# 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 *y ^{2}* 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.”*

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 n^{th} 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 2^{nd} 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 2^{nd} 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* o