The first world solution in the history of Computer Algebra Software, to the Visualization of Higher Dimensons greater than four, was solved by Maple!
- The Maple Algebra Packages are available on Maplesoft's Application Center:
- Quaternions by Michael Carter
- Quaternions, Octonions, and Sedenions by Michael Carter
- The CayleyDickson Algebra from 4D to 256D by Michael Carter
- The C++ Source Code, for the Visualization, is avaiable at the Computer Science Department of Purdue University. Feel free to request a copy. Copyright waiver release delivered to mailbox of Purdue University Professor Aditya Mathur, PhD on Thursday 18 August 2011.
A very exciting break-through that could not be done without the Computer Algebra and programming language of Maple. Maple solved it! This Cayley-Dickson Algebra has never been created on any other Computer Algebra Software except for now, on Maple. The modern physics theories of the universe, relativity, quantum gravity, string theory, M-theory, ang Guage Theory all uses dimensions higher than three dimensions. By providing physicists with visualization of higher dimensions on how space-time-light behaves, with higher degrees-of-freedom, the development of new insights in understanding both the problem and the solution to new physics may be made. The Cayley-Dickson hypercomplex numbers are a generalization of the complex numbers. Fractals will be the method of expressing the visualization of these numbers. There is already a means to visualized ordinary complex numbers (2D) and quaternion numbers (4D) via fractals. With higher dimensional hypercomplex numbers there appears to be no method that does not lose a great deal of information. This new contravariant/covariant technique, in this research, will minimizes the lost of information by a divide-and conquer technique. This contravariant/covariant technique is breaking the problem into stacks of 4D slices of hyperspace. Squashing higher dimensions into three deminsions still loses a lot of information. However, with this new contravariant/covariant technique enough information is preserved to give a quality never thought could be possible. One will noticed fibers, spinning tops, waves, involutions, droplets, splashes, and other twisting structures -- both ridgid and sheer.
What the big deal about hypercomplex numbers? Let us look at a quaternion. In 1824, according to the Abel-Ruffini Theorem there were no current tools to solve a general quinitic polynomial. No one would waste their time searching for something that cannot be found. However, in 1843, a new tool came about. It was William Rowan Hamilton's Quaternion.
For example, the traditional way to factor
y = x5 + x4 + 2x3 + 2x2 + x + 1
is y = (1 + x2)(1 + x2)(x + 1).
However, the Quaternion way to factor
y = x5 + x4 + 2x3 + 2x2 + x + 1
is y = (x*i - j)(x*i - j)(x*i - j)(x*i - j)(x + 1)
for a factor of
y = (1 + 2x2 + x4)(x + 1)
which is also
y = (1 + x2)(1 + x2)(x + 1).
This is a clue that we now have a new tool to find a general solution for quintic polynomials, and higher, via hypercomplex number. Cayley-Dickson hypercomplex algebras are a generalization of the ordinary complex algebra. These abstract algebras are defined according to their dimensions and how the basis components multiply with each other. Each successor hypercomplex algebra doubles in dimensions. Matrices, Banach, Tensors, and Clifford Algebras are all Associative Algebras. Henceforth, Cayley-Dickson and Associtive Algebras diverge after the quaternion algebra. An octonion (8D) is not a Clifford algebra (because an octonion is a non-associative algebra) but an octonion is still a Cayley-Dickson algebra. The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) (see figure 1). The doubling process can continue ad infinitum. The coining of these names were made by Robert P. C. de Marrais  and Tony Smith. It is an alternate naming system, a relief from using the difficult pronouncing Latin names, such as: trigintaduonions (32D), sexagintaquattuornions (64D), centumduodetrigintanions (128D), and ducentiquinquagintasexions (256D). I found that 512D should be named hyper-complex Voudon. And, 1024D should be named hyper-quaternion Voudon. I found no new mathematics after 1024 because higher hypercomplex numbers greater than 1024 are cyclic (they repeat all over again). I do not offer dimensions higher than 256D to the publc Maple Application Center because the mathematics is very slow and time cosuming past 64D. However, I did keep the code up to 256D in the public Maple Application Center. You are invited to used all three of my Maple Applications for your own research and verifications.
Each hypercomplex number has a scalar part and a vector part. These are not the familiar vectors constructed by Heaviside’s and Gibbs’ vector algebra. Heaviside's and Gibbs' Vector Algebra does not have division defined, hence, it cannot be a number (according to how we define numbers). In Cayley-Dickson Hypercomplex numbers, the scalar part is just a real number. The vector part is composed of imaginary basis components with real number coefficients. Each of their vector’s imaginary components is equal to negative one when that basis component is squared. These numbers can all be added, subtracted, multiplied, and may even be divided.
In addition, Cayley-Dickson Hypercomplex numbers are an extension of the concept of numbers. It is found that a real number is a one-dimension number that can be represented on a number line and a complex number is a two-dimension number that can be represented on a plane. Extending that logic, one may also find that one can produce more numbers by continuing doubling the dimensions into higher dimensional spaces.
The hypercomplex number called the quaternion (4D) is a non-commutative division ring algebra. This is what we call a four-dimensional number. Here is an example of a quaternion: 5 + 2i + 3j + 4k. The first term is called the scalar component; it is simply a real number. The other terms consisting of, i, j, k are called the imaginary basis components with real coefficients. All the imaginary components are called the vector part of the quaternion. Heaviside's and Gibbs' Vector algebra uses the same name vector as the quaternion algebra, but with a different meaning. I repeat, although Heaviside's and Gibbs vector algebra and Tensor Algebra are an offspring of Cayley-Dickson Algebra, they are not numbers. In addition, it is impossible to divide a Heaviside's and Gibbs' vector (see figure 2 for multiplication of imaginary bases of a quaternion, which is a true number).
The higher dimensions of the Cayley-Dickson hypercomplex algebras cannot render visualization without a data structure. Hence, a great deal of research was needed to find a data structure. It was discovered that one can used matrices and hypermatrices as the data structure. It is possible to represent all the Cayley-Dickson algebras with an isomorphic mapping representation of hypermatrices.
This new technique, in this research, utilizes an isomorphic mapping of hypermatrices to represent these hypercomplex numbers up to 64 dimensions. This new technique may be used ad infinitum. A hypermatrix is a matrix that has matrices for its elements (see figures 3 and 4). Those hypermatrix elements may also be hypermatrices/matrices as well, and so on. Note, a “m x n x p … x z” matrix is not a hypermatrix. It is just a higher dimensional matrix.
Starting with complex numbers, one loses the permanence of the trichotomy property (less than, equal to, greater than). The permanencies one loses with quaternion numbers are the trichotomy property and the commutative property under multiplication. Furthermore, the imaginary units are anti-commutative under multiplication. Anticommutative means the sign of the imaginary unit changes when one transpose the two operands under multiplication.
If one set the coefficients of the imaginary elements j and k to zero, the quaternion number becomes an ordinary complex number; which are all the algebraic and transendental numbers. One can deduce all of the Real numbers from the quaternions simply by setting all the coefficients of the imaginary units to zero.
It is well known that an isomorphic mapping of a 4x4 matrix over the real numbers can represent the quaternions. One can also represent the quaternions with an isomorphic mapping of a 2x2 matrix over the complex numbers. To make a complete hypermatrix multiplication one uses matrix multiplication, the cross product and the dot product. After one completes the hypermatrix multiplication one will graph from the top row of the hypermatrix (see the top row of figure 3 and figure 4).
The octonions (8D) are neither generally commutative nor generally associative. The sedenions (16D) are neither generally commutative, associative nor a norm division algebra, because they have zero divisors (a × b = 0, where a is not equal to zero and b is not equal to zero).
A unique point to demonstrate is that these algebras may be constructed from matrices and hypermatrices. The quaternion’s imaginary components are usually i, j, and k. And, for the octonions the imaginary components are i0, i1, i2, i3, i4, i5, i6, and i7, where i0 is the Real component and i1 to i7 are the imaginary components. For the sedenions the components are e0, e1, e2, e3, e4, e5, e6, e7, … , e15, where e0 is the Real component and e1 to e15 are the imaginary basis components. This doubling process may continues ad infinitum.
The Contravariant/Covariant Technique
The quaternion (4D) has one unique space, the octonion (8D) has two unique spaces orthogonal to each other, and the chingon (64D), has sixteen unique spaces, which are all orthogonal to each other (because there are intervals of 4D components). Information is still lost with this new technique; however, there is retention of much more information than with the traditional technique by Norton  and Hart  for visualizing higher dimensional space above quaternions (4D).
Starting with the octonion (8D), one has two unique spaces but only one level deep (Real elements only, not a hypermatrix). For the two 4D spaces of the octonion, there is a quaternion matrix. The real components are using the 3-cross and 3-dot products with matrix multiplication (over only the top row). This visualizes the traditional quaternion fractal, which initializes as a sphere ( see figure 5 for 3-cross products).
Then we repeat over the same set of Reals components (the top row only) using the 7-cross and 7-dot products with matrix multiplication; this initializes a 4D-slice of an octonion (8D) to an anti-sphere, which is a tractrisoid. Figure 6 shows an example on how to do a 7-cross product.
The key technique used to make these nybbles is in the multiplication. One formula that can be used is: zn+1 = zn2 + c where z and c are both hypercomplex numbers. Notice that z is squared. To construct the contravariant multiplication of an octonion, one needs both the scalar product and the 3D-cross product. For covariant multiplication one will need the scalar product and a 7D-cross product. However, the last four components (with indices 4, 5, 6, & 7) are set to zero. The 7D-cross product is only used to rotate the space orthogonal to the contravariant space and therefore creates the covariant space. This assures that each 4D slice is also mutually orthogonal.
The name of the first half of the set is coined the name contravariant nybble and the last half of the set is coined the name covariant nybble. The term nybble is borrowed from computer science because one is using 4D slices rather than 2D slices. The terms contravariant and covariant are borrowed from tensor algebra because they are two independent hyperspaces complementing each other. In this case the two independent spaces are orthogonal to each other.
Continuing for the sedenion (16D), there are four 4D slices. This is a quaternion matrix with nested complex matrices as elements (two levels deep). The first two 4D slices are both contravariant nybbles, which uses the 3-cross and 3-dot products for part of their multiplication. The last two 4D slices are both covariant nybbles because they are using the 7-cross and 7-dot products on the same set of elements (of the top row only).
The 3-cross product and 7-cross product allow the spaces to be mutually exclusive or orthogonal to each other. The 3-cross products are used to build the contravariant component nybble spaces. The 7-cross products are used to build the covariant component nybble spaces.
Continuing for the pathion (32D), we are using a quaternion hypermatrix with nested quaternion matrices as elements (two levels deep). The first iteration is with the 3-cross products, to produce contravariant morphs. Next, the second iteration is with the 7-cross products, to produce the covariant morphs (on the same top row of the hypermatrix).
Finally, for the chingon (64D), one must nest three levels deep within the hypermatrix. The hypermatrix is a quaternion (level 1) with quaternions for elements (level 2). In turn, the deepest elements are complex matrices (level 3). One first iterates with the 3-cross, to produce the contravariant morphs. Finally, one then re-iterate, over the same elements of the top row of the hypermatrix, with the 7-cross product, to produce the covariant morphs.
Each of the hypercomplex algebra represents a closed unique system. Hence, the octonion 4D slices are a totally different type of space than the sedenion 4D slices type of space, and so on for all the succeeding hypercomplex algebras.
Nevertheless, one can only visualized three of the seven components of the 7-cross product, hence, one must set the last four dimensions in the 7-cross product to zero. In addition, each of the hypercomplex fractals is a R4 space (4D space) containing Lobachevskian and Riemannian manifolds as the visualization. A Riemannian manifold is a type of surface topology that has a positive curvature (such as a sphere). A Lobachevskian manifold is a type of surface topology that has a negative curvature (such as an antisphere or tractrisoid). A Euclidean manifold has a zero curvature surface topology or simply just flat space.
These fractals, with this contravariant/covariant technique, go up to the chingons (64D). There are limitations within the GPU Cg pixel shading language that will not allow for the routon (128D) and the voudon (256D) fractals using only the GPU’s 32 registers. Later these fractals could be demonstrated with C++ and parallel processing with multiple CPUs. This research is restricted to only using the GPU with the Cg pixel shading language and OpenGL.
Multiplication is the key in the contravariant/covariant technique. The multiplication is composed of the dot and cross products. Notice, in figure 7, the scalar value, r.w is the product of the scalar values of q1 and q2 minus the dot product of the vector values of q1 and q2. When we use the cross product of a 3-vector, we shall call it the contravariant nybble.
To make the covariant nybble we need to use a 7-vector cross product rather than the 3-vector cross product. However, the last four elements add no additional value, therefore, they are discarded. The dot product of the covariant nybble is composed of the sum of eight products (see figures 8, 9 and 10 for examples on visualizing the 3-cross and 7-cross products, and their joining or superimposing of each other).
In the programming code, the traditional method for representing quaternion fractals is by only using the Reals (see figure 7 for a code snippet on how to multiply a quaternion). For the octonion fractal, there is one contravariant quaternion fractal nybble with one covariant quaternion fractal nybble. The w is the coefficient to the scalar component and the x, y, z are the coefficients to the imaginary components. For the sedenion fractal, there are two contravariant nybbles and two covariant nybbles. In the code, one needs to construct the sedenion fractals using the ordinary complex numbers isomorphic matrix rather than the Reals as elements. For the pathion fractal, there are four contravariant nybbles and four covariant nybbles. In the code, one needs to construct the pathion fractals using a quaternion isomorphic matrix as elements. Finally, for the chingon fractal, there are eight contravariant nybbles and eight covariant nybbles. Here, in the code, one uses a hypermatrix composed of a 4x4 hypermatrix containing quaternions matrices, as members, rather than the Reals. The quaternion matrices contain complex number matrices, rather than the Reals. Each nybble is not just a collection of points. Instead, each nybble is an actual quaternion in that particular hypercomplex space. These nybbles are allowed to morph as each of their components is varied. The most interesting feature of each nybble is the visual demonstration of the various Riemannian and Lobachevskian geometrical spaces of each nybble manifold.
To get an idea of how one must use a mechanistic aide to help us develop our intuition of higher dimensions, one can first use the technique with a parabola function curve, which has a maximum. First, one makes 1D slices parallel to the abscissa axis. One calls the 1D slices that are parallel to the abscissa axis the contravariant cuts. In this context, the term horizon is the same as the abscissa axis. The covariant cuts (perpendicular 1D slices to the horizon) are 1D parallel slices to the ordinate axis. If one are denied to see the entire curve, but are allow to see the cuts, then one will see two points slowly moving toward each other as one approaches the maximum of the parabola for the contravariant cuts (by sweeping the plane slices from down to up). In addition, one will see one point slowly elevating higher and higher over the horizon for the covariant cuts on each slice (by sweeping the plane slices from left to right).
Only where the slices meet at the extrema, is where both the contravariant and covariant cuts are equal. In a firstdegree calculus course, one uses a tool, such as a tangent line, to determine if the curve is increasing, decreasing, or at an extrema. Both the calculus and the contravariant/covariant technique provide visualization hints to the user. Nevertheless, when one move to dimensions higher than 3D, the calculus technique seems to fail us for providing visualizing aides. Yet, the contravariant/covariant technique is still programmable, by a computer, for aiding in visualizing. With 4D slices, one first has an abscissa space and ordinate space. Both spaces share the same abscissa hyper-plane. One shall call the abscissa hyper-plane the horizon, and thus a 4D cut parallel to the horizon is called the contravariant cut. One call each set of fours imaginary components a quaternion nybble (see figure 8, 9, and 10 for example of contravariant and covariant images).
In an octonion (8D), one have eight dimensions, thus, one have two quaternion nybbles (one contravariant and one covariant). Although, nybbles look different when rendered, they are both representing the same object from a different perspective. One may have a filet cut of a fish (parallel to the spine) or a steak cut of a fish (perpendicular to the spine). Nevertheless, one has the same fish. Because each quaternion nybble may look so much different from each other, one may understand the story about the three blind men describing an elephant. One blind man believed the tail was a snake. The second blind man believed the trunk was a giraffe. The third blind man believed the belly was a whale. However, they were all describing a different perspective of the same elephant. This new theory mean that the universe is swirling within three space-time-light regions called, Euclidean, Riemannian and Lobachevskian, thus explainng our region appears to be expanding. The theory also mean that we have three realms within each region called, space, time, and light. Yes, Einstein was right, "GOD does not play with dice!" Relativity and Quantum Physics is now unified with the unification of space-time-light. There are 11 hyperdimenions of different sizes: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Gravitiy is 1D. Time is 2D. Space-Time is 4D. Magnestism is 8D. Electricty is 16D. Weak is 32D. Strong is 64D. And, Space-Time-Light is 1024D. Each of the 11 hyperdimensions are not all the same size! Everything in the universe is just a different form of light. Even matter, space, and tme are all forms of light. In addition, each hyper-dimension is completely geometrical using the transcendental number (Pi) and therefore is not a force or an interaction from the geometry point of view. Remember, the universe is not spinning in a void nor is it expanding into a void, it is just swirling within itself like a clusters of galaxies.
cst := space-time-light in becquerel units (Bq)
pi := transcdental number pi
epsilonr := relative permittivity
kappar := relative permeability
r := the first 256 hypercomplex numbers (hyper-scalar r voudon)
i := the second 256 hypercomplex numbers (hyper-imaginary i voudon)
j := the third 256 hypercomplex numbers (hyper-imaginary j voudon)
k := the fourth 256 hypercomplex numbers (hyper-imaginary k voudon)
Each quaternion nybble in each hypercomplex fractal space is describing a different perspective of the same fractal. The octonion space is not the same space as the sedenion space. Furthermore, the tradition quaternion fractal is actually the contravariant nybble of the octonion space initializes as a sphere. The covariant nybble of the octonion space initializes as an antisphere or tractrisoid (see figure 16).
The images in figures 13 thru 24 are snapshots of the fractals morphing in real-time. These images have their own signature appearance and morphs. Many of the morphs appear to be breathing and swirling. For example, figures 14, 15, 18 and 19 show the splitting of two disklike images. These disks can merge into one disk or divide again to as many as three disks. Figure 23 appears to be two hyperboloid morphs. This could require a great deal of research for just analyzing the meaning of these spaces. Other than conjecture and guessing there is currently no understanding behind these spaces and their morphs.
With the quaternion’s Julia fractals, both Alan Norton  and John C. Hart  held one of the four dimensions constant, while the other three dimensions are allowed to vary. Although one still cannot visualize four dimensions, using 4D quaternion Julia fractals allow one to build an intuitiveness on the nature of four dimensional space with real-time ray tracing and morphing. Even with the fourth dimension held constant there is a fifth dimension involved which is also allowed to vary, and that is time. Hence, with three space dimensions and one time dimension, four dimensions are still being visualized.
Norton  and Hart  technique required that the 4D space be divided into stacks of 3D slices. Just like stacks of 2D planes may build a 3D volume space, stacks of 3D spaces should build a 4D space. Norton’s and Hart’s technique becomes cumbersome for higher dimensions greater than 4D. With octonions (8D), one will be forced to hold five dimensions constant. With sedenions (16D), one will be forced to hold thirteen dimensions constant. With pathions (32D), one will be forced to hold 29 dimensions constant, and so on. Because one is losing so much information, this technique becomes more and more useless for higher dimensions other than quaternion (4D) space.
Before we can truly understand multiplication we must first discuss commutative and associative properties of a binary operator. The commutative property of a binary operator means that both operands are free to be either the first and second operand. 3 x 2 = 2 x 3 is a commutative operation. However, the first operand always have the name, multiplicand and the second operand always have the name, multiplier. One may say, “Who cares?” I will respond that we are dealing with mathematics and not English. English has a great deal of flexibility. We love how poets can bring images to our minds with just words. However, English blessing is also English curse. The English language is an ambiguity language. Some statement can have more than one meaning. The reader must use the context of the script to removed vagueness. One may notice that when a lawyer writes a contract that it seems difficult to understand. This language is an ill attempt to remove the ambiguity from the English language. In addition, Latin is used more often because these words have only one meaning without the need for context comprehension. Words used by laypersons usually have several definitions and must be understood only in context. However, for lawyers, who uses Latin words, are clear when writing other lawyers and judges, but are not so clear for laypersons. Mathematics performs an excellent attempt of being a precision language. It is sloppiness on the teacher or the student who make foolish statements like, “who cares?” These people are usually very poor in their ventures with mathematics. One must care and appreciate the language of mathematics so that the communication can be clear between other readers. Let us first briefly look at the division operation, which is the inverse of multiplication. The numerator is also called the dividend. The denominator is also called the divisor. And, the result of the operation is called the quotient. Note that 2/3 is not the same as 3/2. Hence, division lacks the commutative property.
Note that (3/2) / 5 ≠ 3 / (2/5). Hence, division lacks the associative property. If a ≠ b then “a/b ≠ b/a” is non-commutative. And, if a ≠ b, b ≠ c and c ≠ a then “a / (b/c) ≠ (a/b)/c” is non-associative. The point to be made here is that the order of the operands makes a different. A second, point to be made is that the first operand is always called, “the dividend” and the second operand is always called, “the divisors.
Now we can discuss multiplication with more clarity. The first operand is called, “The multiplicand” and the second operand is called, “The multiplier”. The results of the operation are called the product. If you transposed the two operands then the product will still be the same but the multiplicand is still the first operand and the multiplier is still the second operand. In 3 x 7, the 3 is the multiplicand and 7 is the multiplier. However, 7 x 3, the 7 is the multiplicand and 3 is the multiplier.
The multiplicand is also called the scalar. It is convention to make the scalar the lesser of the two numbers. When multiplying two Real numbers then the scalar is the first operand. When multiply with an imaginary number, a variable or a function then the scalar is the first operand and it will be the multiplicand, for example, we say 5 x 7, 5 * j, 5 * y, 5 * f(x) – the five is multiplicand or scalar in this context.
The multiplicand is also called the scalar, the copier, the hopper, adder, and the counter. We are actually counting when we multiply. We are counting by the multiplicand rather than by one. We only count with the intervals of the multiplier. For example, 3 x 1 is counting one taken three times; we have three copies of one being adding. Also 3 x 7 is counting seven taken three times. In English, we just say, “3 times 7” which is not correct in mathematics.
Remember, English is a poetic language and not a precision language. In English, we are looking for the simplest way to say something so that we can keep clarity. In mathematics, we are looking for the precise way to say something so that we can keep clarity. Your elementary teachers are not mathematicians and hence they are looking for the simplest way to teach you something, which is incorrect in mathematics. You only taught one way to do addition, subtraction, multiplication, subtraction, and division which is the simplest route, which is also the wrong route. If you say, “Who cares?” then you have absolutely no understanding of mathematics.
Here are several ways to multiply:
3 x 7 is seven taken three times.
3 x7 is three scaled seven times.
3 x 7 is seven copied three times.
3 x 7 is seven added three times.
3 x 7 is seven counted three times.
3 x 7 is seven hopped three times.
Now, you can get a better idea what multiplication is all about.
Now we can move on to multiplying a scalar by a pure imaginary unit. 3 x j is j hopped 3 times. The correct question is, “What does all that means?” and not “Who cares?” But, first I must interrupt the flow of this publication to interject a very important digression that is not a stereotype. The world of mathematics had been dominated by men. It is considered a club. The math should have precision but the elucidation of the math should be simply. It is very common for two male mathematicians to communicate in mathematics where the layperson cannot understand them. This is unfortunately in our math textbooks as well. A college male math professor will pick a book that has precision to him but also have poor elucidation to his students. That math professor is now a god. Fortunately there are a few women that were able to get into the club of mathematics. They are not interested in keeping the club closed to only the club members. A female college professor actually have great techniques in understanding the precision of the mathematics and also have a great talent in elucidating the material in its simplest way without loss of precision. This is very common among female college professors to be the true teachers of mathematics. There is a bias against women that opposed them in seeking a career in mathematics. I pray that this bias is overcome and that women are allowed to truly teach mathematics as it should be taught. As an Afro-American I purposely did two things:
1) I seek out female college professors as tutors, teachers, and mentors.
2) I sacrifice my grades so that I can learn the material more thoroughly. I even try to learn from the exams rather than try to score high on the exams.
Note, in mathematics beauty is considered a simply solution for a very complex pattern. However, explaining a complex pattern is obfuscation. Explaining the simply solution is truly elucidation. Now, that I have completed my university education, I am now only interested in mimicking these very talented women educators. I seek to boldly go where no man had every gone before―to be a true elucidator of mathematics.
Space is that final frontier where we seek out new knowledge. We first look at a plane. We divide the plane into two equal halves. We slice the plane at the horizon. The axis line that divides the plan is called the abscissa. Abscissa’s etymology first comes from Latin and then Italian. It is a feminine word, which means, “Lying down or Horizontal”. We used this word in an ordered coordinate systems called a Cartesian coordinated system, named after it discoverer, René Descartes (31 March 1596 – February 1650). We divided the abscissa into columns and named these columns after integers. The second term, ordinate, also divides the plane into two equal halves. This term is masculine and its etymology is in Latin and then in Italian. It means “Stand up in order or Vertical rows”. This ordinate axis has notches on it that divides the plane into rows. The abscissa is also known as the “x” rectilinear coordinate of a point. The ordinate is also known as the “y” rectilinear coordinate of a point. In Biology we used the letter “x” for a feminine gender and a broken “x” known as “y” for a masculine gender. Hence, unfortunately only the male gender is consider a proper plane rectilinear coordinate system. However, “z” is used as a deity to represent “GOD” and henceforth only a three dimension coordinate system is consider a complete system: feminine, masculine, and GOD. We used all three coordinates then we used it has length, width, and height or L x W x H. Length is feminine, width is masculine, and height is GOD because GOD is high in the sky. When one look at the ocean from a beach then the horizon is the length across, the width is the distant from the beach to the horizon, and the heaven is the distant from the horizon to the filaments. When we have only two coordinates we let “y” rise. When we have three coordinates we let “z” rise. To reach “GOD” you must rise straight up on the “z” axis to infinity of the Father, move out of the plane (within line of sight) on the “y” axis to the infinity of the Son, and flow to the right on the “x” axis to the infinity of the Holy Spirit. The question is not, “Who cares?” The question is, “What does this all means?” It means that Rene Descartes made a terrible error by using gender and religion to define space. It caused two flaws:
1) It limited space to only three coordinates.
2) It personified space when it should have been completely secular.
The mathematic community wanted to eradicate these religious interpretations out of mathematics. Starting after 1930 all mathematics moves toward secular reasoning only. This was supposed to be successful even until this modern time. There was an explosion of idea that modernized our world called, “Earth”. However, without the religious elements mathematicians have slowly lost their inspiration. Mathematicians do not seek education for learning the mind of GOD. They seek education to do their research or to just get a job. Capitalism is now the new “god.” Although Capitalism is a very good thing for our planet it seems to be the only motivator for learning mathematics. Now, students ask the question, “Why do I need to know this?” Or, student ask the question, “Who cares?” We are now in a 2nd dark age of mathematics. Now you may ask the question, “Who cares?”
Other Related Works
The father of fractal geometry is Benoit Mandelbrot (1924-), who is a French mathematician. Mandelbrot’s work  was inspired by another French mathematician named, Gaston Julia (1893-1978). The first fractal geometry rendering of a hypercomplex number, the quaternion (4D), was by two Americans in the 1980s, first Alan Norton  and then later by John C. Hart . Fractal geometry takes advantage of the unique nature of the imaginary components of the ordinary complex numbers and hypercomplex numbers. Figure 11 and 12 show two typical samples of quaternion Julia fractals.
Conclusion and Contributions
The contribution of this new 4D contravariant/covariant technique is to provide an intuition of higher dimensional space. The excellent work of Norton  and Hart  loses too much information for fractal geometry above 4D. However, with the contravariant/covariant technique, one is trying to take spaces that have higher degrees-of-freedom and restrict them into slices of 4D spaces. Even with this new technique, information is loss. Nevertheless, this does not negate the fact and importance that a great deal of information is still retained.
The two new contributions are the addition of the Maplesoft algebra package  for the Cayley-Dickson hypercomplex algebras up to the voudons (256D) and the visualization of those hypercomplex numbers on the Apple Mac Book Pro.
The discovery of this technique came from studying electron orbitals and Dr. Hanson’s excellent book, “Visualizing Quaternions” .
 Carter, Michael. The Cayley-Dickson Algebras from 4D to 256D. http://www.maplesoft.com/applications. Maplesoft. April 23, 2010.
 Crane, Keenan. Ray Tracing Quaternions Julia Set on the GPU. http://www.cs.caltech.edu/~keenan/project_qjulia.html
 Culbert, Craig. Cayley-Dickson algebras and loops. Journal of Generalized Lie Theory and Applications. Vo1. 1, No. 1, 1-17,2007.
 de Marrais, Robert P. C. Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions.
 Hanson, Andrew J. Visualizing Quaternions: Series in Interactive 3D Technology. New York, NY: Morgan Kaufmann. 2006.
 Hart, John C. and Sandin, Daniel J. Ray tracing deterministic 3-D fractals. Computer Graphics 23(3), (Proc. SIGGRAPH 89), pp. 289-296. July 1989.
 Kotsireas, Ilias S. and Koukouvinos, Christos. Orthogonal Designs Via Computational Algebra. WileyInterScience. May 4, 2006.
 Mandelbrot, Benoit B. Fractals: Form, Chance and Dimension. W.H.Freeman & Co Ltd. 1977.
 Norton, Alan. Generation and display of geometric fractals in 3-D. Yorktown Heights, NY: IBM Thomas J. Watson Research. 1982.