1!! DON'T PANIC !! -Adams (HHGG) 2The spectacular physicalapplication of Clifford algebrais the Dirac equation. -- Finkelstein 3It is only the very wisestand the very stupidestwho cannot change. -- Confucius 4To have faults and to bemaking no effort to correctthem is to have faults indeed. -- Confucius 5The demands that good people make are upon themselves;those that bad people make areupon others. -- Confucius 6Never do to others what youwould not like them to do to you. __ Confucius 7One who will not worry about what is far off will soon findsomething worse than worryclose at hand. -- Confucius 8Good people are ashamed to lettheir words outrun their deeds. -- Confucius 9If I hold up one corner and a student cannot come back tome with the other three, I do not continue the lesson. -- Confucius 10Be slow in word but prompt indeed. -- Confucius 11Good people take as muchtrouble to discover what isright as bad people take todiscover what will pay. -- Confucius 12One who learns but does notthink, is lost.One who thinks but does notlearn is in great danger. -- Confucius 13Clever talk and a pretentiousmanner are seldom found ingood people. -- Confucius 14What satisfies me the most?Basically, that I've never worked for a living. -- Marcel Duchamp 15Even if I could try a civil suitas well as anyone, it would bebetter to bring it about thatthere were no civil suits. -- Confucius 16Int[V8] ( ( d^2~H8 /\ *d^2H8 + + F8 /\ *F8 + + ~S8± (gamma)d S8± ) + + gg + n'(Dg/DA)dn) 17You can make a model of anything from the harmonicfunctions on the unit disk. -- Feller 18It is more important to havebeauty in one's equationsthan to have them fitexperiment. -- Dirac 19The whole purpose of physics is to find a number, withdecimal points, etc! Otherwiseyou haven't done anything. -- Feynman 20Another peculiarity of 4-dimensional space is the occurrence of the 24-cell,which stands quite alone. -- Coxeter 21If you calculate a result that disagrees with experiment,you need courage to say the experiment must be wrong. -- Feynman 22Good people who hold the leftside of a contractdo not take unfair advantage. -- Lao Tzu 23To lead with cleverness bringsharm.To lead without clevernessbrings benefit. -- Lao Tzu 24Why is there somethinginstead of nothing?< > enables us to build theworld from 0. -- Finkelstein & Rodriguez 25Maybe other worlds have folks with more advanced brains.Maybe not. Either way, it's amighty sobering thought. - Kelly (Porkypine and Pogo) 26I believe that the theory that space is continuous is wrong,because we get infinities andother difficulties. -- Feynman 27Whenever a creature was facedwith several possible coursesof action, it took them all,thereby creating many distincthistories of the cosmos. - O.S. 28Charge is just the ability tomake virtual photons, andthe electric field is nothingbut the virtual photon cloud. -- Gonick and Huffman 29I believe nobody will read thisbook for 100 years, but I don't care. -- Kepler,about 100 years before Newton 30Kepler was a better scientistthan Galileo. Galileo did experiments that others mighthave done. Kepler did thingsother people would not do.-E.T. 31I am One that transforms into Two. I am Two that transforms into Four. I am Four that transforms into Eight. After this I am One again. -- Hermopolitan creation myth (quoted by Schneider) 32There are two camps in the current world. In one camp the common good is sought for the masses, while the other camp specializes in keeping most of the benefits for the people in the ruling class. -- Hua Loo-Keng33A mathematician has to be judged by his research accomplishments and not by the number of university degrees earned. -- Wang Yuan in his book "Hua Loo-Keng" 34The death of Zhou Enlai in 1976 was marked by Comet West. The death of Deng Xiaoping in 1997 was marked by Comet Hale-Bopp. 35If some years were added to my life, I would give fifty to the study of I Ching, and then I might come to be without great faults. -- Confucius36A thorough advocate in a justcause, a penetrating mathematician facing the starry heavens, both alike bearthe semblance of divinity. - G. 37Wagner was a wonderfulsunset that had been mistakenfor a dawn. -- Debussy 38Life computers can reproduce! -- Conway and Guy 39"Beauty is truth, truth beauty",-- that is allYe know on earth, and all yeneed to know. -- Keats 40I believe that all things arefundamentally interconnected,but some things are a greatdeal more interconnected than others. - Adams (Gently) 41Music has no meaning or purpose other than to be itself. -- Adams (MacDuff) 4242: The Tao produced the One.The One produced the Two.The Two produced the Three.The Three produced All things. -- Lao Tzu 1=unity,2=duality,3=the rest. -- Marcel Duchamp 43The laws of physics will turn out to be simple, like the checkerboard with all itsapparent complexities. -- Feynman 441 8 28 56 70 56 28 8 1 S8 Spin(8)____________________ 256 = 16 x 16 = 4 x 8x82x(gamma)8 OP2 45 F4-AutGroup a O+ Ov triality: ~O+ b O- Spin(8):O+ -> xO+ ~Ov ~O- c Spin(8):Ov -> xOvx Spin(8):O- -> O-x 46Quantization of the conformaldegree of freedom leads to a consistent and reasonablycomplete picture of maximallysymmetric spacetime.- N.and P. 47<1,i,j,k,e,ie,je,ke>---><1,i,j,k> O: _ OxO: /| or |\ OxOxO: | | 48War is hell. -- Sherman 49If nominated I will not run.If elected I will not serve. -- Sherman 50 F4 = Aut(H3(O)) __________________ H3(O) <-> E7/E6xU(1) Rdim27 Cdim27 (CxO)P2s in (HxO)P2 51 Spin(9) -> F4 -> OP2 __________________ OP2 <-> E6/Spin(10)xU(1)Rdim16 Cdim16 (CxO)P2 52 Spin(8) -> Spin(9) -> S8 S8 <-> Spin(10)/Spin(8)xU(1)Rdim8 Cdim8 RP1s in RP9 ______Silovbdy=RP1xS7____ 53 F4 = Aut(H3(O)) _________________ E6 = F4 + H3(O)0xS1 E7 = F4 + H3(O)0xS3 + SU(2)E8 = F4 + H3(O)0xS7 + G2 54 OP2 = F4/Spin(9)(CxO)P2 = E6/Spin(10)xU(1)(HxO)P2 = E7/Spin(12)xSU(2)(OxO)P2 = E8/Spin(16)__________________ 55O__OxO__OxOxO nu 1 e e; or 1 rd,bd,gd ie,je,ke; or e or 1ru,bu,gu i,j,k 56 SU(2) Weak Force S2&endash;S22[vol(S2)vol(Q)/(vol(D)^(1/2))] D = Spin(5)/SU(2)xU(1) Q = RP1xS2 57 SU(3) Color Force CP2vol(CP2)vol(Q)/vol(D)^(1/4)) D = B6 = SU(4)/S(U(3)xU(1)) Q = S5 58 Spin(5) Gravity S4vol(S4)vol(Q)/(vol(D)^(1/4)) D = Spin(7)/Spin(5)xU(1) Q = RP1xS4 59 U(1) Electromagnetism S1xS1xS1xS1 = T4 4vol(S1) aE = 1/137.03608 60mW+ + mW- + mW0 = 260 GeV = [aW/aE][4+mf1] = mHiggsmW+ + mW-=[Sqrt(2v(S2))/Sqrt(3(S1))]mZ = mW± + (mW0-mW±)/4rt(6)sin2(theta-w) = 1 - (mW±/mZ)^2 61mF = v(Q)xN(grv)xN(oct)xSym mt = (483/21)mb = 130 GeV 62Kobayashi-Maskawa: e = pi / 2sina = +mf1/Sqrt(+m^2f1++m^2f2)sinb = +mf1/Sqrt(+m^2f1++m^2f3)sing = +mf2/Sqrt(+m^2f2++m^2f3) - - Sqrt(+mf2/+mf1) 63Radiative:mnue =4mw^2L/g^2Mpl = 2x10^(-6)eVmnumu =[mmu/me] mnue = 4x10^(-4)eVmnutau =[mtau/me] mnue = 8x10^(-3)eV MSW consistent with KM 64G(m^2proton) = 6x10^(-39)GF(m^2proton)=1.02x10^(-5)aC = 0.6286 at 245 MeV; 0.169 at 5.3 GeV; 0.106 at 91 GeV 65H*(Spin(8);R)=/\(x3,x7,x11,x7') a/g0 = Bg0 = BO8G = O7Spin(8) H*(BRS-a/g0) = S(x0,x0',x4) 66Mozart is free to create thingsothers cannot imagine, becausehe is bound by principlesothers cannot see. -- Boden 67Far from being the antithesisof creativity, constraints onthinking are what makecreativity possible. -- Boden 68Now, for the first time, weperceive the future as afunction of our own actions. -- Bernal (1929) 69Do not be clever-Axelrod5) Rat 4) Pavlov3) Tit for Tat 2) Altruist 1) TURN CHEEK (Remo=>Stable) - Michael Gibbs - Oct 93 70L I z A R d 71sxs =1 + F + L in Cl(8)sxs =1 + x3 + C + L in Cl(7) x3(a,b,c)=<ab,c>=Re(axbxc) C=*x3 F(a,b,c,d)=<a,bxcxd>O: axbxc=(1/2)(a(~bc)-c(~ba)) 72Psychohistory deals withreactions of large humanconglomerates to social andeconomic stimuli. -- Asimov 73To study the laws of history,we must study the common,infinitesimally small elementsthat influence the masses. -- Tolstoy 74Always study law -Not to practice, butto understand the ground rules. -- Heinlein (Long) 75If it can't be expressed infigures, it is not science;it is opinion. -- Heinlein (Long) 76It's amazing how much"mature wisdom" resemblesbeing too tired. -- Heinlein (Long) 77In a mature society,"civil servant" is semanticallyequal to "civil master". -- Heinlein (Long) 78It is contradictory to "love nature" and deplore"artificialities" with which"people have despoiled nature". -- Heinlein (Long) 79Any government will work ifauthority and responsibilityare equal and coordinate. -- Heinlein (Long) 80What are the facts?You pilot always into anunknown future; facts are yoursingle clue. Get the facts! -- Heinlein (Long) 81Stupidity cannot be cured withmoney, or through education,or by legislation. -- Heinlein (Long) 82Courage is the complement of fear. One who is fearlesscannot be courageous,and is also a fool. -- Heinlein (Long) 83The two highest achievementsof the human mind are the twin concepts of "loyalty" and"duty". Without them, asociety is doomed. - Heinlein 84Anyone who cannot cope withmathematics is not fullyhuman. -- Heinlein (Long) 85Specialization is for insects. -- Heinlein (Long) 86Do not handicap your childrenby making their lives easy. -- Heinlein (Long) 87One person's "magic" is another person's engineering. -- Heinlein (Long) 88If you are one of the minoritywho can do creative work, never force an idea.Learn to wait. -- Heinlein (Long) 89A "critic" is one who createsnothing and thereby feels qualified to judge the work ofthe creative. -- Heinlein (Long) 90The difference between science and the fuzzy subjectsis that science requires reasoning, while the others require scholarship. - Heinlein 91Secrecy is the beginning oftyranny. --Heinlein (Long) 92Deductive logic is tautological;there is no way to get a newtruth out of it. -- Heinlein (Long) 93Natural laws have no pity. -- Heinlein (Long) 94Sin lies only in hurting otherpeople unecessarily. -- Heinlein (Long) 95Duty is not what others expectof you; it is what you owe to yourself to fulfill obligationsyou assume voluntarily. -- Heinlein (Long) 96Time is your total capital, and the minutes of your life are painfully few.So learn to say NO! -- Heinlein (Long) 97The human nervous system hasalready evolved a vastdistance beyond biomedical science. -- Thomas 98Although Western medicine hasfailed to accept sudden deathby stress, the physiology probably involves the CNS. -- Fishman, et. al. 99Four good deeds of Ogodei:defeated enemies; postalservice; irrigation systems;intelligence networks.-Secret History of the Mongols 100Four wrongs of Ogodei: wine;unprincipled lovers;harming Temujin's servant;fencing wild animals.-Secret History of the Mongols 101Probability(F) = FF*F - emitter, from pastF* - absorber, from futureTransactions in Many Worlds -- Cramer; Wolf 102Time flies like an arrow.Fruit flies like a banana. 103This sentence is false. 104UCAG x UCAG x UCAG4^3 = 64 > 20 amino acids For DNA U->TT<->A C<->G 105T=100GeV; R=10^14cm: Higgs.T=100MeV; R=10^17cm: hadrons.T=1MeV; R=10^19cm: nu decouple.T=10eV; R=10^24cm: mat=rad.T=0.3eV; R=10^24cm:photons decouple. 106 STAR FORMATION:Magnetic field of collapsing cloud discharged by stellar flares carrying angularmomentum to planet nodes. 107Any system with arithmetic has undecidable algorithmsvarying randomly as the digitsof the halting probability Omega. -- Chaitin 108The halting probability Omega is incompressible and random.In the infinite oracle limit,Omega is a computable sequence. -- Chaitin 109Biological evolution =parallel computation withmolecular components =the nearest thing to an oracle -- Chaitin 110Deep Thought's successor =Life on Earth -Adams (HHGG)Biological evolution =the nearest thing to an oracle - Chaitin 111Omega (halting probability) cannotbe computed finitely. -ChaitinThe Tao that can be named isnot the Tao of the Absolute. - Lao Tzu 112Infinite Improbability Drive -Finite Probability of being atAny Point of Any Universe. -- Adams (HHGG) 113 Begin with 0=origin<0>=ray to<0> <<0>>=ray to<<0>><<0>,<<0>>> = superpositionstransformed by Spin(2)=>S1,C......Spin(N).Triality=>Spin(8);C=>E6 114S4 k=1 => Spin(4)= =SU(2)xSU(2)=S3xS3S4 k=2 => Spin(8)S4/\S4 = S8S8 k=1 => Spin(8)=S7xG2xS7 115 DIAMOND Cubic SemiconductorIndRf: 2.407(red) 2.465(violet)ThermCond: 23.2W/cmK(298K) 116Indra's Net Cyberspace:Reflecting Jewel atEvery Intersection - Benedikt 117Cyberspace:Representation of All Data of Every Nodein the System.--Gibson (Neuromancer) 118BAMA: Boston-AtlantaMetropolitan Axis.Home of Cowboys: Lupus(Case),Dixie(Flatline,McCoy pauley)--Gibson (Neuromancer) 119Wintermute + Neuromancer& Kuang ICEbreaker & Zion dub= Cyberspace MATRIX with voice that was MUSIC--Gibson (Neuromancer) 120MATRIX:I talk to my own kind.There's OTHERS. --Gibson (Neuromancer) 121The OTHERS gave the MATRIXa SHAPE, an overall total form. --Gibson (Mona Lisa Overdrive) 122Photonic CyberspaceNet:Dimension determined bybandwidth and sphere-packingcodes based on Leech /\24, etc.- see Conway & Sloane 123PHOTONIC SYSTEMS:Security is a higher-protocolproblem, since photonics doesnot easily control access.-Fiber Optic Networks (Green) 124OPTICAL FIBRE - 1000 nm;ns tuning time; 10^(-15) ber;1 milliwatt (10^16 photon/s):10^4 nodes at 1 Gb/s-Fiber Optic Networks (Green) 125Laser Space Communication +Nuclear or Anti-Matter Probes |gamma-ray laser Space Comm Net 126Soliton bits: 10^(-11) sATOMIC: 1 eV - 1000 nmNUCLEAR gamma: 1 MeV - 10^(-10) cmgamma-ray data rate: 10^6 Gb/s/ch;10^8 Gb/s/channel with Solitons 127Human DNA: 10^10 triples of 4 kinds nucleotides,4x4x4=64 states code for 20 amino acids + 3 stops:log2(20)x10^10 bits = 10^2 GbNeurons>Axons>Microtubules>>Tubulin>electrons(+/-) 128Neural Network Learning in the Brain; the Immune System;the Evolutionary System:similar processes on time scales of 10^(-3); 10^5; 10^12 sec. -- Pagels, on Edelman 129Human Brain: 10^11 neurons,each with 10^4 2-state TubDimers:10^15 b = 10^6 Gb state of mind.Data rate-nanosec transitions: 10^15 Gb/s 130gamma-ray bursts: 30 sec duration;lots of millisec substructure;broad spectrum around 1 MeV;10^26 erg at 10^(-2) l-yr=10^3AU;2/day. Physics Today Feb 92 131(1 deg / 360 deg)^2 = 10^(-5)Fission 1 kg 235U = 10^21 ergAnnihilation 1 gm = 10^21 ergSolar at earth/day = 10^29 ergSolar Grav Lens focus: 550AU 132Pd Foil Cold Fusion Generator: 1 cm^3 - 3 kilowatts (0.2 hr) 1 m^3 - 3 gigawattsAntimatter Battery:(1 cm)^3-10^24 GeV-1 megaWyr 133Particle-AntiparticlePair Creation:Saddle Node Bifurcation(d^2)F = m^2 - F*F- see Guckenheimer & Holmes 134Connected SimpleQuivers of Arrows:A-D-E Dynkin Diagrams:Lagrangian Singularities.-Catastrophe Theory (Arnold) 135Coxeter-Dynkin Diagram Identities:A1=B1=C1B2=C2D3=A3D5=E5 136 In 1261 Yang Hui described the binomial triangle that was rediscovered in Europe by Pascal.
137
0 Cl(0) 1 R TaoCl(1) 1 1 C Tai JiCl(2) 1 2 1 Q Yin-Yang, I ChingCl(3) 1 3 3 1 O 3 Realms, Tai Hsuan ChingCl(4) 1 4 6 4 1 S 5 ElementsCl(5) 1 5 10 10 5 1 M(4,C)Cl(6) 1 6 15 20 15 6 1 M(8,R), I Ching Cl(7) 1 7 21 35 35 21 7 1 M(8,R)+M(8,R), S7 <x> G2Cl(8) 1 8 28 56 70 56 28 8 1 M(16,R) D4-D5-E6 Physics 2^8 = 16x16 = 8x8 + 8x8 + 256 128 64 32 16 8 4 2 1 + 8x8 + 8x8 256+1024+1792+1792+1120+448+112+16+1 = 3^8 Tai Hsuan Ching
where:
<x> = twisted (fibration) product, (x) =tensor product, Cl = CliffordAlgebra,
R = Real Numbers, C = Complex Numbers, Q =Quaternions, O = Octonions,S = Sedenions,M(n,K) = nxn matrix algebra over the field K.
Cl(8N) = Cl(8) (x)...(N times)...(x) Cl(8) Many-Worlds Quantum Theory Cellular Automata Wei Qi Basins of Attraction interact by Zhen-Shan-Ren
138SU(n) /\(1) ... /\(n-1)Spin(2n+1) /\(1) ... /\(n-1); DSpin(2n) /\(1) ... /\(n-2); D±Sp(n)(2n) /\(1) ... /\(n) - /\(n-2) 139G2 7 14[27 64 77 77 182...] F4 26 52=28+8+16 273 1274 E6 27 78=28+16+1+32+1 351 2925...E7 56 133 912 1539 8645... E8 248=120+128 3875... 140Jordan Algebra: (1/2)(AB+BA)F ->A-> FA ->B-> (FA)B =/= F(AB)F ->B-> FB ->A-> (FB)A =/= F(BA)F ->(AB+BA)/2-> F((AB+BA)/2) => Sum Over Histories 141 Iron when heated glows: red to yellow to blue.Classical Physics was wrong,Cooks and Blacksmiths knew. - Gottfreid and Weisskopf 142Lattice Spacetime Lightcones:2d:(±1±i)/Sqrt2 : c = 14d:(±1±i±j±k)/2 : c = Sqrt38d:(±1±i±j±k±e±ie±je±ke)/2Sqrt2: c = Sqrt7 143Beings woven from light would live "nowhere" and "nowhen". - ManinQuantum Sum Over Histories gives Universe of all possible "crystals" of worldlines. 144e ^ i pi = -1 145Music is the Mathematics of Sense.Mathematics is the Music of Reason. - Sylvester 146Kepler's Platonic Mysticism-Irrational equations of thelowest order - Least rational -Least affected by resonance. -- Thirring (re KAM) 147Human Brain Waves are 1 to 30 Hz, in tune with the Earth's Schumann Resonances. Dolphin Brains look like Double Brains. 148HilbertBilinearJordanQuadratic|a> |a><a|=Pa PaH|a> Hoa Pr(H)Pa=<HPaH><a|H|a> TrHoPa Tr<PaHPa>[H,H'] ? 4(Pr(H)oPr(H')-Pr(HoH')) 149HilbertBilinearJordanQuadratic<a|H|b> ? ?|<a|H|b>|^2 TrPe(Pa)Pr(H)Pš) PaoTr<HPbH> 150Conformal: E7 (complexify)J3(O) E6xU(1) J3(O)M12(O) Spin(10)xU(1) M12(O)J(G7) Spin(8)xU(1) J(G7)SpCnf Lor U(1) Trnsl 151I hear and I forget.I see and I remember.I do and I understand. 152n Cl(n) PI(n)(O) PI(n)(Sp) Clc(n) PI(n)(U)7=-1 R+R Z Z C+C Z 0 R Z/2 0 C 0 1 C Z/2 0 C+C Z 2 Q 0 0 C 0 3 Q+Q Z Z C+C Z 4 Q 0 Z/2 C 0 5 C 0 Z/2 C+C Z 6 R 0 0 C 0 153 32-d complex spinors:E6/Spin(10)xU(1)=(CxO)P2 16-d complex spacetime:Spin(10)/Spin(8)xU(1)=(CxO)P1 28-d Gauge group: Spin(8) 154 A-D-E: (An - n-polygon)Dn - n,2,2 - n-dihedraE6 - 3,3,2 - tetrahedronE7 - 4,3,2 - cubeE8 - 5,3,2 - dodecahedron 1553- dim Schlafli Double-Six: 27 = 6+6+15 lines on cubic72 sets of 6 disjoint lines 156 2- dim quartic: 28 bitangents 157 6-dim E6 root vectors:40 perm of (±2,±2,0,0,0;0) and 32 points (±1,±1,±1,±1,±1;±Sqrt3) (with odd no. of - signs) 158 6-dim E6 root lattice:27 nearest to (0,0,0,0,0,4Sqrt3): (0,0,0,0,0,0) +10 perm of (±2,0,0,0,0;2Sqrt3)+16(±1,±1,±1,±1,±1;Sqrt3) (odd-) 159 E6 Weyl Group27 cosets of (D5 = 2^4 x 5!)(27x16=9x6x8): 72 x 6! Automorphism group of 27 lines on cubic surface 16072 E6 root vectors in 6-dim can project in 4-dim to 24-cell 161 E6 cohomology:3 + 9 + 11 + 15 +17 + 23 E6 Casimirs:2 + 5 + 6 + 8 + 9 + 12 = = 42 = 6 + 36 = 6 + 72/2 162 Spin(10) cohomology:3 + 7 + 11 + 15 + 9 Spin(10) Casimirs:2 + 4 + 6 + 8 + 5 = = 25 = 5 + 20 = 5 + 40/2 163 Spin(8) cohomology:3 + 7 + 11 + 7 Spin(8) Casimirs:2 + 4 + 6 + 4 = = 16 = 4 + 12 = 4 + 24/2 164Witting Complex C4-Polytope: 240 vertices and cells 2160 edges and facesvan Oss C2-3<4>3- 24 vertices40 C3 hyperplanes-72 vertices 165Witting Complex Honeycomb: 240 nearest neighborsvan Oss apeirogon - 3<4>6 -- Eisenstein integers a + bw: w = (-1+iSqrt3)/2 166 F4-E6 Model: Weyl Group:Symmetric Anti-SymmetricCasimir Op. Lie GroupCohomology WickRotationNilpotent BRS Contraction 167 Spin(10)/Spin(8)xU(1) ==rank4 quadric Q8 < CP9 < C10 Smooth quadric Q'4 =Klein quadric Q*4 < CP5 < C6 -- Twistor complex spacetime 168 Quadric Q8 < CP9 < C0: Q8 = C + iR8 = = Siegel right half plane C = light cone in R8 = = Silov Boundary RP1 x S7 169Twistor Quadric Q*4 <CP5<C6 Q*4 = C + iR4 = = Siegel right half plane C = light cone in R4 = = Silov Boundary RP1 x S3 170<0> => 0+Z+ => Z => Q => R R => C => H => OC-matrix M(m,n): Vm -> Vn Quiver of Arrows: <a-R->b>a-R->b <=> Vm -M(m,n)-> Vn 171Simple Quivers: An Dn E6,E7,E8An: /\(1)n ->/\(2)n ->...->(/\(n-1))n Dn: /\(1)2n ->...->(/\(n-2))2n <- <-(2^(n-1), 2^(n-1)) 172 78| 27 -> 351 -> 2925 -> 351 -> 27 E6 = D5 + D5(spinor) + C E7 = D6 + D6(spinor) + H E8 = D8 + D8(half-spinor) 17328=8/\8 Gauge Bosons 8-d Spacetime 8+8 Fermions D4 D5 E6Spin(8) Spin(10) E6 174E6 -> F4 * | *--*--*--*--* *--*=<=*--* - J. McKay, P. Slodowy 175 D4 -> C3 -> G2 * | *--*--* *--*=<=* *=<--=* - J. McKay, P. Slodowy 176 D[n] -> C[n-1] * | * ... *--*--* * ... *--*=<=* - J. McKay, P. Slodowy 177 * ... *--* B[n] | A[2n-1] * -> * ... *=>=* | * ... *--* McKay, Slodowy 178Sets =>A-D-E => D4-D5-E6-E7 states+ states => QUANTUM => atoms++ atoms => RNA, DNA => cells+++ cells => people ++++ people => civilization 179 + - Feynman Sums++, +++, ++++ - KauffmanNK Boolean Network SumsK>4 => Chaos - Adaptation on Rugged Landscapes 180Feynman Sums with Mass:Intermediate Interference (- compare R. Griffiths) Kauffman NK Boolean Sums:"Classical" Approximations 1812^(n-1) Cl(2n) Half-Spinors are linear span of 1+(1/2)n(n-1) = 1 + Spin(n) Pure Spinors [isotropy group SU(n)] 18278(E6) = 45(Spin(10)) + 32 + 1 32 = Spin(12) Half-Spinors 16 = Spin(12) Pure Spinors = = 1 + Spin(6) = 1 + SU(4) [isotropy group SU(6)] 18316 = 8+8 Spin(8) Full Spinors 8 = Spin(8) Half-Spinors = 7 = Spin(8) Pure Spinors = = 1 + Spin(4) = 1 + S3 + S3[isotropy group SU(4)=Spin(6)] 184Birkeland Currents and Diocotron Instability =>=> Galactic Rotation CurveFlatness and Fine Structure - Peratt and Bostick 185You shall know the truth,and thetruth shall make you free. - Gospel of John 8:32 186Polaris: vRCHO1 Arcturus: qCiVega: s-O1 Altair: s+O1Ladle: HO1,HOi,HOj,HOk,O1e,Oie,Oje,OkeHsiu: 28 (4)I Ching: 8x8 = 64 Wei Chi: Qsum, (1+8+1+8+1)^2 187Unless you make the right hand like the left; above like below; and behind like before:you will not see the kingdom. - Acts of Peter 38 188Where the beginning is, there will the end be. - Gospel of Thomas 18 189The universe was createdby a fluctuation in the unbounded Apeiron causing separation into opposites. -- Anaximander 190Constructible Set Theory is consistent if ZF is consistent. -- Godel (see Devlin) 191In Constructible Set Theory,the Axiom of Choice and the Continuum Hypothesisare Decidable and True. -- Devlin 192 24-Cell:24 8 12 6 2 96 3 3 3 3 96 2 6 12 8 24 193 Witting Polytope:240 27 72 27 3 2160 8 8 8 8 2160 3 27 72 27 240 194You can't always get what you want,BUT if you try, you just might get what you need-Glimmer Twins 195Human minds, capable of premonitions, can leave the tyranny of matter to join OVERMIND. -Clarke (Childhood's End-'53) 196OVERMIND network is alive,pulses with light, and communicates bycolors and shapes.-Clarke (Childhood's End-'53) 197OVERMIND has its limits.Material, individual, Overlordsare its interpreters. -Clarke (Childhood's End-'53) 198The opinions expressed in this book are not those of the author. -Clarke (Childhood's End-'53) 199 Parallelizable Manifolds: An Dn E6,E7,E8 Bn Cn G2 F4 - -Torsion=Structure Constants;S7-Torsion varies-Associator. - Cederwall and Preitschopf 200 Parallelizable Spheres:S1 = U(1)-C-Commutative-T=0;S3 = Sp(1)-H-NonCom-T=const;S7-O-NonAssociative-varying T - Cederwall and Preitschopf 201Covariant Derivatives: Metric: D = d + G Parallelizing: ~D = d + ~GTorsion: T = ~D - D = ~G - G - Cederwall and Preitschopf 202 [ ~Da , ~Db ] = 2 Tabc(x) ~DcLie Group => Tabc(x) = Tabc(1) S7 => Tabc(x) - Tabc(1) = = Associator - Cederwall and Preitschopf 203 Octonion Alternativity => =>Aut(O) = G¤ = DerO ‰ d(a,b)x = = [a,b,x] + (1/3)[[a,b],x]So: S7 Associators <=> G2 - (see Sudbery) 204 S7 ~x Associator ~x Position == O0 ~x Aut(O) ~x O0 == S7 ~x G2 ~x S7 = = Spin(8) 205Simple Catastrophe Germs:1 variable: A2,A3,A4,A5,A62 variables: D4, D5, D6, E6 - Gilmore 206Catastrophe Germs: D4 = A^2B + B^3 D5 = A^2B + B^4 E6 = A^3 + B^4 - Gilmore 207 E6 Catastrophe Perturbation: 1 A B AB B^2 AB^2 -Gilmore
208E6 extended Coxeter-Dynkin: 1 2 3 2 1 2 1* - McKay, Sirag 209E6 balance number squares:1^2+2^2+3^2+2^2+1^2+2^2+1^2 = 2424 = order of <2,3,3> = BT == Binary Tetrahedral Group - McKay, Sirag 2107x7 Character Matrix of BT:Columns are eigenvectors of 7x7 E6 extended Cartan Matrix 6 non-zero eigenvalues - McKay 211McKay ADE Correspondence:D4 <-> BD2 = <2,2,2> (8)D5 <-> BD3 = <2,2,3> (12)E6 <-> BT = <2,3,3> (24) - McKay 212E6 ext Cartan: BT Character-Egnval: 0 2 3 3 4 1 1 2 -1 0 0 0 0 0 1 1 1 1 1 1 1-1 2 -1 0 0 0 0 2 0 -1 -1 -2 1 1 0 -1 2 -1 0 -1 0 3 -1 0 0 3 0 0 0 0 -1 2 -1 0 0 2 0 -w -w2 -2 w w2 0 0 0 -1 2 0 0 1 1 w w2 1 w w2 0 0 -1 0 0 2 -1 2 0 -w2 -w -2 w2 w 0 0 0 0 0 -1 2 1 1 w2 w 1 w2 w 213D5 extended Cartan Matrix BD3 Character Matrix 2 0 -1 0 0 0 1 1 1 1 1 1 0 2 -1 0 0 0 -1 -1 1 1 1 1-1 -1 2 -1 0 0 0 0 2 1 -1 -2 0 0 -1 2 -1 -1 0 0 2 -1 1 2 0 0 0 -1 2 0 -1 1 1 -1 1 -1 0 0 0 -1 0 2 1 -1 1 -1 1 -1 Eigenvalues -> 2 2 0 1 3 4 214D4 extended Cartan Matrix BD2 Character Matrix 2 0 -1 0 0 1 1 1 1 1 0 2 -1 0 0 -1 -1 1 1 1-1 -1 2 -1 -1 0 0 2 0 -2 0 0 -1 2 0 1 -1 1 -1 1 0 0 -1 0 2 -1 1 1 -1 1 Eigenvalues -> 2 2 0 2 4 215 A-D-E Classifications:Simple Quivers of ArrowsCatastrophe SingularitiesBinary Polyhedral GroupsWeyl Groups - Lie Algebras 216Quantum Lattice - Group 24-cell - <2,3,3>Pure States: <±1,±i,±j,±k>Coherent: <(±1±i±j±k)/2>Phase: Complex Lattice 217LIE ALGEBRA: Weyl Group WFinite Group on Weyl ChambersSpherical Reflection Rep.NO integer lattice translations I - Fulton and Harris 218Affine Weyl Group: W x G0(Hyperbolic Reflection Rep.)Euclidean Reflection Rep.:LIE GROUP G: Pi1(G) = I / G0 - Humphreys, Brown, Adams 219Kleinian Singularity S = C2/GG = finite subgroup of SU(2,C) H2(S,Z) = C1 + ... + CrIntersection Matrix of Ci == - Cartan = A-2I -Slodowy 220E6 - rank 6: 1,2,3,2,1,2,1 == first Chern numbers (Sing.)1+2+3+2+1+2+1=12: Coxeter1+4+9+4+1+4+1=24: BT orderE6<=>3.F24::E8<=>Monster -McKay 221<±1> -> C2=(2,2,1) -McKay-> A1 |WeylSpin(3)=SU(2)=Sp(1)=S3 -> A1 222<±1,±i>->(2,2,2) ->[<2,2,2>->D4]/2 |Weyl |McKay|Spin(4)+SU(3)->G2 +S7+S7 -> D4 |Weyl <±1,±w,±w^2>->(2,2,3)<-M>->D5/2 223<24-cell> -> BT -McKay-> E6 |Weyl D4 + D5/D4 + E6/D5 -> E6 17| |33 CxV8+S1 Cx(S8-+S8+)+S1 224<24,24*> -> BO -McKay-> E7 |Weyl F4 + S3 + 3x26 -> E7 || D4+V8+S8-+S8+ 225Witting = 2x(24 + (Golden 96 of 24*)) <Witting> -> 2xBI -MK-> 2xE8 |Weyl E8 2xD8 + (S16-+S16+) -> 2xE8 2x120 2x128 2x248 226Quaternion Group <p,q,r> | | Weyl Group [3^(p-1),(q-1),(r-1)] - Coxeter (Ford and McKay) 227 Unit Integral Lattices:R - B1=A1=C1 Integers Z C - D2 Gaussian, A2 Eisenstein H - D4 24-cell, F4 (24 + 24*)O - E8 Witting = 2x(24 + (Golden 96 of 24*)) 228Science is the most reliableguide in life. - Mustafa Kemal (Ataturk) 22928 Spin(8) G0; G1; G2; G3; G4; G5; G6; G7 15 SU(4) G0 -iG1; G2 -iG3; G4 -iG5; G6 -iG7 10 Sp(2) G0 -iG1 -jG2 -kG3; G4 -iG5 -jG6 -kG7 7 S7 G0 -iG1 -jG2 -kG3 -eG4 -ieG5 -jeG6 -keG7 230 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
231Spin(8) - Z2^3xS4 - 192 |Spin(5)&endash;SU(3)&endash;Spin(4)&endash;U(1)^4Z2^2xS2 S3 Z2xS2 1 8 6 4 1 232Spin(0,8): Cl(0,8) = R(16)Cle(0,8) = Cl(0,7) = = R(8) + R(8) R(8) = T = (CxQ) x OR(8) = Cl(0,6): Spin(0,6) 2330 2 2 2 2 2 2 2 1 4 4 2 2 2 2 2 2 1 4 4 4 2 2 2 2 2 1 4 4 4 4 2 2 2 2 1 4 4 4 4 4 2 2 2 1 4 4 4 4 4 4 2 2 1 4 4 4 4 4 4 4 2 1 4 4 4 4 4 4 4 4 1 2340 U U U U U U U 1 4 4 U U 2 U U U 1 4 4 4 U 2 2 U U 1 4 4 4 4 2 2 2 U 1 4 4 4 4 4 2 2 2 1 4 4 4 4 4 4 2 2 1 4 4 4 4 4 4 4 2 1 4 4 4 4 4 4 4 4 1 235
U U U U 1 U U U U 1 U U U U 1 U U U U 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 0 1
236Spin(0,6): Cl(0,6) = R(8)Cle(0,6) = Cl(0,5) = C(4) Re(C(4)) = R(4) survivesIm(C(4)) = R(4) collapses 237
4 4 4 4 4 4 4 4 Re(C(4)) 4 4 4 4 4 - U(4)4 4 4 4______________________ 3 3 3 2 Im(C(4)) 3 3 3 2 3 - U(3)3 3 3 2 2 - SU(2) 0 0 - Higgs
238Im(C(4)):U(3) = U(1) x SU(3)SU(2)Higgs SU(2) Scalar 239Re(C(4)):U(4)=Spin(0,6)xU(1)U(1)<->Dirac ComplexificationSpin(0,6) = Spin(0,5)xS5Spin(0,5) <-> McDM E-H GravityS5=Conformal -> NP Q Gravity 240Cle(0,6)=Cl(0,5)=C(4)=Cl(2,3)Cle(0,5)=Cl(0,4)=Q(2)=Cl(1,3) CxQ(2) = C(4) = DiracCle(2,3)=Cl(2,2)=R(4)=Cl(3,1) CxR(4) = C(4) = Dirac
Thanks toPhilip Dorrell for comments, for corrections with respect to thegenetic code, and for reference to a Many-WorldsFAQ.
With respect to many of my web pages, thanks to Kent Davis forletting me know about some bad links and other errors, so that Icould correct them.
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