Index #
C! -sphere, 53
S! /D! group, 24-25
S! /D! isomorphism, 21
S!" /D!" group, 45-46
S!" # symmetry group, 315
S!$ /D!$ group, 25
W%(1) Lambert W function, 86 Y& , Y! generators, 311
Basel problem, 276
Bessel function of the first kind, 279 bijection, 22 bijective, 48, 360 bilinearity, 31
binomial factorial series, 269 binomial theorem, 269
bivector, 125-131
bivector rotations, 138
Borel regularization, 274-275 boundary, 289
acceleration vector, 62 adjoint matrices, 44
adjugate, adj(M), 22 al-Haytham, 334 alternating harmonic series, 272-274 alternating inverse harmonic series, 275 alternating natural numbers, 274
Ampere’s law, 136
analytic continuation, 355
Angle Space Ring, 99
angle transformation, 101
angular frequency, 111, 277, 365-377, 390
anti-diagonal symmetry, 47, 312
anti-form, 371, 385, 425
anti-identity matrix, 312
antimatter, 371
antiproton, 371
anti-spacetime, 374
antisymmetric wedge product, 125 arc length, 104
arc segment distance, 103
arclength, 104
asymptote, 129-130
asymptotic cones, 137, 141-142
asymptotic functions, 134
asymptotic Lie algebra, e'(⁄) , 130
asymptotic light, 381-382, 389
asymptotic light algebras, 390
asymptotic light functions, 129, 138-139
Asymptotic Light Functions, 124
asymptotic light numbers, 413
Athena, 417
Athene noctua, 417
axis number determinants, 181 axis numbers, 171 B
Baker-Campbell-Hausdorff formula, 269, 281
ball, n, 263
cardioids, 357
Cartan algebra, 44-47, 168-169, 312, Cartesian coordinate system, 98
Cartesian plane, 98
Cartesian-to-polar form, 196
Cauchy-Riemann equations, 122-123
Cayley’s Theorem, 22, 188
characteristic polynomials, 187, 317-325, 331-332, 406
chiral Lie algebra, 192
chirality, 190
chirality change mapping, 193
chord distance, 157 chord matrices, 164 circle metric, 97 circular arcs, 108 circular curvature, 115 circular curvature - wave equation, 109 circular curve in the plane, 107 combinatorics, 269 commutative, 16, 53 commutativity, 75 commutator, 20 non-commutative, 20 commuting binomial theory, 269 complete inner product space, 38 complex angle space, 144 complex conic section, 197 complex conic space, 155 complex conjugate, 124, 132 complex curvature, 117- 119 complex powers, 20 complex signed curvature, 124 complex-angle space, 99-102 complex-angle space vectors, 101 complex-angled space sphere, 101 complex-differentiable, 122 composite number, 344
cone equation, 155
cone function spirals, 205
Cone Group, 166
cone number determinants, 180
cone number elements, 178
cone number geometry, 166
cone number permutation matrices, 169
cone numbers, 176, 191, 394
cone section generator, 414
congruent modulo, 346
Conic Group, 180 conic sections, 155
continuous Lie groups, 63 continuous mapping, 100 continuously differentiable, 123 convergent series, 275 coprime, 336
cosine and sine integrals, 259
cross product, 144
curl, 363
curvature, κ: curvature, 97, 103-109, 399-403
curvature - positive, 96
curvature -negative, 96
curvature of a complex function, 123
curvature of Light, 148
curvature of the asymptotic parametric curve, 149
curved geometry, 79
cutting planes, 155-156
cyclic Z+ :
cyclic groups, 16, 68, 177, 312-318, 338, 407
cyclic modulo shift, 52-53
cyclic shift, 74, 318
cyclic subgroup, 44
cyclic trigonometric function derivatives, 60
cyclic trigonometric functions, 59-60, 73, 425
Dirac delta function, 277-280 directional integration and differentiation, 268 distance metrics, 98
divergence, 137, 363
dodecahedron, 396
dot product, 125, 128, 260, 370
Dyson, Freeman, 358
eccentricity, 283 eigendecomposition, 34, 71, 80
eigenvalue matrices D+ , 340
eigenvalues, 398
Einstein, Albert, 399
Einstein-Minkowski spacetime, 129 electric field, E, 136
electric flux, 136
electromagnetic waves, 114, 135
electron, 371
ellipse, 155
Emerald Tablet, 425
energy, E, 367
energy-momentum relationship, 377, 389, 392
equilateral triangle group, 315
Euclidean:
Euclidean, 79, 96, 404
Euclidean distance, 104, 118
Euclidean geometry, 98
Dde Broglie wavelength, 365 de Broglie, Louis, 365
delta impulse, 282
delta impulses on the singularity, 282
derivative algebras, 56-53, 67-72
derivative subtangent group, 91 determinant, 68, 126, 193, 414
determinants of the exponential derivatives, 68
diagonalizable Lie algebra, 80
diagonalization, 63
diagonalized matrix, 410
diagonalized matrix of eigenvalues, D, 69
differential forms, 56
differentiation of a matrix group, 56
digamma function ψ(%)(z), 258, 337
dihedral forms:
dihedral cone group, D!" ⊞ , 183
dihedral cone group, D!" △ , 169
dihedral differential forms, 62
dihedral differential group, D!0 ', 68
dihedral differential group, D!$ ', 66
dihedral group tangent spaces, 77
dihedral group, D!" , 29
dihedral group, D+ , 13
Dirac delta distribution formula, 281
Euclidean plane, 116, 122
Euclidean space, 97, 196
Euclidean spaces, 382
Euclidean distance metric, 189
Euclidean space, 43
Euler product, 309-314, 355
Euler product formula, 301
Euler’s formula, 260, 277
Euler-Mascheroni constant, 255-257
Euler-Mascheroni constant γ, 337
even divisibility, 343-345
exponential mapping, 63
exponential mapping of cyclic trigonmetric matrix, 65
extended state, 263
extrinsic curvature, 104
factorial function, 337, 344
factorial integration operator, 249
Faraday’s law, 136
Fermat’s little theorem, 335
finite cyclic group Gn, 341
finite geometric numbers, 351
first-order derivative algebra, 64
Flowing Point:
flowing point, 359- 363 quantum, 364
flowing point ideals, 389
force vector, F, 144
form, 371, 385, 425
Fourier analysis, 281
Fourier domain, 278
Fourier transform, 277-283, 389
Fourier transformation and Lie theory, 281
frequency, 367, 377
frequency domain, 113, 277
Frobenius norm, 39
fundamental theory of algebra, 315
G
gamma function, Γ(s), 259, 337, 355, 423
Gauss, Carl Friedrich, 354
Gauss’s law, 136
Gell-Mann matrices, 45-46
generalized group differential, 72
generating set of a group, G, 338
geometric algebra, 125-145
geometric integer, 398
geometric number power-products, 352
Geometric Numbers G+ , 339-346
Geometric Primes, G1 , 339-347, 406-418
geometric prime rotation matrices, 417
geometric product, 125-135, 139-147
geometric series, 260, 269
geometric-dependent energy-momentum relationship, 381-382
geometric-dependent Schrödinger equation, 379-386, 392
geometric-dependent Schrödinger equation wavefunction, 383
group characteristic polynomial, 406-410, 419-420
group order, 338, 341
group velocity, 90, 91
guaranteed primary generators, 340 H
Hamiltonian, 370-371
harmonic logarithm field, 270
harmonic logarithm formula, 257
harmonic logarithms, 255-256
harmonic numbers, H+ , 256-257, 274, 305306, 336-337
harmonic numbers H+ , 339
harmonic series, 255, 272, 302, 306, 336
Heisenberg uncertainty principle, 389
Hermitian symmetric, 40
higher-order algebra derivatives, 73
Hilbert space, 37, 41
Hilbert-Schmidt norm, 39-43
holomorphic, 122
holomorphic function, 122
holons, 364 homeomorphic, 100 homomorphism, 100
hyperbolic geometry: hyperbola, 98, 116, 155
hyperbola asymptote, 399
hyperbola foci, 116
hyperbolic, 379
hyperbolic asymptote, 129-130
hyperbolic curvature, 116-121
hyperbolic functions, 627 372
hyperbolic geometry, 402
hyperbolic geometry of special relativity, 421
hyperbolic group, 61
hyperbolic hypotenuse, 98
hyperbolic Lorentz factor, 373, 390, 400, 403
hyperbolic metric, 97
hyperbolic orthogonality, 401
hyperbolic parameteric signed curvature, 118-120
hyperbolic radius of curvature, 118
hyperbolic rotation, 402
hyperbolic signed curvature, 118
hyperbolic spaces, 49, 96
hyperbolic trigonometric functions, 67-69
imaginary hyperbolic analog, 178
implicit differentiation, 104
incomplete gamma function, 259
infinite cone equation, 203
infinite cone matrix group, 204
infinite differential function, 243
infinite differentiation, 242
infinite differentiation operator, 252
infinite factorial differential operator, 249253
infinite integration, 242, 246, 269, 280-281
infinite integration functions, 54-56
infinite order cyclic groups, 338
Infinite Sphere of Tangent Spaces, 240, 256, 263, 276, 284-293, 371
infinite subtangent space, 256
infinite symmetry group, 273, 280, 283, 293, 285, 359
infinite unit cone, 207
infinitesimal distance, 104-105
injective, 48, 361
inner product, 38-39, 42
inner product on C2×+ , 39
inner product on C+ , 38
inscribed geometry, 332
inscribed sphere(s), 161, 165
integration operator, 247
invariant distance, 98
invariant mass, 389
inverse harmonic numbers, the natural numbers, 274
inverse mapping, 100 involutory, 193
Jacobi identity, 31
Jonquiere’s function, 271
kinetic energy, 368, 370 kinetic energy operator, 370
Lagrange, Joseph-Louis, 335
Lambert W function, 83-86
Lambert W function’s range, 87
Laplacian, 370
Legendre, Adrien-Marie, 354
Leibniz, Gottfried Wilhelm, 335
Leonard Euler, 271-276
Lerch Transcendent, 272-273
Lie Theory:
Lie algebra, 31, 51, 62, 280, 289, 305, 372 generators, 314, 319 primary generators, 324
Lie algebra and differential forms, 71
Lie algebra set, 71
Lie bracket, 269
Lie group, 32
Lie group isomorphism, 130
Lie algebra, 21, 280
Lie group, 21
Light:
light asymptotes, 393, 403
light axes, 144
light cones, 129, 144, 404, 413-415
light cone section rotations, 419
light cone set, 416
light elements, 403
light functions, 140
light numbers, 412-419
light, square plane segment, 139
light-form, 388
lightlike, 129
linear transformation, 15
linearly polarized plane wave, 136 logarithm function, 358
Logarithmic Binomial Formula, 257
Logarithmic Path of Descent, Τ, 93 logarithmic spiral(s), 87-84, 358
Lorentz factor, 374, 378, 399, 402, 403
Lorentz transformations, 400
magnetic field, B, 136
magnetic flux, 136
mass energy relation, 373 mass, m, 368 matrix curl, 202
matrix decomposition matrix diagonalizable, 15 matrix derivative operation, 71 matrix derivatives, 56 matrix determinant, 284-285 matrix divergence, 202
matrix exponential, 21
matrix group conic sections, 155 matrix group exponentiation, 15
matrix group properties: associativity, 13 closure, 13 identity, 13 inverse element, 13
matrix groups: general linear group, 13 orthogonal group, O(n, F), 15 special linear group, SL(n, F) , 14
matrix logarithm, 63, 71 matrix powers, 21 matrix tranpose, 18 matrix-by-scalar derivative, 56 matrix-vector rotations, 320 matter, 371 matter-wave energy equivalence, 392 Maxwell’s equations, 136, 363, 399 metric, 97
minimal element, 177, 294-302, 314-315, 321, 405, 408-411
minimal element cone numbers, 179
minimal element matrix representation, 179 minimal slicing, 176
Mirabilis Group, 92-93
modified Bessel function, 266 modulo operation, 337, 342, 345 momentum, 368
Montgomery, Hugh, 358
natural logarithm, 20
natural number matrices, 349
natural number set, 274
natural numbers, 176, 302, 342
natural numbers, closed form, 274
n-dimensional Euclidean ball, 423 negative curvature, pseudosphere, 103 negative minimal element, 412 neighborhood of the identity, 80 net-zero, 381, 407
net-zero point, 244
net-zero splitting, 423 non-Euclidean, 96
non-Euclidean geometry, 98
null curve(s), 155-157, 166
number theory, 358
numeri mirabiles, 81
odd and even partition geometry, 176
offset logarithmic integral, 355
omega constant, 86 orbitons, 364
outer product, 260-272
outer-product regularization, 275
distribution, 298
geometric prime number algebras, 313 geometric prime numbers, 299, 312-313 geometric primes, 301 polygon vertices, 299 prime decompositions, 326 reciprocal prime factors, 302
pair correlation conjecture, 358 parabola, 155 parallelogram, 125-126
parametrized signed curvature, 106 partial sum, 272-274
particle physics, 271
Pauli matrices, 193
period of a waveform, 277 permutation, 22
phase transition, 265
phase-shift, 62-68, phase-shift dependency, 82
phenomenology, 404
phenomenology of light, 134
photon energy, 390
Pietro Mengoli, 276
pilot wave, 363
Planck’s constant ℏ, 365
Plank length, 371
Plank’s constant, 365
Plank-Einstein relation, 367
p-norm, 38
Poincaré-Hopf theorem, 315
polylogarithm, 271
positive curvature, sphere, 103
positive integer function, 354
positive integer waveform function, 346
positron, 371
potential energy, 365
pre-Hilbert space, 38-40 primality, 349
primary generator(s), 61 341, 346, 397
primary minimal element, 177
prime number theory:
prime characteristic polynomial, 327
prime decomposition, 342
prime minimal element, 297
prime number, 296
polygon vertices, 298
prime number generators, 297
prime number matrices, 349
prime number theory, 334
prime numbers, 294, 310, 328, 343, 345, 353, 354-356
asymptotic distribution, 354
prime-counting function π(n), 354 primordial algebras, 78 primordial light, 391 primordial symmetries, 78 principal-valued logarithm, 21, 70 product log function(s), 83, 92 projective forms, 50-51 projective transformation, 52 proton, 371 pseudosphere, 97 Pythagorean identity, 116 Pythagorean theorem, 382 Pythagoreans, 425
quantum chromodynamics, 45 quantum field, 370 quantum mechanics, 260, 370 quasi-particles, 364 quaternion ismorphism to geometric algebra bivectors, 157 quaternions, 157
radius of convergence, 263 radius of curvature, 103, 108 random matrices, 358 rapidity, 375, 400 ratio of curvatures, 110 reciprocal factorial series, 269 reciprocal prime number, 296 recursive differential operation, 253 recursive integration operation, 250 reduced Planck’s constant, 390 reflections/rotations of inscribed SU(2) sphere, 160 regular n-gon, 78 regular pentagon, 323
regularization of outer product array, 275 regularized infinite outer-product summation, 275
relativistic energy equivalence, 376 relativistic energy-momentum, 376 relativistic mechanics, 371, 373 relativistic velocities, 399
representation theory, 281 rest mass, 372
Riemann Hypothesis, 356
Riemann surface(s), 361-362
Riemann zeta function, 271-272, 307-308,
355
critical line, 356
critical strip, 356
zeta function zeros, 356
Riemann, Bernard, 355
Roman factorial, 256
rotational symmetries polygon, 338 S
scalar momentum, 373
scalar portion, 128
Schrödinger, Erwin, 365
Schrödinger’s equation, 365-369, 386
second-order derivative algebra, 65 semi-major axis, 117
sequence, 268
signed curvature, 104-107, 118
sinc function, 278-280
sine cardinal function, 279
singular value decomposition, 17-19, 36 singularity, 207, 240
singularity boundary, 291
skew-Hermitian, 77
skew-symmetric, 31
skew-symmetric matrix, 76
skew-symmetry, 47, 77
smooth, 76
smooth manifolds, 76
smooth path, 76
SO(3), 33
space and time curvature, 116
space of angles, 98
spacelike, 129
spacetime, 116, 374
spatial domain, 113
special relativity, 116, 389, 399, 402 speed of light, 115, 377, 399 sphere of tangent spaces, 293 spherical Bessel function, 279 spherical delta function, 282
spinon, 364
spira mirabilis, 89
split complex number, 422
split of the identity, 423
split trigonometric functions, 60 square-integrable functions, 38
Standard Model, 158
SU(2) subgroup of the symmetric conic group, 188
SU(2), inscribed sphere of the light cone, 158
SU(2) group, 158, 404
SU(2) set, 163
SU(3) group, 45 subgroup chirality, 192 subtangent space, 245-246, 285-290 superluminal, 403 surjective, 48
symmetric conic group, 178, 193 symmetric conic group, S!" △ , 180 symmetric dot product, 125 symmetric group, S+ , 23 symmetric groups, 22
symmetric/dihedral conic group, S!" △ , D!" △ , 176
symmetric/dihedral group, S!0 /D!0 , 47 symmetric/dihedral group, S!$ /D!$ , 26 symmetry group, S! , 23
Tangent Spaces:
tangent matrix, 56
tangent space, 56, 281-283
tangent space addition, 102
Tangent Space Complex Symmetry Group, 292
tangent space element as polygonal form, 199
Tangent Space of the Cone, 163
Tangent Space Rotation, 200
tangent space velocity, Tv, 82
tangent space, TϵG, 76
tangent sphere, 206-207
tangent vectors, 76
tangential angle, 104
tangential force, Ftan, 145
Tau function, Τ, 308
tensor algebra, 260
tensors, 260
tetrahedral numbers, 307
Theory of Light, 359, 393
third-order derivative algebra, 65 three-way cyclic trigonometric functions, 65
three-way exponential functions, 45 three-way trigonometric functions, 58 time-dependent Schrödinger’s, 377
time-domain, 277
timelike, 129
torque, 144
torque bivectors, 145
torque vector, τ, 134
trace of a matrix, 271
traceless, 193
triangular numbers, 303-307, 314
inverse sums, 307-308
inverse/reciprocal, 305
two-norm, 36
U(3) symmetry group, 30
U(3) unit sphere, 165
unextended state, 636
Unified Field, 370-371, 382, 391
unified geometry, 138
unified light functions, 138, 144-147
unit basis vector(s), 125-126 unit hyperbola, 67, 118, 165 unit hyperbolae, 383 unit impulse, 278-279 unit invariant mass, 389-393 unit normal vector, 103 unit singularity, 289 unit tangent vector, 103 unitary, 17 unitary asymptotic light generators, 391 unitary group, U(1), 21 unitary group, U(3), 32 universe, 425
vertex element set, 343
Waring, Edward, 335 wave domain, 277 wave equation, 114 wave number, 111 367-368 waveform, 274 wavefunction, 136, 194, 365, 370, 386, 392 wavelength, 111, 365, 377 wavenumber, 277, 365, 377, 386, 392
wavenumber-angular frequency relationship, 137
wave-particle duality, 134
wave-particle duality of light, 134 wedge product, 125, 128, 133
Wilson, John, 3334-33
Wilson’s theorem, 334-337, 347, 351
X -type matrices, 313
Y subgroup of the symmetric conic group, 186
zero curvature, 120
zero derivatives, 244
zero-dimensional, 157, 359, 364-365, 371, 423
zero-dimensional point, 54, 391, 423-424
zero-dimensional point of primordial light, 391
zero-point, 245-246, 257, 273
zeta zeros trivial, 356