‘Bessel Type’ Spiral 291
4Dii, function types 332
4th Level Exponent 302
Airy Type Helix 257
Airy Zeros Approximations 288
amplitude, nested variable 379
amplitude, variable 379
Antiderivatives, Helix 304
Archimedes Spiral 81
Astroid 12, 62, 63
barrel 355
base, wavelength relationship 218
Bessel Zeros Approximation 292
Bifurcation 12, 43, 142, 147, 161, 162, 186, 187, 192, 194
Bifurcation, Elliptic Curve 175
Bifurcation, Lame Curve 179
Bifurcation, Mordell 174
Bifurcation, Polynomial 184
Bodily Rotation 404, 408
Cardioid 118
Cardioid derivatives 120
Cardioid Limacon Cubic Surface 360
Cardioid Motion 324
Cardioid orbits 327
Cardioid Petal 325
Cardioid Surface and wave 376
Cayley’s sextic 325
Circular Helicoid Standing Wave 414
circular helix 386
Circular Transverse Standing Wave 413
Circular Waves in Motion 408
closed surfaces in motion 395
closed surfaces in orbit 400
Cochleoid 94
Complex Coefficients – Polynomials in Space 197
complex regional domain 362
complex slope 204
Complex Slope, Table 212
Complex Wavelength 232
Conchoid 95
Conversion From 2D Polar to 3Di Coordinates 94
coordinate conversion, 2Di to 3Di 215
Cornu Spiral 64, 91, 93, 363
Cornucopia Function 297
Cowboy Hat Helix 132
cube lozenge 351
Cube Surface Function 349
Cubic 184
Cusped Helix 62, 91, 117, 242, 245
Cusps 168
cycloid 70
cylinder, pointed elliptical 356
cylindrical Nephroid surface 364
de Moivre Numbers 40, 194, 253
Ellipse-Hyperbola 161
elliptic helix 59, 60, 62, 242
Elliptic Spiral 12, 64, 242, 243
embedded dimensions 19, 366, 384
Epicycloid 68
Epitrochoid 68
Equiangular Spiral 12, 13, 98, 105, 110, 115, 116, 225, 324, 328, 329
Euler formula upgraded 222
Euler Identities 219
Exp/Log Surface 371
Exponential Graph Rotation 51
Exponential Tangents 239
Fermat’s spiral 83
Freeth’s Nephroid 123
Fresnel Integrals 91
Fresnel Surface 363
generalized helix equation 118
geometry function 353, 384, 386, 391, 398, 400, 401, 402, 404, 408, 410
Helical Solar System 406
helicoid, circular 389
helicoid, elliptic 380
Helicoids 377
helitricoid 380
Helix, Derivative Eight 299
Helix, Hovering Arc 299
helixes as polynomial coefficients 153
4Dii, function types 332
4th Level Exponent 302
Airy Type Helix 257
Airy Zeros Approximations 288
amplitude, nested variable 379
amplitude, variable 379
Antiderivatives, Helix 304
Archimedes Spiral 81
Astroid 12, 62, 63
barrel 355
base, wavelength relationship 218
Bessel Zeros Approximation 292
Bifurcation 12, 43, 142, 147, 161, 162, 186, 187, 192, 194
Bifurcation, Elliptic Curve 175
Bifurcation, Lame Curve 179
Bifurcation, Mordell 174
Bifurcation, Polynomial 184
Bodily Rotation 404, 408
Cardioid 118
Cardioid derivatives 120
Cardioid Limacon Cubic Surface 360
Cardioid Motion 324
Cardioid orbits 327
Cardioid Petal 325
Cardioid Surface and wave 376
Cayley’s sextic 325
Circular Helicoid Standing Wave 414
circular helix 386
Circular Transverse Standing Wave 413
Circular Waves in Motion 408
closed surfaces in motion 395
closed surfaces in orbit 400
Cochleoid 94
Complex Coefficients – Polynomials in Space 197
complex regional domain 362
complex slope 204
Complex Slope, Table 212
Complex Wavelength 232
Conchoid 95
Conversion From 2D Polar to 3Di Coordinates 94
coordinate conversion, 2Di to 3Di 215
Cornu Spiral 64, 91, 93, 363
Cornucopia Function 297
Cowboy Hat Helix 132
cube lozenge 351
Cube Surface Function 349
Cubic 184
Cusped Helix 62, 91, 117, 242, 245
Cusps 168
cycloid 70
cylinder, pointed elliptical 356
cylindrical Nephroid surface 364
de Moivre Numbers 40, 194, 253
Ellipse-Hyperbola 161
elliptic helix 59, 60, 62, 242
Elliptic Spiral 12, 64, 242, 243
embedded dimensions 19, 366, 384
Epicycloid 68
Epitrochoid 68
Equiangular Spiral 12, 13, 98, 105, 110, 115, 116, 225, 324, 328, 329
Euler formula upgraded 222
Euler Identities 219
Exp/Log Surface 371
Exponential Graph Rotation 51
Exponential Tangents 239
Fermat’s spiral 83
Freeth’s Nephroid 123
Fresnel Integrals 91
Fresnel Surface 363
generalized helix equation 118
geometry function 353, 384, 386, 391, 398, 400, 401, 402, 404, 408, 410
Helical Solar System 406
helicoid, circular 389
helicoid, elliptic 380
Helicoids 377
helitricoid 380
Helix, Derivative Eight 299
Helix, Hovering Arc 299
helixes as polynomial coefficients 153
Hexagon Lozenge 353
hyperbola-ellipse 39
Hyperbolic Spiral 85
Hypocycloid 68
Hypotrochoid 68
i-base Exponential 235
Imaginary Logarithmic Surface 338
imaginary numbers 15
Imaginary Rotation and Concavity 151
imaginary slope 78
Imaginary Translation 149
Imaginary Wavelength 229
inner amplitude 368, 370, 372, 378, 381, 383, 389
Input and Output alternatives 309
inverse functions, surfaces 346
inverse imaginary slope 210
inverses, polynomial unctions 201
i-rotation 15
jump discontinuity 160
Lambert W 53, 55, 56
Limacon 126
List of Animations 12
Lituus 88
Logarithmic Surfaces 341
Lozenges 351
mode, object 374
mode, wave 375
Morphing Concept 310
Morphing, 2nd Derivative 317
Morphing, Exponential 313
morphing, helix 330
Morphing, Lissajous 316
Morphing, Parabolic 315
Morphing, Polynomial 311
Morphing, Skewed Spirals 319
Natural logarithms, new geometry 333
Negative Wavelength 228
Nephroid 127
Newton’s Diverging Parabolas 142
nonlinearity 39, 160, 194
Nonlinearity 12, 41, 43, 148, 161, 162, 179, 184, 194, 195, 253, 309
Nonlinearity, conic 39
nonlinearity, helix and spiral 253
Nonlinearity, Polynomial 194
Nonlinearity, quadratic 147
Objects and Their Associated Waves 367
observable dimensions 19, 366, 384, 395
outer amplitude 368, 370, 372, 377, 381, 383
periodic coefficient 322
Polynomial Space Trajectories 208, 395, 398
projection planes 27
Quadrifolium 65
Quartic 191
Quintic Roots 193
Reciprocal Sphere Surface 369
Reciprocal Spin 411
Rose Curves 65
Rotating Exponentials 53, 237, 338
Saturn Lozenge 353
Semi Cubicle Parabola 142
Semicubical Parabola 172
Serpentine 136
simple rotator 322
Sphere Surface Function 347
spin coefficient 404, 408, 410, 411, 412
Spira Mirabilis 98, 104, 225, 333
Standing Waves 413
surface operator 352, 357
Surface, transverse wave 383
Surfaces, circular transverse wave 393
Surfaces, geometric torus 391
Table of Contents 7
Talbot’s Curve with Derivatives 129
trajectory function 398, 405
Triangular Cornu 93
Tricuspoid 12, 62, 65, 330
Trifolium 65
Tschirnhaus’ Cubic 142
Variable Wavelength and Location of the ‘Zeros’ 263
Wavelength Interpretations 223
wavenumber 218
Witch of Agnesi 136
x base 248
hyperbola-ellipse 39
Hyperbolic Spiral 85
Hypocycloid 68
Hypotrochoid 68
i-base Exponential 235
Imaginary Logarithmic Surface 338
imaginary numbers 15
Imaginary Rotation and Concavity 151
imaginary slope 78
Imaginary Translation 149
Imaginary Wavelength 229
inner amplitude 368, 370, 372, 378, 381, 383, 389
Input and Output alternatives 309
inverse functions, surfaces 346
inverse imaginary slope 210
inverses, polynomial unctions 201
i-rotation 15
jump discontinuity 160
Lambert W 53, 55, 56
Limacon 126
List of Animations 12
Lituus 88
Logarithmic Surfaces 341
Lozenges 351
mode, object 374
mode, wave 375
Morphing Concept 310
Morphing, 2nd Derivative 317
Morphing, Exponential 313
morphing, helix 330
Morphing, Lissajous 316
Morphing, Parabolic 315
Morphing, Polynomial 311
Morphing, Skewed Spirals 319
Natural logarithms, new geometry 333
Negative Wavelength 228
Nephroid 127
Newton’s Diverging Parabolas 142
nonlinearity 39, 160, 194
Nonlinearity 12, 41, 43, 148, 161, 162, 179, 184, 194, 195, 253, 309
Nonlinearity, conic 39
nonlinearity, helix and spiral 253
Nonlinearity, Polynomial 194
Nonlinearity, quadratic 147
Objects and Their Associated Waves 367
observable dimensions 19, 366, 384, 395
outer amplitude 368, 370, 372, 377, 381, 383
periodic coefficient 322
Polynomial Space Trajectories 208, 395, 398
projection planes 27
Quadrifolium 65
Quartic 191
Quintic Roots 193
Reciprocal Sphere Surface 369
Reciprocal Spin 411
Rose Curves 65
Rotating Exponentials 53, 237, 338
Saturn Lozenge 353
Semi Cubicle Parabola 142
Semicubical Parabola 172
Serpentine 136
simple rotator 322
Sphere Surface Function 347
spin coefficient 404, 408, 410, 411, 412
Spira Mirabilis 98, 104, 225, 333
Standing Waves 413
surface operator 352, 357
Surface, transverse wave 383
Surfaces, circular transverse wave 393
Surfaces, geometric torus 391
Table of Contents 7
Talbot’s Curve with Derivatives 129
trajectory function 398, 405
Triangular Cornu 93
Tricuspoid 12, 62, 65, 330
Trifolium 65
Tschirnhaus’ Cubic 142
Variable Wavelength and Location of the ‘Zeros’ 263
Wavelength Interpretations 223
wavenumber 218
Witch of Agnesi 136
x base 248
