‘Bessel Type’ Spiral 291

4Dii, function types 332

4

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

4

^{th}Level Exponent 302Airy 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 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, 2

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*78Imaginary 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, 2

^{nd}Derivative 317Morphing, 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