Welcome to “A Three Dimensional Coordinate System for Complex Numbers”.
The new coordinate system is called “3Di”, an acronym for Third Dimension Imaginary which makes everything simpler. A basic introduction is provided by:
A New Geometry of i
part 1: Elementary Algebraic Functions in 3Di Coordinates
Imaginary numbers are discussed in relation to human experience and a new coordinate system, “3Di” – an acronym for “Third Dimension Imaginary” is introduced and defined. A three dimensional function of the form: [y + iz = f(x)] is then defined. Polynomials, conics, elliptic curves and hyperelliptic curves are graphed in these coordinates and shown to exist in three dimensions by way of a new and simple geometric definition of an imaginary number. This definition becomes intuitively obvious and therefore provides a potentially high school level of understanding of complex numbers. Thus, the new coordinate system itself potentially facilitates a grand enhancement to mathematics education. The two dimensional functions become three dimensional functions with the notion of a ‘bifurcation point’. Usually occurring at vertexes, cusps and turning points, a bifurcation point is where multiple branches, both real and imaginary, of the various algebraic functions meet and make various types of turns onto orthogonal planes.
Historically and currently Euler’s formula:
enjoys wide agreement as to its algebraic interpretation and meaning. In this six-part blog we present its many and varied new geometric interpretations and meanings made possible with “3Di” coordinates.
In blog #1 Euler’s formula is used as a rotation of the exponential graph through an imaginary circle.
In blog #2 Euler’s formula is used as an ‘orbit operator’.
In #3 as a helix with both fixed three-dimensional geometry, and in four dimensions with an animation variable.
In #4 we show that Euler’s identity actually maps a point on an ‘exponential/natural log surface’!
In #5 a ‘circular helicoid standing wave’ is shown in which Euler’s imaginary exponential is used no less than five times in the equation: as a surface operator, an orbit operator, a synchronizing rotator, a spin coefficient, and as a reciprocal spin coefficient!
In #6 we present the equation that updates Euler’s formula for all bases: positive, negative, real, imaginary and complex.
“3Di Coordinates” is a completely new concept that makes everything easier to understand. The eBook contains nearly six hundred line and surface graphs illustrating the equations so you can settle in for an enjoyable as well as informative experience. Thanks!
Greg Ehmka’s math e-book demonstrates the real potential of this publishing medium; an active table of contents, an active list of animations, and active cross-referencing make browsing quick and studying delightful. There are also links to external sites that provide instant access to relevant sources. Adding enjoyment and inspiration are over eighty brief videos that animate many of the ground-breaking math equations to colorfully express their resulting objects. This doesn’t begin to speak of the mind-blowing content, which you will just need to see for yourself! Ehmka mingles whimsy with his genius rendering this work both astoundingly brilliant and accessible.
- M.C.W., editor