A New Geometry of i
part one: “Elementary Algebraic Functions in 3Di Coordinates”
by Greg Ehmka
Abstract
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.
Contents
Introduction
Imaginary Numbers and Human Experience
Algebraic Functions in 3Di
The Imaginary Circle
The Imaginary Hyperbola
Conic Nonlinearity
The Imaginary Semicubical Parabola
The Imaginary Mordell Curve
Imaginary Elliptic Curves
Elliptic Curve Nonlinearity
Skewed Elliptic Curves
Imaginary Hyperelliptic Curves
The imaginary Parabola, Quadratic Nonlinearity and Roots Graphs
Imaginary Polynomial Branches
A Cubic Roots Graph
A Quartic Roots Graph
A Quintic Roots Graph
A Sextic Roots Graph
Summary
Further Reading
Introduction
This article is organized into three parts. Part one is the introduction and definition of “3Di Coordinates” along with a discussion as to how an imaginary number may be seen more clearly in a geometric manner that complements the wide agreement which imaginary numbers have algebraically. This is then carried further and relates imaginary numbers to human experience.
The second part surveys elementary algebraic functions and presents their usual equations extended to three dimensions along with their graphical curves in 3Di coordinates. Included are animations that rotate the three dimensional curves for improved perception and understanding. Special emphasis is given to the imaginary branches of these curves that exist in space off the real (x, y) plane.
The third part is a short summary of the basic points and principles previously covered in the article.
3Di coordinates is very simply constructed by beginning with the usual, real plane and designating the x-axis as the horizontal and the y-axis as the vertical. To this is added the depth axis positive to the front and negative to the back. As will be described, these designations horizontal, vertical and depth are designed to more closely match observations from a human perspective with their mathematical representations.
The depth axis is named the z-axis and it is then defined to be imaginary just as the vertical axis is defined to be imaginary in the complex plane.
The notion of ‘projection planes’ will often be implied wherein three dimensional curves are projected to one of the three two-dimensional planes often designated in engineering or architectural terms as front, top and side. In this way the various three dimensional curves in 3Di coordinates are then projected to these planes giving a two dimensional view of the three dimensional curve with interesting and even surprising results.
The algebraic curves in part two are most of the usual elementary functions that in two dimensions are written as:
and graphed on the usual real plane. In 3Di Coordinates these functions take the form of, what we could call, a three dimensional equation. That is, a single equation that has three variables in it, each of which are graphed on the three aforementioned axes. The functions then take the form:
In this form the function has one input variable and two output variables. In this article all of the curves in part two will have this form. Meaning, the input variable x will be real and the two output variables will be one each, real and imaginary. Or in other words the functions have a real number input and a complex number output.
A typical example is the two functions associated with the top and bottom halves of a circle. These are usually written:
In 3Di coordinates these two functions are written:
The fundamental advance of 3Di Coordinates is that they allow the domain of x to expand to include values that would normally produce imaginary numbers as output. Now that we have a definition and an axis for these imaginary output values they can be graphed right along with the real output ones. As we will see conics, polynomials, elliptic curves to include hyperelliptic curves will exist in three dimensions wherever complex numbers are generated as output in the functions.
‘Bifurcation points’, defined as; “the point at which the curve abruptly transitions either on to or off of one of the projection planes (usually)” is the term we will use to describe where a curve changes from being a three dimensional one to a two dimensional one or viceversa.
Also included at applicable points in the article is an exploration of a certain type of non-linearity. That type being described as “an equation or function with exponents other than one on the dependent variable”. A typical example is the y2 term in an elliptic function and implied with the Semicubical Parabola. This type of non-linearity also generates additional imaginary branches to the curve.
Imaginary Numbers and Human Experience
Generally speaking imaginary numbers are often thought to be, at worst, an annoyance, at times, a reluctant necessity and, at best, strange but of undeniable usefulness. At the same time there is a striving to know just what exactly an imaginary number is. A short survey of various on-line forums shows interesting discussions wherein one writer typically asks about an intuitive understanding of the imaginary unit, i, and other writers attempt an answer. For example, here, or here. Or if the forum is somewhat more science oriented, the discussion centers around visual/physical representations of imaginary numbers. For example, here.
The source of these types of discussions stems from the wide agreement as to the algebraic definition of the imaginary unit i and the perceived, by some, insufficiency as to the geometric definition of i.
The algebraic understanding is, of course, this definition:
along with comments such as this one from Leibniz and similar others:
“From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised.”[*]
The current geometric understanding of i is as units along the vertical axis of the complex plane along with the intuitive sense of a rotation. These are described by the Argand Diagram [*] [**] and extended via complex numbers to a circle by Euler’s formula:
about which there is a similar duality as to wide agreement on the algebraic meaning and an insufficient understanding as to the geometric meaning. This duality is typified by comments such as this one by Benjamin Pierce:
“Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it,
and we don’t know what it means. But we have proved it,
and therefore we know it must be the truth.”
and more recently this one by Scott E. Brodie[*]:
“An intuitive understanding of Euler’s formula for the complex exponential,
remains elusive, notwithstanding hundreds of years of contemplation.”
[See: Euler’s Formula Upgraded – blogs 1 thru 6]3Di coordinates offers a significant, and as will be seen, a very satisfying advance in the geometric understanding of imaginary and complex numbers. One that implies a direct connection to human experience. As suggested in the introduction to this article, imaginary numbers can now be seen to have these two additional definitions:
In three dimensions, imaginary numbers are measurements in the ‘depth’ direction. Real numbers are measurements on the horizontal and vertical directions. Mathematically, depth is imaginary.
In three dimensions think of an i rotation as going from the horizontal or vertical to the front rather than from the horizontal to the vertical as in the two dimensional complex plane.
In order to establish this direct connection to human experience it is useful to note that imaginary and complex numbers have historically not been seen to have a direct connection to human experience. The philosophical concerns relative to imaginary numbers no longer generate much discussion but the direct connection to human experience is still insufficient.
This is seen in Leibniz’s above quote and in these two comments by Nobel Laureate Eugene Wigner:
The complex numbers provide a particularly striking example for the foregoing.Certainly, nothing in our experience suggests the introduction of these quantities. [*]
Surely to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. [*]
3Di coordinates seeks to establish this direct connection to human experience beginning with some simple visualizations:
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 with an ever so slight sense of depth due to binocular vision. People who have only one eye functioning are acutely aware of the predominance of this two dimensional screen.
For 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 facade is a two-dimensional view which is even more pronounced if you close one eye.
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 facade 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.
In another example, a friend once told me of an acquaintance of his who only had one eye and who liked to play tennis. As the story goes the one-eyed tennis player 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.
An even more striking example of this can be found in flight training for pilots. If a pilot is flying under VFR (visual flight rules as opposed to IFR, instrument flight rules), as another aircraft is observed, it is critical to determine whether or not there is a relative motion between the two aircraft. Here is one quote of the principle that can be found in many places:
if another aircraft appears to have no relative motion, but is increasing in size, it is likely to be on a collision course with you [*]
Quite likely some interesting thought experiments may be conducted as to under what conditions it is possible to know whether or not a tennis ball is of constant size and approaching or at a constant distance and growing in size!
We can also ask ourselves; “What is the nature of the distance that appears to be unobservable to the pilot between the two aircraft that are on a collision course?” Extending this inquiry just a bit further; “How does the introduction of radar or a laser beam, which can and does determine the distance precisely, differ in terms of measurement from the type of measurement possible in the horizontal and vertical directions which, traditionally, uses “measuring sticks”? At the very least, a radar or laser beam requires time and a measuring stick does not.
Transferring this general inquiry directly to mathematics we might ask; “Does the real x, y plane assume a one-eyed view? Can the common human experience of binocular vision, or in certain cases the absence of it, be suggestive of imaginary distance?
If it’s true that we must imagine what goes on behind a building’s facade in the z or depth direction of perspective or imagine the distance to an object, moving or not, that is directly in front of us then possibly imaginary numbers can be combined with the real plane to form a new three-dimensional coordinate system more closely representative of what we actually observe.
In other words; 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.
As a final example, special relativity theory with its concept of length contraction in the direction of travel or parallel to it [*] also is suggestive of a different quality to the third or depth dimension.
Although quite interesting in and of itself, how much of the foregoing discussion is factual or truthful is not what we wish to determine here. The introduction of 3Di coordinates only takes the very small step of declaring that the third or depth dimension is sufficiently different from the horizontal and vertical directions to warrant a slightly different mathematical treatment. And that slightly different treatment is only to define the third dimension as being imaginary analogous to the y-axis being defined imaginary in the complex plane.
As it turns out this small step of defining the third dimension as imaginary brings wonderful new results.
Algebraic Functions in 3Di
One of the simplest confirmations of the validity of 3Di coordinates is the appearance of the imaginary circle/ellipse between the vertices of the standard hyperbola.
The Imaginary Circle
The standard equation of the hyperbola:
with:
gives the two dimensional graph:
Rearranging gives the two functions:
with the positive root function in black and the negative root function in red, in three dimensions, this would look like:
Bringing our attention to the area between the vertices, notice that input values:
will generate imaginary numbers and these are not normally allowed as part of the graph. When x is in this part of the domain the functions actually generate complex numbers whose real part is zero. Now that we have a definition for these output imaginary values as being along the depth axis, we can include the domain of x which generates them and graph the two functions as follows. Positive root in black, negative root in red:
The imaginary circle appears! But it is only visible in 3D perspective or the top view. The standard two dimensional graph is of the front view and the imaginary circle, although always there, is not visible in front, i.e. real plane, view. (The graphing software being used here is Pacific Tech’s Graphing Calculator which has some difficulties at certain cups, vertices, etc.)
The vertices of the hyperbola are examples of what we have previously mentioned as ‘bifurcation points’. These are defined as the points at which the graph makes an abrupt turn on-to or off-of an orthogonal plane. In this case, on-to and off-of the front real plane (FRP) and the top imaginary plane (TIP).
The Imaginary Hyperbola
The standard equation of the circle/ellipse:
with:
gives the two dimensional graph:
Rearranging gives the two functions:
with the positive root in black and the negative root in red, in three dimensions, this would look like:
In this case input values for x that are outside the interval:
meaning that x is either less than -1 or greater than +1, will generate imaginary numbers and again these output values are complex numbers with real part zero. As before, now that we have a definition for these output imaginary values as being along the depth axis, we can include the domain of x which generates them and graph the two functions as follows.
And the imaginary hyperbola appears. In this case the circle/ellipse appears on the front, real plane and the imaginary hyperbola appears on the top, imaginary plane.
From this we can see that the standard equation:
is actually of one three dimensional object and the sign only serves to determine whether the ellipse or the hyperbola appears on the front, real plane.
Conic Nonlinearity
We can now define a certain type of nonlinearity. That being; “integer exponents other than one on the dependent variable”. Meaning, with x as the independent variable and y as the dependent variable, for the hyperbola:
Just as when n = 2 there were two functions generated, when n = 3, 4, 5 etc. there will be three, four or five etc. functions generated. The separate functions are determined by what is known as “Demoivre Numbers” or numbers that give the nth roots of unity. [*] These are complex numbers of the form:
where k is the kth root of the n roots of unity. E.g., for the cube roots of unity, n = 3 and k = 1, 2, 3. So, for the hyperbola with an exponent of 3 on the dependent variable we have:
This makes y complex so the three individual cube roots functions resulting from each are:
As with the hyperbola in the previous section, when taking square roots it is, of course, common to place the +/- sign in front of the function. I.e.:
The plus and minus signs are actually the two Demoivre numbers for square roots.
and are, of course, the square roots of unity. Whenever roots of a function are taken the Demoivre numbers are what allow each root to be its own separate function. In a sense, they can be thought of as being generated automatically by the operation of taking roots. Taking the cube roots of a function will give three Demoivre numbers. Taking the fifth roots of a function will give five Demoivre numbers and so on. These are then used to graph the separate roots functions.
Going back to our three cube root functions of the hyperbola:
The three Demoivre numbers or the three cube roots of unity have the following values:
The three roots graphs then become:
First, an enlarged view of the “elliptic” segments of the three graphs for the interval:
And then extending the interval to:
Note the segments of the graphs that appear on the front, real plane (FRP). They are the “elliptic” segment of the red graph and the “hyperbolic” segment of the black graph. Other than the bifurcation point no part of the blue graph appears on the FRP.
In two dimensions, the real x, y plane would show:
Note the resemblance to a “Mordell Curve” which will be discussed in a subsequent section.
The Imaginary Semicubical Parabola
Just as with the imaginary circle/ellipse and the imaginary hyperbola, the imaginary semicubical parabola appears on the top, imaginary plane (TIP) as follows. The basic equation is:
With square roots, the Demoivre numbers are +1, -1 and these generate the two dimensional graph with the positive root in black and the negative root in red:
And in three dimensions with x only taking positive values:
Just as with the previous curves certain values for x generate imaginary numbers i.e. complex numbers with real part zero, and are not usually shown. In 3Di coordinates we can allow x to take on negative values resulting in the following graph:
The cusp at x = 0 is, we might say, a smoother bifurcation point as the graph transitions from being on the front real plane for positive values of x to being on the top imaginary plane (TIP) for negative values for x.
The imaginary semicubical parabola lying on the (x, iz) TIP, appears in two dimensions as:
The Imaginary Mordell Curve
Adding a constant term to the semicubical parabola results in a Mordell Curve:
taking square roots and switching to 3Di coordinates gives:
When C = 0 the semicubical parabola is the result. When C is positive the “elliptical” segment of the graph lies on the FRP. When C is negative the “elliptical” segment of the graph lies on the TIP.
The positive root is in black and the negative root is in red. For each value of C, the 3D view is shown first and the real plane view is shown second:
C = 0
C = 1
C = 2
C = -1
C = -2
Animation Varying the Constant C
Imaginary Elliptic Curves
Elliptic Curves are Mordell Curves with an added linear term:
The breaks that are seen in some of the usual graphs of Elliptic Curves where the curve appears to be in two completely separate parts are because those graphs only show the FRP. The breaks in the graph are where the graph has transitioned off the FRP and onto the TIP, or the other way around. As with the previous curves part of the graph lies on the top, imaginary plane and part of the graph lies on the front, real plane.
Taking square roots and switching to 3Di coordinates gives the two roots functions:
As before, the positive root function is in black and the negative root function is in red,
with:
Notice the graph appears to be in four segments, and that the two functions meet at the three vertices or cusps. The two functions are on the same plane together in each segment and cross over one another at the vertices. (For a better view, this graph is oriented positive horizontal to the left.)
In two dimensions below are the FRP real values with each of the two functions (red and black) contributing the real segments.
In two dimensions again, overlaying the TIP view with the FRP, the blue and aqua functions contribute the imaginary segments that are orthogonal to the red and black functions which contribute the real segments.
Elliptic Curve Nonlinearity
As defined previously, nonlinearity here means exponents n other than one on the dependent variable:
So in 3Di coordinates with, for example, n = 5 gives the five roots functions:
The five Demoivre numbers would be the five fifth roots of unity. In decimal and TIP graph form they are:
And the five roots graphs in three dimensions:
Skewed Elliptic Curves
Skewed elliptic curves represent an asymmetric variation of elliptic curves wherein the two functions are not mirror images of one another. Additionally the line segment which connects the axes of the graph does not lie on the x-axis but has a slope.
This can be generated by using the concept of nonlinearity and extending it to a quadratic form for the dependent variable. (It can also be extended to a cubic or quartic form.)
Using an equation like:
and, applying the quadratic equation for y:
with:
and coefficients:
will move the “loops” off the usual planes and out into space, plus the graph is given a slope.
Changing the coefficient c alters the ‘skew’ of the curve. The two dimensional real plane graph is on the left and the three dimensional graph is on the right for some values of c. The 3D graph is oriented positive to the left to compensate for the graphing software’s difficulties at the cusps:
c = 1 (the above example)
With c = 2:
With c = -2:
Imaginary Hyperelliptic Curves
Whereas elliptic curves are of degree three for the independent variable and of degree two for the dependent variable, hyperelliptic curves are of degree greater than three for the independent variable. For example:
Positive root in red, negative root in blue, positive x to the left
Front real plane view:
Top imaginary plane view:
Here is another example using a polynomial of degree six for the independent variable:
Front real plane view:
Top imaginary plane view:
The imaginary Parabola, Quadratic Nonlinearity and Roots Graphs
As is well known, under certain conditions the solutions to a 2nd degree polynomial equation give real numbers, and under other conditions the solutions give complex numbers.
For example:
will give the real number solutions:
while:
gives the complex number solutions:
In the first example the polynomial equation:
gives the graph of the function:
And, it is easy to see that the two roots are real and equal to:
They are, in fact, located at what are usually called ‘the zeros’. And, this corresponds with the original equation being set to zero and having two solutions or roots.
The second polynomial equation equation:
meaning the function:
gives the graph:
The graph itself does not intersect the x-axis, and so “the zeros”:
do not fall on the x-axis and do not fall on the real plane at all.
In 3Di we know exactly where these roots do fall. Notice the red dot:
The red dot represents the two roots that are located at:
They are not on the real plane and are “the zeros” of the equation:
Both roots show up if we look at the TIP:
Polynomial equations such as these can be a little confusing due to the fact that when solving for x, it represents the solution(s) of the equation, rather than input to a function. So, in the function :
the input is x and the output is y but in the polynomial equation:
x is the solution and input and output, as such, do not strictly apply. But, this is actually somewhat untrue, and grounds for further confusion, because the equation:
is identical to the function:
If we apply the quadratic equation to the above polynomial:
it becomes:
Now here is the interesting part: if we, similarly, apply the quadratic equation to the whole function, we have:
And then, if we find the solutions, x for each value of y , we will be building the three-dimensional parabola backwards from y to x, rather than forwards from x to y, and the notions of input and output have actually been reversed!
This means that y is the input which specifies where the solutions are located for that value of y. Consequently, x is, in this sense, the output of a multivalued function of y. Graphing the parabola in this way allows us to include the portion of the graph that is below the vertex which lies in the third dimension.
As an example, if we were to graph the roots of some successive polynomial equations that differ only by a constant, for example:
this would be the same functionally (multivalued) as y being the input and x being the output, as in:
In the 3Di top view, this would look like the following:
(Colors denote pairs of solutions.)
y = -2 in red,
y = -3 in blue,
y = -4 in black.
In FRP these would look like the three graphs:
If we extend this idea to allow y to take on all values, then the three-dimensional graph includes both the portion above the vertex and below the vertex, and looks like:
’Rotate Quadratic Roots Graph’
From the above, we can see that in the usual solving of a polynomial equation when we find the roots, which is equivalent to finding the zeros, – we are in actuality sliding the, would be, parabola (were we to graph one) up and down in accordance with the constant C, since the dependent variable is always set to zero in a polynomial equation. In so doing, we often find complex numbers.
But, in graphing the function itself there are no complex numbers generated, and so the graph, seemingly, “does not exist” below the vertex and so only lies in two dimensions on the real plane.
When we reverse the roles of x and y by using the quadratic equation in this three-dimensional way, if complex numbers are generated, then the imaginary part, along with x, becomes part of the output and will lie on the iz axis. This gives us:
which, as said, reverses the input and output roles of x and y in the usual functional equation:
and this provides the imaginary numbers for a three-dimensional graph.
The quadratic equation, of course, gives two solutions and so we have the two functions:
These are the quadratic ‘roots graphs’. The positive one is in red and the negative one is in black:
Imaginary Polynomial Branches
A Cubic Roots Graph
Just as with the quadratic equation applied to 2nd degree polynomials, the procedures for finding roots of cubic and quartic equations will also generate examples of bifurcation, non-linearity and separate roots graphs.
Analogous to quadratic non-linearity, the point in the polynomial roots graph of cubics and quartics where there is an abrupt turn onto a different plane, occurs at what are usually called the ‘turning points’ – the maximums and minimums. Additionally, these turning points, bifurcation points, vertices, etc. are the points at which two or more functions (roots graphs) meet.
A fairly typical cubic function like:
will, for example, with y = -3 generate the polynomial equation:
and will have the three roots:
which can be seen in the TIP, in red, in the following way.
These points for the roots will show up like this by first solving the equation in one of the usual ways, and then graphing , on the vertical as the input y (in this example at y = -3), and graphing x + iz as the output on the horizontal and depth axes respectively. This is the same procedure as for the quadratic roots graphs previously.
In the FRP these three points will be the three zeros of the graph of equation:
Only one of the three points (red dot) is actually on the FRP. The other two (violet dots) are projected to the FRP, and so only the real component can be seen:
Ordinarily, this function has x as input, y as output, and has a two dimensional graph exactly on the FRP. If the function is slightly rearranged to:
we can see that y is the vertical coordinate and, as set, equal to zero.
Then by applying the Cubic Formula(*)(**)(***) and varying y for as large an interval as we desire in this case:
and then also continuing to graph y on the vertical, as above, along with x and iz as above, then a remarkable graph results. The three roots obtained with the cubic formula for each y as input give three separate functions. When they are all graphed the following picture results:
Note first the three separate functions specified by the Demoivre numbers for cube roots;
in blue,
in red and
in black.
Then notice that the same exact graph that is produced in two dimensions on the FRP is also produced exactly on the FRP in three dimensions; but, each of the three separate functions provides only a segment of the FRP graph. Notice also that the blue and the black graphs do not meet each other, whereas each of them meet the red graph. These meeting points are the polynomial bifurcation points, where the individual graphs make their abrupt qualitative changes in addition to encountering one another.
When using the various cubic formula approaches, since there are two square roots for each cube root, there are six separate values generated in this way; and various choices are generally made as to which ones may be discarded along the way as duplicates. While it is true that taken all together, the points on the graphs of the three branches will be the same regardless of the discarded values, in terms of continuity, they are not graphed in exactly the same way. Different choices of values along the way will contribute different continuous segments to the branches. So, it would appear (a more rigorous analysis would be necessary) that there are six different potential functions, even though they will perfectly overlay so as to generate only three distinct lines.
For contrast, here is another set of colored segments that more generally uses the negative square roots in contrast to the positive square roots above. (Note that the graphing software detects discontinuities – the straight horizontal lines. And so, the discontinuities also reflect that, for continuity, certain choices within the calculations would need to be made before hand. This particular software is not programmable in the sense that it can make decisions within a routine.)
For:
meaning the negative square root relative to the three cubic roots; and, for the cube roots:
in blue,
in red and
in black.
If the polynomial has complex coefficients (discussed in a future article), turning points for any given polynomial that has them may lie in space. For example, with:
the local maximum can be seen here to be off the FRP and displaced in the positive iz direction.
When the cubic has only one real root for all portions of the graph, the other two roots will form branches wholly off the FRP. For example:
In this example, the black line is the FRP graph:
A Quartic Roots Graph
Although more complicated, the same basic principles apply to the Quartic.
Graphing a more or less typical Quartic function:
in two dimensions in the FRP:
we can see already where the bifurcation points will be: at the three turning points. These points will be where two or more of the four separate roots functions will meet.
Then, rearranging our equation to place it in polynomial form:
we then allow y to take on all values on some interval and use the quartic formula (here). Using the same graphing procedure as for the cubic above and the quadratic previously, there are four separate roots functions generated. In the quartic formula there are square roots, cube roots and fourth roots, which, of course, makes a too complicated picture to sort out here. So, the following four functions, taken from the quartic formula would only be part of the total picture.
Note, as before, that separate roots functions contribute separate segments to the real FRP graph. (Also note where the graphing software detects discontinuities by virtue of the different 2nd, 3rd and 4th roots and tries to accommodate):
A Quintic Roots Graph
For the quintic below and the sextic following an online polynomial solver was used. Just as before if we take a quintic function:
the two dimensional FRP graph is:
The red horizontal lines enclose the interval:
which is used for the 3Di roots graph below. y takes on values on this interval in half unit steps and a numerical polynomial solver(*) is used to find the five roots which are then graphed at the corresponding value of y.
So with the function converted to polynomial equation form:
and y taking values on the interval in half steps, the quintic roots graph forms as follows:
Each of the dots is a single solution to the equation at its corresponding value of y. The black dots are the real solutions lying on the FRP and the blue dots are the complex solutions lying in space. Note the bifurcation points where the complex solutions and the real solutions meet. These meeting points or bifurcation points will move around relative to the coefficients in the original equation.
A Sextic Roots Graph
Continuing the same process for a sextic we take a typical sextic function:
and generate its two dimensional FRP graph:
As with the quintic above the red horizontal lines enclose the interval:
which is used for the 3Di roots graph below. y takes on values on this interval in half unit steps and the numerical polynomial solver(*) is used to find the six roots which are then graphed at the corresponding value of y.
So converting the function to polynomial equation form:
and letting y take values on the interval in half steps, using the numerical polynomial solver the sextic roots graph forms as follows:
The black dots, again, are the real solutions and the blue and red dots are the complex solutions. In this example the blue dots connect to the bifurcation points and the red dots form wholly independent branches.
Summary
1) The 3Di coordinate system comes about by beginning with the usual real plane and adding to it a third dimension of depth. This dimension of depth, positive to the front and negative to the rear, is then named z and defined to be imaginary. Consequently the coordinates of 3Di are:
2) Functions in 3Di take the form:
and include all of the usual functions which normally have the form:
and, in fact, these usual functions, which lie wholly on the real plane, can be interpreted as a subset of 3Di functions that have imaginary part equal to zero. I.e.
3) The advance of 3Di coordinates is that the input domain of x can be extended to include values that generate imaginary or complex values as output. The imaginary part of the output values, if it exists, is then graphed on the depth or iz axis which then allows for the parts or branches of the graph that exist off the real plane to be included in a complete three dimensional representation of the function.
4) Elementary algebraic functions, to include, conics, polynomials, Mordel curves, elliptic and hyper elliptic curves can be interpreted to have imaginary or complex branches that lie either in space or on the Top Imaginary Plane (TIP).
5) A ‘Bifurcation Point’ is the point on a graph where the graph makes a transition onto or off of the Front Real Plane (FRP) or the Top Imaginary Plane (TIP). It is also the point where two or more roots graphs meet.
6) A ‘roots graph’, typically for polynomials, is generated by reversing the input and output roles of x and y. That is to say y becomes the input and x (more precisely x + iz) becomes the output. The roots are found by:
Where F is either the quadratic, cubic, quartic equation or one of various numerical polynomial solvers. For example, using the quadratic equation:
As y takes on all values, this generates two distinct roots graphs that meet at the vertex of the parabola and result in an imaginary parabola along with the real parabola. For higher degree polynomials the roots graphs may meet at the global or local turning points or form wholly independent branches.
Further Reading
1) The second part in this series will likely be “Complex Slope in 3Di Coordinates”. Very briefly, real slope occurs in the FRP and can be likened to climb or descent of an aircraft. Imaginary slope occurs in the TIP and can be likened to the aircraft’s heading. When the two occur together they generate complex slope. The algebra is a straightforward extension of the usual point-slope equation of a line.
2) See the eBook: “A Three Dimensional Coordinate System for Complex Numbers” for the original material and a wide array of other topics including historical functions, transcendental functions, surfaces and both 4Di and 4Dii animations in time.
3) A gallery of interesting surface graphs from the eBook, with some of their equations, is available on Facebook.
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