Highlights 4

A) Historical Curves

An extraordinary new look at the “Historical Curves.” The list of historical curves presented, for example, here has over sixty listings. With the ideas presented in this book there are entirely new equations for dozens of them and a fresh, three dimensional look at a large majority of them. These are mostly found in section 3.0.

B) Cusps

Many ‘cusps’ that appear in two dimensions are actually transitions of the graph on or off the front view. For example, using an base helix function when projected to the front view shows the cusp at (x, y) = (0, 1).

The curve is actually a 3D helix on the positive side and an asymptotic curve exactly on the front real plane for negative

C) Second Level Exponents – “Airy Zeros”

Using second level exponents on the helix and adjusting coefficients, good, differentiable approximations to Airy function zeros and Bessel J₀ and J₁ function zeros and derivatives are possible in elementary function form.

For example, in two dimensions the zeros for Bi(x) can be approximated by:

	1	2	3	4	5	6	7
Bi (b)	-1.1737	-3.2711	-4.8307	-6.1699	-7.3768	-8.4920	-9.5381
Helix Re + Im	-1.1657	-3.2639	-4.8227	-6.1668	-7.3820	-8.5076	-9.5657

D) Helix Functions

With the 3Di coordinate system, Euler’s Formula becomes a helix and amounts to giving rise to a whole new class of elementary functions, i.e. helix functions.

One example from section 7.63 are two versions of a “Cornucopia Function.”

In 3Di:

And in front and top view projections:

Previous Highlight

Quotes From the Book

The notion of “3Di:Third Dimension Imaginary” is probably a good example of what Victor Hugo called: “An idea whose time has come.”
Greg Ehmka,

To the degree that mathematics assists us in understanding what we perceive as we look out at life, we must acknowledge the fact that what we perceive includes both seen and imagined components. Consequently both real and imaginary values must be included in the mathematical representation of it.
Greg Ehmka,

It’s like a coffee table picture book for math people!
M. W.,

Consider an aircraft. Real slope is its climb or descent. Imaginary slope is its heading. Complex slope is the sum of the two.
Greg Ehmka,

A negative wavelength can be defined as a clockwise development of the helix rather counterclockwise which would be a positive wavelength.
Greg Ehmka,

This e-book is dedicated to all high school students, past and present, who found a spark of interest in mathematics and then decided that she or he was too math-phobic, or not smart enough, to go further. It took me nearly thirty years to write this. Don’t give up on what you love!
Greg Ehmka,

I’ve, personally found useful the realization that there are essentially only four graphed or observable dimensions - horizontal, vertical, depth and ‘animation’- And theoretically there could be many ungraphed, unobserved or embedded dimensions.
Greg Ehmka,

If it’s true that we must imagine what goes on behind the façade in the z or depth direction of perspective, then possibly imaginary numbers can be combined with the real plane to form a new three-dimensional coordinate system.
Greg Ehmka,

In section 6.0 we take Euler's Famous Formula which is stated for base e: e^iθ = cosθ + isinθ and upgrade it for all bases; positive, negative, real, imaginary and complex.
Greg Ehmka,

The exponential-natural logarithm function forms a surface on which each point on the surface has the coordinates, (x+iθ or ln(y + iz) = x +iθ
Greg Ehmka,