Hermann Schubert Probes the Fourth Dimension

Mathematical Essays and Recreations

Hermann Schubert was a German teacher and textbook author. Mathematical Essays and Recreations is a small collection of articles ranging from the foundations of the number system, the foundations of algebra up to an extensive essay about the fourth dimension.

In the first article: Notion and definition of number, Shubert gives a brief account of how the concept of number evolved within the human mind.

In another of his essays: Monism in arithmetic, the author explores and writes about the elementary rules of algebra and about the importance that mathematical systems can operate with defined rules for any kind of number without making exceptions for some of them. For example, we can define the commutative rule of addition for the natural numbers, as in 2 + 3 = 3 + 2, and the same number should apply if instead, we use natural an imaginary numbers combined, like 2 + 3i = 3i + 2. Following his exposition, we can easily see why the class of complex numbers is such a robust field in mathematics, and how it can be derived from the natural numbers following easy and consistent steps. On the other hand, we can also see why the quaternions—when compared to the complex numbers—lack such popularity and why this class of number is so limited in applications and acceptance.

The third essay in this Datum edition is about the fourth dimension. But contrary to other science authors that rarely touch the fourth dimension from the metaphysical point of view, Schubert is not afraid to confront both doctrines. When dealing with the fourth dimension from the mathematical point of view, Schubert goes carefully from the definition of the point to the definition of dimension to a many-dimensioned space.

What Are Ordered Pairs?

An ordered pair is the intuitive idea that objects can be flipped in different positions in such a way that the order in which we take them can make different entities.

This "definition" may sound a little abstract, but a few examples should bring the idea comprehensibly.

When we think about the basic Cartesian coordinate system of two axes, we immediately think of two "real number lines" intersecting at 90 degrees.

Every plane curve is a set of ordered pairs

The figure above shows an example of how we intuitively use ordered pairs when we plot graphs of real functions.

In this example, The function is any abstract one-one (1-1) rule Y = f (x). When the variable x on the X-axis assumes or takes the value a, then the function f assigns the value b on the Y-axis to that choice x= a on the X-axis.

Hence, we are necessarily and intuitively talking about the ordered pair (a, b). This entity (a, b) is an ordered pair because the function f explicitly and uniquely assigns the value b to the unique value a.

The notion of ordered pair is not limited to the usage of the real numbers.  We can choose the second entry of the ordered pair to be an imaginary number. In that case, the Y-axis is no longer a real-numbers axis, but an imaginary numbers axis. In that case, the ordered pair is simply a complex number.

The ordered pairs are very useful when we deal with transformations, especially transformations of plane figures.

For example, the following transformation made up of two parametric equations:

transforms a circular area of the plane into a  dome in space.

To dramatize the results, a picture of a cat is shown before this transformation, and after the parametric equations are applied to the cat's photo.

A flat image transformed into a 3D-image.

In this example we are implicitly using triplets, that is, ordered pairs of three entries, like (x, y, z). The first two entries of the triplet are for the locations of the points of the cat's photo, and the third entry of the triplet is for the amount of "deformation" applied to each point of the photo.

Transformations and ordered pairs are very interesting subjects because they are not so abstract after all.

Interested in more examples of transformations as in the example above? Then download this free E-Book:

The Golden EBook of Graphs
of Mathematical Functions.

Interested in an in-depth development of the foundation of the complex numbers from the standpoint of the ordered pairs?

Foundations of Transcomplex Numbers.

Complex Numbers: How Complex Are They?

The "history" of the integer numbers is a simple one. From the natural numbers 1, 2, 3, ... we move to the positive integers 0, 1, 2, 3, ... then we add the negative integers ... -3, -2, -1, 0, 1, 2, 3, ....

Then we escalate to fractions and decimals and non-terminating decimals (although historically was not in this order). The ladder continues to the irrational numbers and to the algebraic and transcendental numbers. This is the "world" or universe of the real numbers.

Every real number has a specific place on the Number Line.

But mathematics is a product of our minds so this "universe" or field can be further expanded to suit our needs.

The next heaven after the real numbers field is the imaginary numbers; numbers that in combination with the reals make the complex numbers field.

But how complex are the complex numbers? Curiously, they are as simple as the "preceding" ones.

The negative numbers haunted the mathematicians and philosophers for many centuries; no wonder the misnomer "negative". Even the number zero took a long time before it was accepted in the kingdom of the mathematics (in Europe, where it was later accepted.) It was unacceptable to count "backward".

The imaginary numbers suffered the same fate: no wonder the epithet of "imaginary". The square root of minus 1 was impossible to compute because no number times itself is equal to minus 1.

Take a read at this article: "The imaginary numbers are not so imaginary and the complex Numbers are not so complex" and you will see how easily and beautifully the complex numbers emerge out of the real numbers.

Transcomplex Numbers

Foundations of Transcomplex Numbers.

Integer numbers, negative numbers, fractions, real numbers, transcendental numbers, irrational numbers, and imaginary numbers are a few of the number types we usually find in mathematics.

Is there no end to this? Is there no "final" type of numbers?

From the standpoint of number fields, all of them can be encompassed into one type called the complex number field.

Wikipedia, in a short background, mentions how the complex numbers emerged:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

Complex numbers can also be understood and developed from the standpoint of view of ordered pairs. Today, developing the complex number system from the foundations of set theory and the concept of ordered pairs is possibly the most intuitive approach we have at hand.

For a rigorous development of the complex number system download the free EBook: Foundations Of Transcomplex Numbers: An Extension Of The Complex Number System To Four Dimensions.

This mathematics book is about a way of extending the complex numbers system to four-coordinate variables, maintaining the usual operations attributed to the complex numbers.

Foundations ... is a fully illustrated EBook. See --and Click-- for example, the following figure about how to multiply two ordered pairs:

Transcomplex numbers are an extension of
the common complex numbers. 

Complex numbers are usually plotted using the familiar plane Cartesian coordinate system, but transcomplex numbers are four-entry ordered pairs, also called 4-tuples, so they belong to a four-dimensioned space.

In a nutshell, transcomplex numbers are complex numbers whose elements are ordered pairs.

In the following simple illustration, also taken from Foundations ... we can see that out of a four-entry complex number system we can extract four 3-dimensional spaces like "ours".

The transcomplex numbers need a 4-dimensional
coordinate system to be represented.

The chapters of the book are divided as follows:

• Ordered Pairs. The whole theory of transcomplex functions is based on the ordered pair concept: from the two-dimension plane up to the four-dimension space.
• Complex Numbers. The complex numbers system is derived from the ordered pair's concept.
• Transcomplex Numbers. Here starts the extension of the complex numbers into ordered pairs of complex numbers, arriving at the concept of transcomplexs.
• The Coordinate System S4. This chapter is devoted to deriving a suitable coordinate system to plot transcomplex functions.
• Transcomplex Functions. Functions of complex variables evolve to make space for functions of four-entries ordered pairs.
• Transcomplex Surfaces. A radical and totally new perception of surfaces generated by complex variables.
• Theorem Proofs. This chapter collects all the proofs of the theorems stated along the book.