At last count there were some 425 pages and about 30,000 words. This means there is, on average, only about one large paragraph of “reading” per page. The rest is taken up with over five hundred line and surface graphs and the functions that generate them. To that is added nearly ninety, short, ten- to twenty-second animations of the graphs that represent the functions – for a total of almost six hundred illustrations.
With a few notable exceptions like Euler’s Famous Formula Upgraded, Helix Antiderivatives, and a few others, all of the equations are straightforward functions.
The New Coordinate System for Complex Numbers is called ‘3Di’ which is kind of an acronym for ‘Third Dimension Imaginary’ and is based on the following definitions and characteristics:
1) Essentially, imaginary numbers are measurements in the ‘depth’ direction. Real numbers are measurements in the horizontal and vertical directions. Mathematically, depth is imaginary.
2) In three dimensions think of an rotation as going from the horizontal or vertical to the front, rather than from the horizontal to the vertical.
3) A ‘dimensional approach’ is taken to functions. Meaning, functions are categorized by the number of dimensions the particular function has. By ‘dimension’ what is meant is essentially a variable.
4) One of the primary points of view is that imaginary dimensions are as equally important, and as equally present, as real dimensions.
5) The primary purpose of the e-book is to present a wonderful array of the many new functions made possible by ‘3Di’ and to graph these functions.
To make it a little clearer, with this model we could say that the usual real plane might be designated as ‘2D,’ the complex plane as ‘2Di,’ the usual real space as 3D,and this system as 3Di.
For the categorization of functions, the presence and number of i’s in the designation denotes the number of imaginary dimensions or imaginary variables. Building from the above, we may have for example, 3Di, 4Dii, 5Dii, 6Diii and so on.
The functions that are graphed, therefore, associate the number of variables – including both input and output – to the number of dimensions and give rise to the following useful categorization of types of functions:
|3Di||helixes, polynomials, conics|
|4Dii||complex natural logs|
|4Dii||complex imaginary logs|
|4Dii||closed and open surfaces|
|6Dii||closed surface objects in motion|
With five and six variables, becomes exclusively an output variable, whereas, in lower dimensions it is usually both an input and an output dimension.
As we stand and peer out at life we “know” that we “see” a three dimensional reality. But, if we are rigorous about what we are actually seeing then we don’t actually see anything except a two-dimensional visual screen. People who have only one eye functioning are acutely aware of the loss of depth perception. There is a story of a one-eyed tennis player who had to train himself to observe the growing size of the tennis ball as it moved toward him to know the proper distance at which he should hit the ball for the return.
In another example, if you walk down the street and stand squarely in front of a building you don’t actually see anything other than the front of the building. What we refer to as the ‘the façade’ is a two-dimensional view which is even more pronounced if you close one eye since the three-dimensional perspective of binocular vision is lost.
If we don’t move, and therefore, continue looking only at the front of the building through one eye, we can draw a two-dimensional coordinate system in our mind’s eye upon which the front of the building could be sketched. Designers and architects do this all the time. They are sketching what one would actually see.
Mathematically, the exact point at which the eye would look, without moving, would be the origin. And, we could draw a horizontal x-axis and a vertical y-axis from that origin point.
Now, if you happened to have had your eyes closed while someone else guided you to this exact position, standing squarely in front of the building, then how would you “know” that you were looking at a whole building, rather than just the backdrop of a theater set or just the façade of a Hollywood set?
The answer, depending on how well the film or theater set was constructed, is that you would NOT know. The “rest” of the building, or whatever is behind the set, would have to be imagined.
With this awareness we can speculate that imaginary numbers may, in fact, be a way for mathematics to map the specific third-dimension of ‘depth.’ Accordingly, this book proposes and explores a new three-dimensional coordinate system wherein the horizontal and the vertical axis both take on real values, but the ‘depth’ axis – before and behind us – takes on imaginary values.
So, essentially, imaginary numbers are measurements in the ‘depth’ direction. Real numbers are measurements in the horizontal and vertical directions. Mathematically, depth is imaginary.
In 3Di we will primarily be graphing functions of the following form:
i. The real input will be graphed on the horizontal axis.
ii. The real output will be graphed on the vertical axis.
iii. And, the imaginary output will be graphed on the “depth” axis.