 92.03 - Energy Momenta tensors 102 Copyright ABRAHAM [2008] - All rights reserved _y __ _ y 10 ·-------.__\\.\ 9 ｾ＠ 7 6 5 -; - 4 3 :!! ｾ＠ 1 ＦｩｾＭ L -- ---....__ ...·········· ... ·--------------. ｾＲ＠ c Ｍｾ v ODD SQUARE quanta quanta c ＧﾷＭｾ＠ Linear correction factor Scalar correction factor c Wavelength contraction \... Lorentz factors are UNBAR and SCALAR velocity related corrections ,./ to the relativistic mass-energy momenta content of any physical system accelerated by BM Forces '•' '• '•'' ''' ' ' ' ' ' ' ' ' ' ' " " " " " " " " " " Relativistic momentum Ｇ ｾ＠ ｾＮＢＭＧ＠ ,. ' ' •'' ------/ Relativistic mass-energy Lorentz corrections Tetryonics 93.01 - Relativistic Lorentz corrections Time Dilation 103 Copyright ABRAHAM [2008] - All rights reserved v Equilateral energies Any measurements of mass without a Space-Time co-ordinate system are measurements of Energy Photons form the basis for all charged energy geometries and their associated quanta distributions 112 1J2 9 1 25 112 1J2 11 1 312 2 512 3 712 3 512 2 312 1 1J2 112 1 312 2 512 3 712 4 712 3 512 2 312 1 112 Tetryonics 93.02 - Equilateral energies 312 1 112 36 81 112 1 105 Copyright ABRAHAM [2008] - All rights reserved Electromagnetic mass 2D mass-energy is the surface integral of3D Matter 1 Y 2 11:v. Photons ____ / 4 ••• ••..... _s:;._z_____ ••• --·· Time specific measurements of equilateral ENERGY momenta geometries forms the basis for EM mass ｾ Ｚ＠ ｾＯＲ＠ 1/2 ··. .......... ·····- ..... ______ ｾＭ ,, .. "'"' /................ ｾ Ｎ Ｖ＠ ........1. ｾＭ＠ ··.. 1/2 1 3/2 2 3/2 ········...... ..... ｾＭﾷＯ＠ 1 ＱＭ Ｍﾥ 2 Ｍｾ ｾ＠ ｾ 1/2,/ Ｍ ·····-.... 512 ｾ 2 Ｍ ｾ＠ 1/2 Ｍｾ ••••••·••...........\ ｾ ＺＧ＠ Ｎ｟Ｍ 3/2 ....... \\9 ｾＭ ·····- ... ---------- ｾ＠ 3/2 ｾ 1 Ｍｾ＠ ｾ Ｚ＠ \.._11 "' 1 "' 2 "' 3 "' 2 "' 1 .. ｾ＠ ...... 112 ···-..... Ｍｾ Ｍｾ ,•' /.............······· /.........········· mass-less panicles is a physical misnomer 411' \ W 1 112 2 112 3 712 3 112 2 112 1 112 / ···········........... ｾＭ ..................... ｾ Ｕ＠ ｾ＠ .. 11.1 1 • 2 • 3 71J1 4 71J1 3 112 2 • .................. . Ｍｾ 1 w: .... ··············/ \ 17 .... ,,.· Ill 1 Ml 2 Ml 3 lllll 4 ·············.......... Tetryonics 93.03 - ElectroMagnetic mass ｾＭ Ml 4 lllll 3 Ml 2 Ml 1 Ill ... --·············/ :: All measurements of energy in spatial co-ordinate systems are measurements of mass Matter-less is the appropriate terminology 106 Copyright ABRAHAM [2008] - All rights reserved Standing-wave Matter v ｂｯｳＬ｢ ｟ｾ Ｍ ｾ Ｍ ﾷ ［ Ｍ ｾｲｧｹ＠ Matter is a higher order 3D topology created by standing wave mass-energies ···-.., _______ ······.., _____\_\\ ｨｾ Ｎ＠ \\\ y2 ......__ Photons .../ ···· .........ｾＭﾷ＠ The equilateral mass-ENERGY content of 4nn Tetrahedral standing-wave topologies forms the basis of Matter ｾＭ ＯＲ＠ｾ ······· ...... ___ ＮﾢｾＭﾷ＠ ,, ..... "' ｾ Ｏ＠ ///// \? ｜ ｾ＠ ｬ Ｎ ｾ＠ ｜ＮＬ ｾ Ｍ 1 3/2 2 ｾＭ＠ 1 ｾ ＭＯ＠ 3/2 ｾ＿ ﾷ＠ ｜Ｌ ｾ＠ ___, ······· ..... ____ ﾢＮｾＭﾷ＠ /,//_, ..----··' \ 1 ｜ ｾ＠ ＭＢ ｾＰ ｾ ｾ Ｌ＠ ···-.. , _________\\\ 3 .. 2 .. '· ·. . . . . . . . .'ｊｾＮ＠ _ _ _ _ ,,/ //_,/_/_, •• -··'' 1 ·· ..• ····..,_____\\\ ,, .. "' ｾＱＲ＠ ｾ＠ \_\ ｾ＠ ·1.1 ｜ ｾ Ｍ 1/2 . ﾷＭｾ＠ ···- .., _________ ////_,.-·' ｾ＠ 1 "' 2 "' 3; "' 2 "' 1Ｏ ｾ Ｏ＠ ·····.......... ___ ｣ＮｾＭﾷ＠ ///_, ..---··' ' 'f•' '··-...., _________\ \ \ Only Ｍ ',/\1\1i\l'\/ \,' 1 .. 2 .. 3 ｾ＠ Ｎ Ｚｾ ＭＯ ····-........ ___ ＮﾢｾＭﾷ＠ 1 1 ""' 2 .,iz 2 ""' 1 1/2 •, ···-.., ________\\\ ···-.......... ///// ｾＭ＠ 1/2 ﾷＭｾ＠ ,, .. "' ,\._\ 1! 1/2 ｾ Ｚ＠ ! 49 \ 4'!. 1/ : _./ 13 64/ ［ｾ＠ .64 ｾ Ｍ 1 .. 2 .. 3 " t .. 3 .. 2 ..__ :•••Ｍ ｾ Ｍ ' ',' '•'•' ' . . . . .Jｾ＠ """"',. / \.17 81 ｜Ｎﾷ ｾ ＭＮ Ｚ＠ . .. 2 .. . - • ··-----..... _______｣｟ｾＭﾷ＠ Tetryonics 93.04 - Standing-wave Matter mass-ENERGIES contained in the tetryonic fascia of standing-wave topologies contribute to weight 107 Copyright ABRAHAM [2008] - All rights reserved Periodic element nuclei v ........ -.. .......... ｾＭ ···········... v·.. ···... Photons .. / , y3 ···· ......... ＮＨＺｾＭﾷ＠ The mass-ENERGY content of 4n7t Tetrahedra] standing-wave topo1ogies forms the basis of a11 periodic e1ements ....... ----:.. -- .......... ﾷＭｾ＠ ﾷＭｾ＠ -- .......... ,.·········· ····· ............. ＮｃｾＭﾷ＠ ····-............ Ｎ｣ｾＭﾷ＠ ·····-....... ___ Ｎ｣ｾＭﾷ＠ -........... . . ······· .... ____ ､ｾＭﾷ＠ ...... ----:---- ...... ,.··········· 0 ALL ﾷＭｾ＠ periodic elements are comprised of n level deuterium nuclei ﾷＭｾ＠ ·····.......... ___ ｾＭﾷ＠ Tetryonics 93.05 - Periodic element nuclei 108 Copyright ABRAHAM [2008] - All rights reserved n 1 -Charged Tetryon Templates Tetryonics 94.01 - n1 [Charged] 109 Copyright ABRAHAM [2008] - All rights reserved n 1 - Neutral Tetryon Templates Tetryonics 94.02 - n1 [Neutral] 110 Copyright ABRAHAM [2008] - All rights reserved n2- Charged Tetryon Templates Tetryonics 94.03 - n2 [Charged] 111 Copyright ABRAHAM [2008] - All rights reserved n2- Neutral Tetryon Templates Tetryonics 94.04 - n2 [Neutral] 112 Copyright ABRAHAM [2008] - All rights reserved n3 - Charg d Tetryon Templates Tetryonics 94.05 - n3 [Charged] 113 Copyright ABRAHAM [2008] - All rights reserved n3 - Neutra Tetryon em plates Tetryonics 94.06 - n3 [Neutral] 114 Copyright ABRAHAM [2008] - All rights reserved n4- Charged Tetryon Templates Tetryonics 94.07 - n4 [Charged] 115 Copyright ABRAHAM [2008] - All rights reserved n4- Neutral Tetryon Templates Tetryonics 94.08 - n4 [Neutral] 116 Copyright ABRAHAM [2008] - All rights reserved nS -Charged Tetryon Templates Tetryonics 94.09 - n5 [Charged] 117 Copyright ABRAHAM [2008] - All rights reserved nS - Neutral Tetryon Templates Tetryonics 94.10 - n5 [Neutral] 118 Copyright ABRAHAM [2008] - All rights reserved n6- Charged Tetryon Templates Tetryonics 94.11 - n6 [Charged] 119 Copyright ABRAHAM [2008] - All rights reserved n6- Neutral Tetryon Templates Tetryonics 94.12 - n6 [Neutral] 120 Copyright ABRAHAM [2008] - All rights reserved n7- Charged Tetryon Templates Tetryonics 94.13 - n7 [Charged] 121 Copyright ABRAHAM [2008] - All rights reserved n7- Neutral Tetryon Templates Tetryonics 94.14 - n7 [Neutral] 122 Copyright ABRAHAM [2008] - All rights reserved n8- Charged Tetryon Templates Tetryonics 94.15 - n8 [Charged] 123 Copyright ABRAHAM [2008] - All rights reserved n8- Neutral Tetryon Templates Tetryonics 94.16 - n8 [Neutral] 124 Copyright ABRAHAM [2008] - All rights reserved Periodic element geometries 0 ＭＢｾ . . 42-42 An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on further and further from the nucleus. The shell letters K, L, M, ... are alphabetical Each shell can contain only an integer number of whole deuterium nuclei [Proton, Neutron & electron] Each shell consists of one or more subshells, and each subshell consists of one or more atomic orbitals. Q 8 7 RE ----------------------------------------------------· r :v-.:. :v·J·---------------- Vl Q) ..c V') c 0 ｾ＠ +-' u Q) UJ ·----· t"' ·--------------------------------------------------- ----------------------------------------------------·------------------- -----------------"- ---------------------------------------------------- KE !Mo ........................................................................................................... .................................... VALENCE SHELLS The electrons in the partially filled outermost [or highest energy] shell determine the chemical properties of the atom .................................... ........................................................................................................ f d p s +3 +2 +1 0 p -1 d f -2 -3 Sub-orbitals Tetryonics 95.01 - Periodic element geometries 6 5 4 3 Vl ｾ＠ Q) ..c E :::J z E :::J +-' c ro :::J a ro c.. c u 2 1 Each atomic shell equates to a specific Energy level for the dueterium nuclei that comprise it, in turn affecting the angular momentum of electrons in that shell 125 Copyright ABRAHAM [2008] - All rights reserved Periodic Harmonic motions Much of the math in of modern physics is predicated on the assumption that rc [where it appears] is related to the properties of a circle x =A cos (wt + <p) Circular motion F= -kx Linear motion Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis Principal Quantum Numbers circular harmonic motion ' ' 'I 'I I : Ｚ ｵ ｾ Ｚ Ｚ［ Ｚ ﾷＭ ＭＫ Ｍ c © -- ｉ ｾ］［ｊｆ u ｾＭ -s; g Q) Qj I - -----::Itt¢:1: :::_:: ｾ＠ ｾ＠ ------+--- Vl Qj Circular motions describe the motion of a body with a changing velocity vector [the result of an acceleration force]. ' 'I --+------ 6 ｾ ﾷ ＭＺ ｾ ｉｆＭＫｾ＠ ｾ ｾ ｾ ｆ］Ｍｾｉ ｾ ｾ ｾ ］［ｊｦｙＹ＠ Ｍ ｾ ｾ ltMtl ------- [b ------- -------+--- [K\ ------- Ｍ simple harmonic motion Mo : ｾＭ ＧＢＭｦ Ｌ ｾ ｾＭ Ｚｾ ｾ Ｌ＠ 5 ---4 1------ - 3 ｾ＠ -- ---+------+------ 2 :KE: ---+-------+------+------- 1 ' ' ' ' ' ' Sub-orbitals Tetryonics 95.02 - Periodic Harmonic motions Nuclei per shell in elements follows a'periodic summation rule' that is reflective of photonic energies 126 Copyright ABRAHAM [2008] - All rights reserved Baryon . rnass-energ1es E The quantum level mass-energies of Baryons determines the kinetic energies of electrons Atomic nuclei mass-energies electron KEM field Each element's weight [mass- Matter in a gravitational field] is the result of the total quanta comprising that element 1 n -1 e19v M /. . . . . - - - - -· · - - -. . . . . . \. \ ---V7----------1-5028o4921-e15----- NO \ p+ ｜ ｾ Ｍｾ ＭＮ＠ e.../V:_/ ﾷＭｾ＠ The nuclei forming each atomic shell have specific mass-energy quanta -----56,-448-----------n28--- ｾ＠ ｾ＠ --V4---------8.17-2424232-e14----- 8 Baryon rest masses ITU [ ------52,-4 88-----------n - -7-- V3----------4-596988631 e14 --- lepton rest mass KEM [72 (n) ]+[12e19 ]+[mev J] l 2 2 Deuterium mass-energy per shell ｾ＠ ｾ＠ Despite having differing mass-energies each Deuterium nuclei has the same velocity invariant Matter geometry [84n] spin orbital coupling in synchronous quantum convertors Electrons act as quantum scale rotating armatures in atomic nuclei and can only have specific energies reflective of the electron orbital energy level of the Baryons in which they are found 1.2e20 Compton frequencies 930MeV electron rest Matter Is velocity Invariant 496 keV spectral frequencies 13.6 eV They acheive these energy levels by absorbing or emitting photons to acheive the specific angular momentum required Tetryonics 95.03 - Quantum level nuclei masses 127 Copyright ABRAHAM [2008] - All rights reserved KEM fields Baryons 930.947 MeV electrons 13.525 ev + + 496.519 keV Mapping Planck mass-energy contributions to elementary Matter and isotopes general form quatratic equation E nv polar aujbau Bohr's atomic orbitals polar energy spirals courtesy of Rene Cormier ldentifying electron rest Matter topologies as velocity invariant we can re-arrange the component Planck mass-energy geometry formulation of periodic elements to + v Spectral lines + reveal a quadratic formulation for all Z numbers Tetryonics 95.04 - Quadratic mass-energies 128 Copyright ABRAHAM [2008] - All rights reserved z nuclei per shell 120 element number [K\. ._ . 1 Periodic Summation Periodic summation follows the atomic shell electron config 2 -----------o 3 4 4 3 ·------------------- v-v 8 18 2 STEP TWO Periodic summation is a notation developed forTetryonic theory to model the geometric series addtion of Z[n 2] energy level Deuterium nuclei that form the periodic elements STEP ONE Periodic elements build up following the aufbau sequence 8 ｾ ｒ ］＠ 2 2 nuclei [74.496 ea] 7 ｾ ｑ ］＠ 8 8 nuclei [69,780 ea] 6 ｾ ｰ＠ = 18 18 nuclei [65,232 ea] = 32 32 nuclei [60,852 ea] = 32 32 nuclei [56,640 ea] = 18 18 nuclei [52,596ea] 8 8 nuclei [48,720 ea] 2 2 nuclei [45,012 ea] 120 Unbinilium + 118 Ununoctium + 110 Darmstadtium + ＲＨｸｾ＠ 32 ｾ 32 ｾ 92 Uranium 60 Neodymuim 28 Argon 10 Neon 2 Helium 0 Hydrogen + ｎ＠ + 18 2 Ｐ＠ 3 ｾ 2 ｾ ｍ＠ + Each atomic shell can hold only a fixed number of deuterium nuclei 8 Ｍ 2 ................ ........................ , .................... Ｎ Ｍ ｾ Ｍ 'Mo ' lf @] +3 +2 ｾ Ｚ ＧｔＢｴ ｫＢＧＭ Ｎｪ ｴＢＺＧｉＮ ｾ＠ + ﾷｉｊ ｾ ＭＺＱ Ｍ ｨ ｾ ｪ Ｍ ＮＭ ｩ Ｍ iKE ! .. ................................ .................... , .............. . ｾ＠ -2 0 ｌ＠ + ｾ ｋ＠ -3 THe LHS of the notation determine the number of nuclei in each atomic shell, from the periodic mass-energy levels for atoms, and the RHS follows the aufbau building principle to determine the rest mass-Matter of any specific element Aufbau Each periodic element is made of Z [n2 energy] deuterium nuclei Protons Neutrons electrons [24-12] [18-18] [0-12] J n1-8 Tetryonics 95.05 - Periodic Summation Planck mass-energies form the surface integral of rest Matter topologies for each periodic element 129 Copyright ABRAHAM [2008] - All rights reserved Atomic Nuclei Numbers Proton - Neutron Curve The graph below is a plot of neutron number against proton number. It is used as rule to determine which nuclei are stable or unstable. All periodic elements have an EQUAL number of Protons, Neutrons & Electrons with their molar mass-Matter being determined by their quantum level mass-energies Plot of Baryon numbers based on excess Neutron model of periodic elements sta'b i'lity Iine ... ' N"' ........... ｵｮｳｴ｡ Ａ ｢ｾ･＠ ｾ＠ ｾ＠ ｾ ﾷ＠ .ｲｾ＠ t·: · • ｾ＠ ...:- nud icfes x: - - - - - - ｾ＠ +.;, .... ｾ＠ ｾ＠ ｾ＠ F ｾ＠ ｾ＠ 8 Deviation from Tetryonic plot is the result of the intrinsic mass-energies of each particle comprising the atomic nucleus 7 6 V') Plot of Baryonlc nudel numbers based on Tetryonlc topologies of periodic elements •" V') Q) 5 <IJ @ ..c V') > Q) ｾ＠ u E [f\D 4 0 O"'l !o.... Q) c Q) +-' ro 3 . ---------- ....------------- .,-----. .. .. . ... . . ｾ＠ . :M : a·1 ---------T-... . 0 40 50 60 {f @] +2 70 Ｆ ＧｕＢｴ Ｍ ｆ ｾ Ｍ ｾ ｉｯｦ ｾ Ｍ .. .... +3 .. ｈｏ [?) + Ｍ Ｍ ｾ＠ ｾ Ｍ ..., ..ｾ Ｍ ... g) ...ｬ ｋｅ .. @] ｾ Ｍﾷ＠ {f 0 -2 -3 ... .... 2 1 orbitals Proton Number [Z] Historically, Proton-electron numbers are viewed as being equivalent in neutral elementary matter with the excess molar mass measured being the result of 'excess or extra' Neutrons in the atom Tetryonic modelling of the charged mass-ENERGY-Matter topologies of elementary atoms and the nuclei that comprise them, reveals a DIRECT LINEAR relationship for the number of Protons-electrons-Neutrons in all periodic elements and nuclear isotopes Tetryonics 95.06 - Proton-Neutron curve 130 Copyright ABRAHAM [2008] - All rights reserved Planck mass-energy contributions to elementary Matter and isotopes electron z KEM Deuteron En= -13.52 eV The mass-energy content of Deuterium nuclei creates the molar mass of elements [not extra neutrons in excess of the elemental number] 291,166 + En= -10.35 eV En = -7.60eV 291,107MeV 285,065 262,158 22,903 MeV + Elemental mass-Matter [in MeV} 3,972 keV En= -82.8 eV 48,262 MeV 8,937 keV En = -136.8 eV + En = -5.28eV 59,580 keV 213,887 80,1 74MeV 15,888 keV En = -168.96 eV e En= -3.38eV + 74, 40MeV 133,697 ,888 keV = -108.16 eV + En = -1.90eV ＳＹＬﾫＱ＾ 58,940 ﾮｾ＠ 565.11 keV eV 8,937 k 2,171.8MeV En=- eV KEM field mass-energies + 26 En= -0.84 eV 19,840 [in eV} 16,111 MeV 3,972 keV En= -6.72eV 2,013.9MeV KE + 25 En = -0.2 1 eV 3,725 1,861.9MeV 496.519 keV [Ji) Quantum levels w?i. 2 + 8 18 32 32 18 8 + + + + + + 2 Bohr's atomic orbitals nuclei number per shell [Ji)l] Elementary nuclei are comprised of equal numbers of Protons, Neutrons & electrons with varying energy levels Baryons electrons 930.947 MeV + 496.519 keV KEM fields + 13.525 ev The mass-energy content of Matter topologies is velocity invariant The mass-energy content of Baryons determines the KEM field of electrons Tetryonics 95.07 - Planck mass-energies in Matter 131 Copyright ABRAHAM [2008] - All rights reserved All elements are comprised of n level Duetrium nuclei Baryons The atomic shell energy levels of Deuterium nuclei in elements z electrons KEM fields Z [[72n 2 ] + [12v 2 ] + [1.2e20]] 13.525 eV 1,861,949 MeV electron 496,519 keV KEM Deuteron ............................. Elemental mass-Matter [in MeV} .-.. -.. ｾ＠ ...... . .......... ｾ＠ En= -13.52 eV ------ Ｍ ＳＬｾＵＭｯ Ｚ ｍｾｶ ! __ . / 30 En= -7.60 eV V') Q) En= -5.28 eV ..c V') u E En= -3.38 eV 0 +-' ro En=-1.90eV En =-0.84eV 2,013.9 MeV 25 En =-0.21 eV 1,861.9MeV 496.519 keV Ｍ Ｎ ＭＯ Ｍ ＲＬｾＸ ｾ＠ Ｍ Ｍ ! ! [L ｬ Ｉ ＱＳ［ＸＰ ＮＬ ｾ Ｎ＠ [K\ Ｍ Ｑ Ｍ ＸＶｴ : ｾＧＬ＠ i . ｝＠ + [1.l2e20]] ﾢＹｴ＿ Ａ ｓＱＹ＼ ! 2 2 ] + [1 f*7v ] + [1.2e20]] ＧＱ［ＶｾＭ･ｶ Ｍ Ｍ z [[72f 28n 1+ [1ＮＳｾＭ･ｶ＠ f *4v 2 Ｍ ｍｾｖ ＭＺｯＮｳｩＴ･ｶ z i ｛＿Ｒｾｬｮ＠ ···· ... ｾ ＺＮ＠ : ｾ ｶ Ｍ ｜Ｎｾ Ｍ i ··· ... i Ｒ Z ｛ＷＲ ｩＭ ｾ ＲＵｮ ｝＠ + ｛ＱＮｾ･ＲＰ｝＠ -\ - i : Ｒ : ｯＮ｡ＡＱＭ･ｖ ＭＭ : : ｩＭ ｾ Ｍ ＴＹｾｩＧｴ ｝＠ + [12*1v 2 ] + [1.2e20]] ·········· ... ____ ....... __ .... Ｍ ｃ Ｎ ｾ ＭＧＢ＠ -----.-.. 1 .- _/,../ Ｌ ｳ Ｍ ｴＹ Ｍ ｫ ｾ ｶＭ Ｍ Ｏｾ Ｍ i _,-/ !../ : Ｍｫ ｾ ｖ Ｍ ,,, i ｾＨＲ＠ V') ｾ＠ + a. ｾＨＲ＠ + ｾﾮ＠ u :::J c + 80,174MeV 15,888 keV En= -168.96 eV + 74,740MeV 15,888 keV En = -1 08.16 eV + 39,092 MeV 8,937 keV En=-34.2eV ® + (2 + 16,111 MeV 3,972 keV En= -6.72 eV + 3,724MeV 993 keV En =-0.42 eV . is the sum of the mass-energies of all atomic nuclei and spectral lines that comprise its mass-Matter topology as measured in any spatial co-ordinate system per unit of time Tetryonics 95.08 - rest mass in Atomic Matter Q) ·cu The relativistic rest mass-energy-Matter of all periodic elements Determines the spectral line [KEM field energies] of electrons bound to them Q) ..c + ! + 48,262MeV 8,937 keV En= -316.8 eV + ' ' ' ｾ Ｍ ＴＹＶ 3,972 keV En= -82.8 eV ｾﾮ＠ \ ｾ ｶ Ｍ . 496,519 koV + 22,903MeV + \\ . Ｚｶ Ｍ 993 keV En= -27.05 eV ® --------- ｾ Ｍ ＴＹＶ ［ ｳｩＹＢｴ｣ 496,519 ｫｾ 6,101 MeV + ! : + l1 -re20]] 'l + [12*3v 'l + [l.k elOll : Ｍ ] ' ' ' ' ' ' Z [[72f 26n 2 ] + [1 ｾＪＲｶ : 2 (2 ｾ ｶ Ｍ --------- Z [[72*29n'] + [1h* v'] + [1.2e20]] ! ｾ Ｙ Ｍ ｾ ｍｾｖ ----- t-------------- Ｍ ｾ Ｚ V-----------------------5.2· eV----------·--------------- ｾ＠ -Ｍ ＴＹＶ Ｌ ｳＭｴＹｫ '-\.,2, 1·71.7 MeV Ｍ Ｍ Ｒ Z [[72*30n'] + [1 f*6v'l + 1.2e20ll IT\D Ｍ ｾ＠ . .-..---- ＲＬｾ Ｚ ＳＵ Ｍ ＮＶ＠ ｍｾ Ｚ ｖ＠ [}YX} Ｍ ･ｶ + [1 ｾＪＸｶ Ｍ i -i----- -2 ;S05 ;4 Ｍ ｍｾ＠ ] Ｍ Ｎｳｾ : .,, ﾷ Ｍ ＴＹＶＬＵＱＺ Z [[72*31 n Ｍ ｍ Ｍ ｊｶ : 2 ----- ﾷＭＱｴＮｾｳ･ｖ : Ｍ ｬ［Ｒ / (Q) Z [[72f 32n ｍｾｶ .:/ : 2,6811 MeV --------- t-- ------ Ｍ Ｑ Ｍ Ｓ ! @ ------- ＲｾＬＹ＠ En= -10.35 eV 2,5862.9 MeV Ｍ y [1.2e20]] e the rest mass-Matter of bound photo-electrons is velocity invariant 132 Copyright ABRAHAM [2008] - All rights reserved photo-e1ectronic energy momenta lonisation energies Higher energies v Sup .. KE RE .. Fractiona1 quantum differentia1s 12 48 4 9 108 KEM / / Spectral 16 192 36 432 300 RE :: 11v...__ＮＭｾ Ｎ＠ ＮＬ ｟ｾ a down 768 ..· ··. S 588 ｶＲ＠ M Mapping photo-electron transition energies to Tetryonic energy momenta geometries reveals many key facts about the ionisation energies of nuclei Lower Energies 13.6Z2 eV . n2 The differing fractional KEM field energy momenta of electrons that results from their transitions to specific energy nuclei in elements results in differing QAM quanta and produces spectral lines and fine line splitting Humph f Photo-electrons absorb/emit spectral energies ｾ ｰ＠ spectral lines are produced by accelerating electrons C2 E [kj! mole] Note: this is an Illustrative schema for modelling KEM field energies All KEM fields possess the same physical spatial geometry in radial-time defined spatial co-ordinate systems B C N 0 F Ne Na Mg .8.1 Si P S Cl 48 C4 I Ｚｾ Ｑ Ｑ ﾷ Ｑ Ｑ Ｑ Ｑ Ｑ Ｑ Ｑ ﾷ Ｑ ﾷ Ｑ ﾷ Ｑ Ｑ Ｑ Ｑ Ｑ Ｑ Ｎ Ｑ Ｑ＠ Be KEM 108 ｾ ｴ ｬ＠ H He Li Spectral line transitions 192 Quantum 300 differentials 432 1 ne E=eV= - 41TEo a 1¥ K Ca Tetryonics 95.09 - Ionisation energies 588 2 hv Mv' hv, R, ｾＮ＠ p, KE Planck, Rydberg, Lorentz, Newton, Leibniz uniting classical physics and relativity through equilateral geometry 768 133 Copyright ABRAHAM [2008] - All rights reserved Eletnent nutnbers Nuclei per shell in elements follow a 'periodic summation rule' that is reflective of photonic energies lonisation energies Principal Quantum Numbers 3 2 :::IR¢L:: _:_ｾ＠ 2 : :_:_::_::.:s :_::_::_ + 8 3 2 + ｾ＠ 18 Ｌｾ＠ + 2(x 2 ) Qj 32 ...c V') + c 32 + 18 --------+---- 6 ----"--------- ｾ＠ 5 --4 @--- e ｾＭ +-' u ｾ Q) Qj ｾ＠ + 8 ｾ ｾ ｾ ｾ ｩ ｾ Ｚ ｾ＠ 3 Ｚ Ｌ｜Ｎ "Ｍ ｾ jＺ ｾ＠ [K\ ____ : _ - ]:j(gj:, ::::::::: " ｾ＠ z Ｗｲ ! ｾ＠ + 2 ｲ ｾ @] ｾ＠ [p) [p) @] 21 y ｾ＠ Sub-orbitals Periodic tnass-E E GY-Matter ｾ＠ ｾ ｕｮＭ｡ｭ･､＠ Following periodic summation rules for shell filling n[1-8] quantum energy deuterium nuclei combine to form elementary Matter 120 Unbinilium 119 Ununennium 118 Ununoctium 87 Francium Humphries series 112 Copernicium 55 Caesium ｾ＠ 102 Nobelium 37 Rubidium 70 Ytterbuim 19 Potassium 30 Zinc 11 Sodium IK\ ｾ＠ 32 Baryon rest masses [ 25 [ lepton rest mass 72 (n) 2 ]+[12e19 ]+[m ev l 8 KEM 2 ] The measured weight of Matter in gravitational fields is the result of planar mass-energies in tetryonic standing-wave geometries 0 Paschen series Balmer series Lithium Helium Deuterium Hydrogen Brackett series Deuterium mass-energy per shell 10 Neon 3 series The periodicity of all the elements, along with their exact molar rest mass-energies and quantum wavefunctions can be described with Tetryonic geometries Tetryonics 95.10 - mass-ENERGY & Matter yman series ( Mv' = KEM =heR " J Photon emission /absorption 134 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 96.01 - Differentials 135 Copyright ABRAHAM [2008] - All rights reserved Fermat's method of Factoring also known as 'the difference of two squares' is used to facto rise large numbers x2 - y2 [x-y]. [x+y] All spectral lines transitions are an example of 3 4 11 36 13 49 Fermat's difference of two squares in action at the quantum level leading in turn to Ryberg's formula y2 5 9 ｾ＠ 7 16 ｾ＠ ｾ＠ [x+y] Fermat knew that every odd number could be written as the difference of two squares or as revealed geometrically through Tetryonic theory's equilateral geometry, every'SQUARE'number is the sequential sum of ODD numbers Tetryonics 96.02 - Fermat’s method of Factoring x2 15 64 136 Copyright ABRAHAM [2008] - All rights reserved Fractions 3/4 24/25 A A fraction is a number that shows how many equal parts there are 40/49 In quantum mechanics fractions appear in quantum steps as a result of the equilateral geometry of Planck energy momenta 4 v 80/81 1 0: = R .987654321 ( 1 1 ) 1 - 81 .987654321 Cos 60 Sin 60 .5 .866025403 1/2 • Tetryonic geometry exp1ains the fractiona1 mathematics of Rydberg's fonnu1a Tetryonics 96.03 - Fractions 137 Copyright ABRAHAM [2008] - All rights reserved Lyman spectral transitions 11hv n1 8 180 768 n1 7 13.525 eV - 756 7 quantum level jump heR KEM Planck Emission 15 756 Rydberg 63/64 - 768 15 48/49 588 + 576 156 n1 6 576 432 + 132 432 420 300 + n1 4 192 + 180 84 .9600 288 108 4 quantum level jump 192 .9722 24/25 5 quantum level jump 288 .9795 35/36 6 quantum level jump n1 5 .9843 15/16 .9375 '"'' m-1 IMEI§§Niedlle-lfl E 13.31395504 eV IIQJI ltl*'l' 1 BP¥10,68 *1''' 0,829.01 ""' v m-1 D iMeNe@IW$Hie-#fl eV E 13.2-4926138 u;;w ltlii!ti*' v m-1 10,600,475.55 iiiiifjMMMie-:pa E 13.14958523 lfl&#id eV IIQJI IIIII BPM!E'M m-1 v 10,467,2 12.43 g iiitpwpgMU-:fM E 12.98427616 eV ltl&tiiiiiHI •a v 8/9 ｾ｀＠ + IIIII 108 n1 2 60 4 48 1 quantum level jump 96 5 E 12.67995718 ｬｴｩｍＧｓｾ＠ Bll•llll v .9722 R( _!.__2_) 1 25 .96 eV 1 1 R( 1-16) up;; .9375 IIIII m-1 9,69 1,863.362 eV E 12.02247792 48 36 3 - R( _!.__..2._) 1 36 Absorption IIQJI R(+-+) .888 5 + 36 .9795 IIIII D F 1•5i**HMn-:w 3/4 36 R( _!._1 _ _2_) 49 m-1 10,22 1,5587.14 ''•ti!EE!fiiiMII-:M ｬｴＧｩｎＡｐｾｗ＠ 2 quantum level jump .984375 IIIII BW*i"P1•6 D R( +- ＶｾＩ＠ 180 7 3 quantum level jump 96 1 BW ''''i1. 5 v 10,732,98 D - .7500 BMM#§d v IIIII m-1 8,177,509.712 D MMHWHII-IFI eV E 10.14396575 ltJI!NBfiif§l up;; 3 Tetryonics 96.04 - Lyman transitions 12 ｾ ｍ ｶ Ｒ＠ R(+-+) .75 138 Copyright ABRAHAM [2008] - All rights reserved Balmer spectral transitions Lihv KEM 180 768 Rydberg Emission - 720 heR 3.381 eV Planck 15 [n2-8l 720 [n8-2l 60/64 - 768 .2343 45/49 588 + 156 [n2-7] 540 [n7-2] .2295 13 32/36 5 quantum level jump 432 + 132 432 11 [n2-6l 384 [n6-2l .2222 21/25 300 + 108 300 9 [n2-5J 252 192 2 quantum level jump 12/16 7 [n2-4l 1 quantum level jump 144 7 [n4-2l ｾ｀ﾮ＠ + 60 .1875 5/9 60 108 192 + 84 .2100 9 [n5-2J 3 quantum level jump 144 5 - [n2-3] + Absorption 60 eV 3.169989296 IIIII Ifill IIIWIMUd' m-1 v 2,5033,319.299 11111111111111111. . . dl E 3.105295637 eV ;;p;; ... m'"''"'*' .,, Ｖ ｾＩ＠ ｒ ＨＫＭ .2343 2,422,965.84 m-1 E 3.()0561948 eV mi&MME "''' 0 Ｓｾ＠ 2,289,702.7 19 E 2.840310409 m-1 Ill rlillilllillllli-IFI eV ;;p;; mws•" IIIEtWEM'F v 2,044,3 77.428 eV 2.535991437 mwgt;+w "''' ••.., v 1,51 4,353.65 E 1.878512175 - Tetryonics 96.05 - Balmer transitions 48 eV ;;p;; mP''"""" ｍ 1 1 R( 425) .21 R(_!__2__) 4 16 .1875 ;;up IIIIIIDJE:JIIIIII-IFI ｾ .2222 m-1 5 [n3-2] R( +- ＳｾＩ＠ IIIII m-1 IIIIIII·I·!BIIFBII-11 E .2295 ;;up llllifMiiiji v R(_!_ __!_) 4 49 ;;up v Ill ++ii#§&iMif-IPI 11 4 quantum level jump 252 E m1111111B1 15 6 quantum level jump 540 111111111&1 iliil m-1 v 2,555,471.1785 Ill ilifdllltifidli-lpj ｶ Ｒ＠ R(+-+) .138 139 Copyright ABRAHAM [2008] - All rights reserved Paschen spectra1transitions 11hv 1.502 eV - 660 768 180 5 quantum level jump 15 cR KEM Planck Rydberg Emission [n3-8] 660 [n8-3: - 588 156 -71 588 480 432 132 3 quantum level jump 16 [n -61 [r [n6- .0907 .8033 16/25 300 E 1.29147712 108 [n - 1 192 .0711 [n 7/16 192 + 16 84 84 [n3 4] 84 m-1 IJIIIHI§@Mn-:p ..., E ..•, eV 1.226783461 908,612.1902 E 1.127107305 rJIMQI@iMII mm1n-t t m-1 :g eV +- ＶｾＩ＠ .09548 R(.2..9 __2_) 49 .0907 .1..1 BIIIIIIJfi v R( eV ..... .........fi'·!. m-1 v 988,965.6492 R lllllllllllli-IA . . . . . . . . .!.!. R( +- ＳｾＩ＠ .0833 ..w1 [n4 3. 1 quantum level jump + Absorption Tetryonics 96.06 - Paschen transitions .0486 ,;,; m-1 v 775,349.069 g DibiS•ifin-:M E 0.961798233 ltJ4i$&ji.ii 2 quantum level jump 192 1,041,118.135 me•auw 25 300 v ｾ＠ + 192 , .... B€§1ii1"i# ltJB+if.i8' 27/36 432 324 .0954 40/49 + . 324 768 15 + 4 quantum level jump 55/64 ..., eV .. ••·t.•ut*i§ ,;,; 0,657479261 ..., m-1 v 530,023.7776 1111111181111111•-=• E mw.;.;•••• eV R(.2..-_l_) 9 25 .0711 ｒ ＨｴＭｾＩ＠ .0486 140 Copyright ABRAHAM [2008] - All rights reserved Brackett spectral transitions KEM 0.8451 eV Planck heR Rydberg 48/64 Emission - -------------------,' -- 768 576 768 33/49 588 396 588 .04209 156 3 quantum level jump 20/36 432 240 432 2 quantum level jump .0347 132 11 240 300 108 300 1 quantum level jump E 0.634001085 m ;p.;.ma.J BIM'tW1 v 108 nm m-1 .._ , • eV ,., R( -fG- Ｖ ｾＩ＠ .0468 108 9 -- + Absorption Tetryonics 96.07 - Brackett transitions 192 .0225 rlrrf 458,941.8716 m-1 1111'1 Uii1 E 0.569307096 eV lti&Hil--t .I WI 11'"" ' :a B1 Wi+§JI§i ,;,; v 378,588.4126 m-1 IIII'''I"'AAu•t.-=M E 0.469630433 ltJIWJWif911 9/25 25 1,956 585876 511,094.357 1111 iiiJ#I 15 180 4 quantum level jump .0468 1111 v Bl '"'W'"tt v 245,325.2914 ..., 030432052 ltJII+f#li -&- Ｔ ｾＩ＠ .04209 ( 1 1 ) R l6- 36 eV .0347 ,,,, ,., m-1 IIJIIIH.-tMNirfi-iM E R( eV R( 1 1 ) 16 .0225 25 141 Copyright ABRAHAM [2008] - All rights reserved Pfund spectral transitions KEM 768 3 qua;,tu 180 eve1 jumo 468 156 1 quantum level jump -zs 768 288 132 11 - 132 + 300 Absorption Tetryonics 96.08 - Pfund transitions Ill 1Wit+i11• v 265,769.0656 lll"*'tdhfin .0243 1iJ IIN:IIII:f'! 24/49 IIIIIIIBIIM: v 213,616.5802 .0195 ll/36 432 132 432 39/64 588 288 2 quantum level JUmp Rydberg Emission - 68 heR 0.541 eV Planck .0122 E 0329678886 111Nii!!U$§1$Hn E 0.264985227 liJIIIfilli§i lllllilll·i·i v 133,263.1212 111 ｬｩｊｉｂｾ＠ E •wn,••wwr. 0.165309071 IIIII m-1 #fj .... R( eV =• eV .... ;4) .0243 IIIII m-1 ｾＵＭ R( 1 ;5 - 49) .01959 IIIII m-1 =• eV UWI R( 1 25- ＳｾＩ＠ .0122 142 Copyright ABRAHAM [2008] - All rights reserved Humphreys spectral transitions KEM heR 0.375 eV Planck Rydberg IJIMjfitiiJH - 336 768 .> ｱ＼Ｎｾ､ＧＱｴｕｬ＠ Emission 180 Ｑｾ＠ 336 588 768 588 ｾ＠ 156 v 0 ｾ＠ 432 Absorption Tetryonics 96.09 - Humphreys transitions IIIII 132,505.9444 Ｑ ＮｾＵ＠ 0 ,. .00733 rmn m-1 .... E 0.099676156 ''"*'"M R ( eV IJII@I•JIIIi• IIIII v 80,353.459 m-1 0 iiM:P#if§ipiNIIMifj m 156 1) ＬＮ E 0.164369815 .01215 m ＧＢｒｾ＠ 13/49 lew l1t..mo 156 1 quantum level jump -zs 28/64 eV "*' 1 1 ) 36 - 64 .01215 R( ｾＶ＠ - ＴｾＩ＠ .0073 143 Copyright ABRAHAM [2008] - All rights reserved Un-named spectra] transition heR KEM 0.276 eV Emission Ｍ 768 1 quantum level jump ］ｾ Ｍ 768 15/64 .0047 ｾＭ Ｍ｣ ｾ Ｊ ｾ ］ Ｍ Ｑ＠ 588 Absorption Tetryonics 96.10 - Un-named transition IJIIQftijl · v 52,152.48541 ;;;,; m-1 IJIIIifij@i§NII-IN E 0.064693658 m ••s•www eV "*' 144 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 96.11 - Atomic spectral series transitions 145 Copyright ABRAHAM [2008] - All rights reserved ninths Sizteenths Fractionals and fractals eigteenths Twenty-sevenths White light {EM radiation] is comprised of many superpositioned frequencies Fractal antennas are tuned to specific wavelength frequencies to match the equilateral geometry of photons Twenty-sevenths Eighteenths Twenty-fourths Thirty-seconds Tetryonics 97.01 - Fractionals and fractals 146 Copyright ABRAHAM [2008] - All rights reserved rev deg rad 0 21ill 0 1 and Angles Polygons 135° 2:2:!5° 1800 sup1pl€! ｭｾｓｴ｡＠ ry .anglll!!ls. rev rev rnmpiL!ml@ntalfY 8illglll!!ls. 116 1!12 deg deg sum= a.ao.o rad rad Tetryonics 97.02 - Polygons and Angles sum= 90° ﾷ ｾｲｬＺ｡ｮｧ ｬ ｬＡｳ＠ 147 Copyright ABRAHAM [2008] - All rights reserved Pi vs Tau bosons T:/2 photons Tetryonics 97.03 - Pi vs Tau 148 Copyright ABRAHAM [2008] - All rights reserved Triangular dissection of an equilateral triangle is a way of dividing up a original triangle into smaller equilateral triangles, such that none of the smaller triangles overlap 19 20 11 11 lowest order perfect equilateral triangle dissected by equilateral triangles 19 9 2 lowest order perfect dissected equilateral triangle, an isomer of the first Tetryonics 97.04 - Triangular dissection of an equilateral triangle 149 Copyright ABRAHAM [2008] - All rights reserved Golden mean Spirals Golden Mean Spiral- This spiral is derived via the golden rectangle, a unique rectangle which has the golden ratio. This form is found everywhere in nature: the Nautilus Shell, the face of a Sunflower, fingerprints, our DNA, and the shape of the Milky Way 0, 1, 1, 2 3, 5 8 13, 21, 34, 55, 89 144 .. . Golden spiral Continued fractions and the Fibonacci sequence The convergence of the continued fractions l 1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 --------------1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ are: i.... ....... 2 1 - 3 2 - 5 3 Ｍｾ＠ 8 5 13 8 - ｾ＠ 21 13 34 21 ｾ＠ 55 ｾ＠ 34 89 55 ｾ＠ 144 89 233 144 377 233 610 377 The Golden Ratio (Golden Mean, Golden Section) is defined mathematically as: i{J == 1 + y'5 2 == 1.61803 39887 . .. . Tetryonics 97.05 - Fibonacci Spiral Golden spirals 150 Copyright ABRAHAM [2008] - All rights reserved Koch fractal Curve Niels Fabian Helge von Koch (January 25, 1870- March 11, 1924) was a Swedish mathematician who gave his name to one of the earliest fractal curves ever known He described the Koch curve, or Koch snowflakes as it popularly known, in a 1904 paper entitled "On a continuous curve without tangents constructible from elementary geometry" The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. Von Koch wrote several papers on number theory. One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem. Actually Koch described what is now known as the Koch curve, which is the same as the now popular snowflake, except it starts with a line segment instead of an equilateral triangle. Three Koch curves form the snowflake. ,a 1 2 - 2 v'i2 The Koch curve is a special case of the Cesaro curve where: which is in turn a special case of the de Rham curve. Tetryonics 97.06 - Koch fractal curve 151 Copyright ABRAHAM [2008] - All rights reserved Flower of Life Koch snowflake vesica piscis 13Sl 780 > . r;; · .· 3 >. VJ ·2fi.5, 153 · \ I \ I star tetrahedron ---space-time The Flower of Life is a name for a geometrical figure composed of multiple evenly-spaced, overlapping circles. Tetryonics 97.07 - Fractal Flower of Life 152 Copyright ABRAHAM [2008] - All rights reserved Unit circles - SlNE WAVES -Photons MaxweU's Equations as ;-+ xE;;:;; -at Classic model of a photon · B= P X H= ao + j ar V• B=O ir/.·· e electric fields 1 eir/6 eiT/ e (..· 3T'/ ). magnetic fields ｩｾ＠ ｾ ｬ ｩ＠ i Ｑ ＭｦｬＺｾｉｊＮＧＡ 0 Tetryonic model of a photon -.1 Magnetic waveforms are 90 degrees out of phase with Electric waveforms Tetryonics 98.01 - Unit circles - SINE WAVES - Photons 153 Copyright ABRAHAM [2008] - All rights reserved Boson distributions in monochromatic EM waves 1/64 ｾ＠ a- 2/64 3/64 4/64 5/64 6/64 ｾ＠ 0\ ::j 0\ ｾ＠ U1 4/64 3/64 2/64 1/64 .. ｾ＠ w r.Jl r-t" "'1 ｾ＠ -·c... w . Ul ｾ＠ Ul Ｍｾ 0 Ci' c: G\ c... rD r-t" rD nr-t" ... -..... ...... ｾ＠ ｾ＠ OD 00 ::j -..... QJ ,..., n ...... w ::j rD Ci' .j::o rD "'1 aq Ul w ｾ＠ "'< 3 ,..., 0 3 ... rD ｾ＠ QJ < rD ::j r-t" "'1 c... :r r-t" rD ｾ＠ "'1 rD ::j n rD -cQJ r-t" r-t" rD --f-) c... 3 N 0 '-1 7\ ｾ＠ "'1 ::j co ｾ＠ 3 ,..., '-1 ::::t') ,..., w U1 c: rr:l 0 .j::o c... s· ... w w ::j ｾＭ 0 5/64 N -c :L a- 6/64 ::j I -· 7/64 rD ::j r.Jl r-t" 8/64 ｾ＠ 0 r.Jl 0 ::r' 0 r-t" 0 7/64 G\ Ul w . N w ｾ＠ QJ N .. < N "'1 ::j -c"'1 0 c... c: n rD c... a"'< ..c c: QJ ::j r-t" c: 3 -·c... r.Jl r-t" "'1 ｾ＠ c: ｾＭ 0 ::j .015 .031 .046 .062 .078 .093 .109 .125 .109 .093 .078 .062 .046 .031 .015 Probability distributions of monochromatic EM waves Tetryonics 98.02 - Boson distributions in monochromatic EM waves r.Jl 154 Copyright ABRAHAM [2008] - All rights reserved ...... Wave interference patterns Matter particles are stopped by theＧ ｢ｾｲｩ･＠ but the [K]EM wave passes through both slits and is diffracted by them producing ｷｾ｡ｫ･ｲ＠ EM waves that then superposition with each other to produ'ce interference patterns The double-slit experiment, sometimes called Young's experiment, is a demonstration that matter and energy can display characteristics of both waves and particles, and demonstrates the fundamentally probabilistic nature of quantum mechanical phenomena and Establishes the quantum interference principle known as wave-particle duality. ' ' Equilateral ｳｵｰ･ｲｯｩｴｮｾ､＠ KEM waves produce constructive and ､･ｳｴｲｵ｣Ｇｩｾ＠ interference waves that have historically been interpreted as being circular wqvefronts It is not the Particle passing through both slits that produces an interference pattern it is the particle's associated K{EM}wave '-;.:n ｾ＠ e.. ｾＭ ....a ('1:) \ ｾ＠ ('1:) ｾ＠ ｾ＠ ("") 1\J ('1:) ｾ＠ @ ('1:) """'( ｾ＠ w ｾ＠ d ｾ＠ :;:::: ("") ('1:) ｾ＠ ｾ＠ ｾ＠ ｾ＠ ('1:) ｾ＠ ｾ＠ ('1:) ｾ＠ ｾ＠ .. 0 ｾ＠ ｾ＠ ;::s-' ｾ＠ en ('1:) """'( ('1:) ｾ＠ c;· ｾ＠ ｾ＠ en ;::s-' --a· en ｾ＠ ｾ＠ s('1:) ｾ＠ 0 w en ｾ＠ :;:::: ('1:) "";'C ｾ＠ N The Compton frequency of any [K}EM wave is comprised of identical wavelength Photons which can combine to produce interference patterns - In the basic version of the experiment, a coherent light source such as a laser beam illuminates a thin plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen- a result that would not be expected if light consisted strictly of particles. However, on the screen, the light is always found to be absorbed as though it were composed of discrete particles or photons. / /...- ｾ＠ c;· 1\J .... ｾ＠ ('1:) I ...-/ / 0 Any detctor placed ｡ｦｴｾ＠ the primary screen will remove energy from phe secondary KEM fields and affect the ｩｮｴ･ｲｦｾ｣＠ patterns produced If one slit is observed for the passage 0f the electron in order to determine its physical state then the its KEM fieldwave will be absorbed by the detector resulting in only one wave ｲ･ｭ｡ｩｽｾｧＬ＠ enforcing a classical particle outcome Tetryonics 98.03 - EM wave interference / ｾ＠ ｾ＠ ....a ｾ＠ ｾ＠ < d ｾ＠ r;j 155 Copyright ABRAHAM [2008] - All rights reserved Quantum computing via EM wave super-positioning BY superpositioning two beams of EM radiation the resultant 'colours' will perform quantum level computations that can be read via the resultant interference patterns produced constructive interference additive in-phase EM waves ln phase Additive ·· . ............................................... Subtractive Out of ohase Various basic operations, such as ADDITION, SUBTRACTION and SQUARE ROOTS etc are all easily computed using EM wave super-positioning destructive interference subtractive out-of-phase EM waves The lines of force Tetryonics 98.04 - Quantum computing via EM wave superpositioning 156 Copyright ABRAHAM [2008] - All rights reserved q By utilising the statlscal distribution ｾ＠ ｾｵｬ｡ｴ･ｲ＠ ｐｊｾｯｫ＠ energy momenta quanta In EM waves Tetryonlc theory provl9esa-pfcictieal geometric solution ﾷ ｴｯ ﾷ ｱｵ｡ｮｴｾｭ＠ computing problems p v.s. NP --- The set of all decision problems for which _a-n·alg;rithm exists which can be carried out by a ､･ｾｲｭｦｮｴｩ｣＠ Turing machine in ｰｯｬｹｮｾｴｩ｡Ｎ＠ time An bosons ｡ｬ ｟ ｧＶｲｩｾ ｨｭ＠ .... the set ｯＯｾＱ ﾷ ､･｣｟ｩｳｯｮ＠ which can be ｣｡ｲｩｾＨＭｧｊＡ＠ problems for which an algorithm exists by a non-deterministic Turing machine in'po.lynomial time oftime complexity O(n) is one which increases in time linearly as the "size [n-n] 2n+l ｯｦｴｨ･ Ｍ ｾｴｑ ｾｉ･ｭＢ＠ E photons 2n-l 0.0156 0.0312 0.0468 0.0625 0.0781 0.093 ...0 =: ｾ＠ ｾ＠ ｾ＠ 0.109 0.125 ｾ＠ '"0 ""'I a 0.109 0.093 0"' ｾ＠ ｾ＠ ｾ＠ Cl'l 0.0781 + 0.0625 ODD + ｾＨＯＰＶ＠ 0.0468 0.0312 t?.rs 0.0156 An algorithm of corilP,Iexity 0(2n) utlises exponential quanta; increasing n by ｨｾＭｬ･＠ the quanta required 2n2 n An algorithm of complexity O(n 2) meaning that if you double n 1t.•ｾＺＭＮｦｏｵｲ＠ ｵｴｩｬ＿ｾ＠ quadratic quanta, times as many quanta. 0 Unlike Math treatise on P vs Np ｴｨｾＧﾷｲＡＮＺ＠ hv exponential polynomial time O(n k) Tetryon1eometry of EM ｦｩＭｾﾷｹｴｈ･＠ ················....... 1 By relating p to the number of equilateral wave fields secona . . . . .· · · · · · · exponential energies per second In·aiiyEM wave-very-large- [Pii2 Jdata sets can be modelled and processed every second Tetryonics 98.05 - P = NP hf 157 Copyright ABRAHAM [2008] - All rights reserved eg: quantum encryption 1024 Quantum Cryptography lf the numerical sequences were applied to amplitude modulation [38 6o oz 64 72 go 5010 47 05 94 o8 48 52 27 26 51 66 01 91 03 zo] 1 their non-repeating numerical sequences would appear to be purely random noise Using Tetryonic geometry advanced non-repeating cyphers of any complexity can be easily developed The level of encryption can They can be based on known letter sources be easily increased (without limit) (books, pages of magazines etc) by increasing the dimensions and can be further encrypted of the cypher geometry through rotations, double encrypting or pictogram substitutions eg: num her of quanta 20 vs 3D geometries Tetryonics 98.06 - Quantum Cryptography 158 Copyright ABRAHAM [2008] - All rights reserved Quantu111 Co111puting The Proton/Neutron geometries of atomic nuclei can be built at the quantum scale to create an atomic nuclei that can operate as a Opto-memory-transistive computing element, many elements can then be combined in lattices to create super computers no larger than bacterium Spin UP Energy can be gated through individual nuclei using the centre Baryon as the base transistor element, in turn effecting the energies of bound photo-electrons Spin DOWN SQUARE ROOTS p N ODDS N p PROBABlLlTlES Q-bits 1 photo-electronic transitions can be used to directly recieve or emit memory states through the absorption and emission of spectral photons of specific energy momenta Tetryonics 98.07 - Organic Quantum computing 0 159 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.01 - mass-Energy geometry 160 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.02 - mass-ENERGY-Matter 161 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.03 - Planck's Constant 162 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.04 - Charged Planck mass-ENERGY-Matter 163 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.05 - The 3 Laws of Tetryonics 164 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.06 - Tetryonic Unified field equation 165 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.07 - Tetryonic theory 166 Copyright ABRAHAM [2008] - All rights reserved Tetryonics 99.08 - Geometrics