## Active Table of Contents

A Three Dimensional Coordinate System for Complex Numbers

Cover Illustration:

Acknowledgements.

Dedication:

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Active Table of Contents.

Active List of Animations.

1.0 Introduction.

1.1 General Characteristics.

1.2 Associated Dimensions and Functions

1.3 4Dii Example: The Complex *Ln/Exp* Function as Rotations in Space

1.4 Other Implications

2.0 3Di Means: “Third Dimension Imaginary”.

2.1 Constructing the Coordinate System..

2.2 A Left Hand System

2.3 Projection Planes

2.4 Real, Imaginary or Complex Function Input and Output

2.5 Projecting a Space Curve

2.6 Conics in 3Di

2.61 Conic Nonlinearity

2.7 Helixes in 3Di

2.71 Beat Helix

2.72 Combining Bases

2.8 Exponentials in 3Di

2.81 Exponential Graph Rotation.

2.82 Lambert W Function

2.83 A ‘Lambert W Spiral’

3.0 Historical Curves in 3Di

3.1 The Elliptic Helix

3.2 The ‘Cusped Helix’

3.21 Tricuspoid

3.22 Astroid

3.23 5-cuspoid, Etc.

3.24 Trifolium

3.25 Rose Curves, Quadrifolium, Etc.

3.26 Epicycloid, Hypotrochoid, Epitrochoid, Hypocycloid.

3.3 Cycloid

3.31 Imaginary Slope

3.4 Historical Spirals in 3Di

3.41 Archimedes Spiral

3.42 Fermat’s Spiral

3.43 Hyperbolic Spiral

3.44 Lituus

3.45 Helix Derivatives, Lituus.

3.46 Square Cornu Spiral/Fresnel Integrals.

3.5 Conversion From 2D Polar to 3Di Coordinates.

3.51 Cochleoid

3.52 Conchoid

3.6 Spira Mirabilis 6 New 3Di Equations for the Equiangular Spiral

3.61 In 3Di Converted Form.

3.62 The “Complex Exponential” form:

3.63 In Logarithmic Form

3.64 In Complex Base Form

3.65 In Negative Real Base Form..

3.66 An ‘exponential rotator’ form:

3.7 Additional Terms in the Helix Equation.

3.71 Cardioid

3.72 Unification Note

3.73 Cardioid Derivatives

3.74 Freeth’s Nephroid with Derivatives.

3.75 Limacon

3.76 Nephroid

3.77 Talbot’s Curve with Derivatives.

3.78 Cowboy Hat with Derivatives.

3.8 The Serpentine, Witch and a “Circle”.

3.9 Additional Historical Curves.

4.0 Polynomials in 3Di

4.1 Quadratic Input and Output

4.11 Quadratic Nonlinearity

4.2 Imaginary Polynomial Coefficients.

4.21 Imaginary Translation

4.22 Imaginary Rotation and Concavity.

4.23 Imaginary ‘Tilt’

4.3 Helixes as Polynomial Coefficients

4.4 A Quadratic Ellipse-Hyperbola in Space.

4.5 Cusps

4.51 Helix Cusp

4.52 Polynomial Cusp

4.53 Semicubical Parabola

4.54 Mordell Bifurcation, Moving the Cusp.

4.55 Elliptic Curve Bifurcation moved down from page above.

4.56 Lame Curve Bifurcation which heading is this supposed to be?.

4.6 Polynomial Bifurcation

4.61 Roots of a Cubic

4.62 Roots of a Quartic

4.63 General Location of the Quintic Roots.

4.64 Polynomial Nonlinearity

4.7 Complex Coefficients Polynomials in Space.

4.8 Inverse Polynomial Functions.

5.0 Complex Slope

5.1 Complex Slope Geometry and Algebra.

5.2 Polynomial Space Trajectories.

5.3 Inverse Imaginary Slope

5.4 Table of Slopes in 3Di

6.0 Euler’s Formula Upgraded, Helix and Spiral Functions.

6.1 Euler’s Formula in Three Dimensions.

6.2 Coordinates in the Normal Complex Plane and in 3Di

6.3 The Helix Base b and Wavelength

6.31 The New Famous Five Identities.

6.32 Euler’s Famous Formula Upgraded.

6.4 Wavelength Interpretations.

6.41 Table of Exponent Effects for Various Bases:

6.42 Positive Real Wavelengths.

6.43 Negative Wavelength Interpretation.

6.44 Imaginary Wavelength Interpretation.

6.45 Complex Wavelength Interpretation.

6.5 Exponentials

6.51 The -base Exponential

6.52 Rotating Exponentials

6.53 Exponential Tangents

6.54 The Elliptic Spiral

6.6 The or Variable Base.

7.0 “2^{nd} and Higher Level Exponents”.

7.1 Helix and Spiral Non-linearity.

7.2 Additional Levels of Exponents.

7.3 Variable Wavelength and Location of the ‘Zeros’

7.31 2^{nd} Level Exponents and the ‘x-base’

7.4 Exotic Graphs Using Second Level Exponents.

7.5 ‘Airy Type’ Spiral

7.51 Airy Zeros Approximations.

7.6 ‘Bessel Type’ Spiral

7.61 Bessel Zeros Approximation.

7.62 Variable Coefficient and Derivative Examples.

7.63 A Cornucopia Function

7.7 Higher Level Exponents

7.71 3^{rd} Level exponents Two Examples.

7.72 4^{th} Level Exponent Example.

8.0 Helix Antiderivatives

8.1 First Level Exponent Imaginary:

8.2 Second Level Exponent Real

8.3 Second Level Exponent Complex.

8.4 First Level Exponent Complex.

9.0 4Di: Function Morphing

9.1 4Di Function Input and Output alternatives.

9.2 Morphing Concept

9.3 4Di Morphing Function Examples.

9.31 The Polynomial Morphing Function.

9.32 Exponential Morphing

9.33 Periodic Exponential Morphing.

9.34 Parabolic Morphing

9.35 Lissajous Morphing

9.36 Derivative Morphing

9.37 Morphing Skewed Spirals

9.38 A Simple Rotator

9.39 The Equiangular Spiral and Cardioid Motion.

10.0 4Dii: Helix Morphing

11.0 4Dii: Algebraic Forms

11.1 Functions in 4Dii

11.2 A New Geometry of Natural logarithms.

11.21 Spira Mirabilis Again

11.22 The Four Coordinate Complex Exp/Log Function.

11.23 The Four Coordinate Complex Exp/Log Surface.

11.3 The Imaginary Logarithmic Surface.

11.4 More Exponential/Logarithmic Surfaces.

12.0 4Dii: Surfaces

12.1 Surfacing a Function

12.11 Inverse Functions

12.2 Closed Surfaces

12.21 The Sphere Surface Function.

12.22 The Cube Surface Function.

12.23 Lozenges, Barrels and Pointed Elliptical Cylinders.

12.3 Open Surfaces

12.31 Constant, Exponential and Polynomial Surfaces.

12.4 Complex Regional Input

12.41 Cornu Spiral/Fresnel Surface.

13.0 4Dii: Object-Wave Duality Surfaces.

13.1 Observable and Embedded Dimensions.

13.2 Objects and Their Associated Waves.

13.21 The Sphere Surface

13.22 The Reciprocal Sphere Surface.

13.23 The Exp/Natural Log Surface.

13.24 Polynomial Surface

13.25 Cardioid Surface

13.3 Helicoids

13.31 Inner, Outer and Variable Amplitude.

13.32 Geometric Helicoids

13.4 A ‘Transverse Wave’ Surface.

14.0 5Dii: Circular Surface Functions.

14.1 Circular Helix

14.2 Circular Helicoid

14.3 Geometric Torus Surfaces.

14.4 Circular ‘Transverse Wave’ Surfaces.

15.0 6Dii: Surfaces in Motion.

15.1 Objects in Polynomial Space Trajectories.

15.2 Objects in Orbits

15.21 A Two Dimensional ‘Solar System’

15.22 Bodily Rotation

15.23 A Three Dimensional ‘Solar System’

15.24 A Helical Solar System

15.3 Circular Waves in Motion.

15.31 Spin Coefficient

15.32 Reciprocal Spin

15.33 Standing Waves

Concluding Personal Comments

Index