Mathematical Preliminary Calculations and Algebraic Properties of an Object

The Beginning

The study of mathematics has been evolving since the prehistoric times when people used stones, marks, and/or knots to count their herd. Egyptian mathematicians made the next advances in 3000 B.C. with the development of a decimal system that did not include a place value (Funk & Wagnall, 2017). Egyptians were pioneers in geometry with the creation of formulas for area and volume. These formulas would prove instrumental when they built the pyramids. By 600 B.C., the Greeks made additional advances with the concept of logical deduction. Pythagoras, a Greek philosopher, believed that everything was understandable as a whole number or ratio thereof. This was deemed incomplete after the discovery of irrational numbers. Euclid, a Greek mathematician, created an entire geometry system by way of abstract definition and logical deduction in 300 B.C. (Funk & Wagnall, 2017). Archimedes, also a Greek mathematician, calculated a highly accurate value for pi in 200 B.C. (Funk & Wagnall, 2017). The Greeks also introduced three dimensions.

Geometry

The previous advances may seem inconsequential, but they were the stepping stones for the 17th century mathematicians who were not satisfied with the generic geometry mathematics. Gottfried Wilhelm Leibniz was the first to have the idea of a higher level of mathematics which we would later know as topology (Drucker, 2013, Background & History, para. 1). The math of choice of his time was Euclidean geometry, but he wanted something more. Euclidean geometry dealt with lines, circles, and simple shapes and the sum of a triangles angles equaled 180 degrees (Casti, 1996, p. 51). Unfortunately, it would be the 18th century before anyone would put the first application of topology-based problem solving to use.

The Founder

Swiss mathematician Leonhard Euler wanted to know whether one could traverse all the bridges in a particular town without repeating. This was known as the famous riddle of seven bridges of Königsberg (McElroy, 2005, p. 88). Euler would use the polyhedra in his studies developing topology. A polyhedra is a basic shape, i.e. a cube, meaning many faces. Euler developed a formula relating the number of edges, faces, and vertices of a polyhedra which would be known as the Euler characteristics (McElroy, 2005, p. 88). He was credited as having founded mathematical physics and having authored topology. Euler also created the graph theory. This is not the graphs we are familiar with that gives points on an axis, but instead is more of a relationship between objects. The relationship models between objects can have issues such as the distance cannot be defined using typical geometry but instead by the number of edges required to get between the points. The questions of these problems can be answered using algorithms, but the answer can get very lengthy. Nearly a century later, Euler's graph theory was followed up on by way of chemistry. It was used to show the transformation of one substance into another which is similar to how topology works. Euler was considered to be a great mathematician who could work anywhere under any condition. His determination to prove himself as one of the greats would cost him dearly. Several leading mathematicians spent months unsuccessfully trying to solve an astronomical problem in which Euler spent three straight days. That prolonged time cost him his eye sight in his right eye, but he proved himself to be the better mathematician. Euler's work with topology would lay dormant until the 19th century when another famous mathematician would pick it up and make great advances in the field.

Transformations

During the 19th century, various geometries (euclidean, noneuclidean, analytic, affine, projective, etc.) were introduced and studied. Felix Klein developed the Erlanger Program in 1872. It essentially said that each type of geometry was different based on how the objects studied handled particular transformations. Klein needed to isolate the propositions that were fixed or invariant when given a transformation. 'Euclidean geometry is set apart from other geometries by properties that remain unchanged when objects in the plane such as circle/triangles are subjected to rigid motions' (Casti, 1996, p. 51). Klein felt that to look at two objects on paper, we can't necessarily compare them. Using a group of transformations such as those that would later be used in topology, two objects can be transformed one to the other.

Topology

According to Drucker (2013), 'Topology is a mathematical study of the relationships and properties of objects in any number of dimensions, independent of size' (Summary, para. 1). It is often referred to as 'rubber sheet geometry' which simply means that it has characteristics that can stretch and bend like rubber (Definition and Basic Principles, para. 2). Although topology can be characterized as a type of geometry, size and measurement are not important like in geometry. The formulas and information to solve problems involved with topology can come from geometry, algebra, and/or calculus. While Euler's focus for studying topology was the polyhedra; the tetrahedron, octahedron, dodecahedron, and icosahedron are also shapes that are commonly associated with topological studies. It uses tools such as groups and rings to describe and classify objects. Triangles and footballs are examples of non-topological objects. A doughnut and a straight line are topological because the transformation meets the one to one requirement. For objects to be considered topologically equivalent, they must be deformable one to the other via a continuous transformation. Topology can be a common link to describe the relationship and/or studies between the various levels of mathematics and various levels of sciences. This relationship between mathematics and sciences would lead us to dimension theory. Physicists based theories on space time of four dimensions. Over the years, fractional and infinite dimensions have been introduced. In 1970, string theory was introduced which conceives of the universe having 10, 11, or 26 dimensions (Crilly, 2007, p. 99).

The Follow Up

Another famous mathematician who contributed to topology was Henri Poincaré. He didn't believe that a human being could understand or do quality work in more than two of the four divisions of mathematics – arithmetic, algebra, geometry, or analysis; without some mention of astronomy and mathematical physics (Bell, 1937, p. 527). While he had a great mind, it worked quicker than his mouth was able to speak and his fingers were able to manipulate and/or write. He once took the Binet test and had he been tested as a child instead of a famous mathematician; he would have been deemed an imbecile (Bell, 1937, p. 532). In the 19th century, Poincaré, began to shape and influence several areas of mathematics to include but not limited to algebraic topology (McElroy, 2005, p. 209). His studies led to different branches of topology. Algebraic properties of an object combined with geometrical properties, gave geometrical topology its beginning.

The study of sets as a collection became known as point set topology. Point set topology is defined as a closed set if it has boundaries and an open set if it does not. The intersections and combinations created the definition of open or closed set. According to Drucker (2013), 'Poincaré's study of physical processes in which classical mechanics seemed unable to describe the results, gave rise to nonlinear dynamics. This is one of several branches contributing to the development of the Chaos Theory nearly 100 years later' (Background and History, para. 2). Poincaré is known for his 1895 paper, Analysis Situs (site analysis), which was a systematic treatment of topology. This paper along with many others over the next decade would continue to develop the area of algebraic topology. Algebraic topology was considered abstract discipline by the 19th and 20th centuries, but before that not all equations could be solved using standard arithmetic. The creation of theories, additional algebraic language, and corresponding laws to real numbers helped to identify and characterize objects. The various types of objects lead to different levels of classification. This information carved the path for exploring the world of geometry. 

Poincaré is most famous for his conjecture of a connected, closed, three-dimensional manifold. It basically says that each object is topologically equivalent to a three-dimensional sphere. This means it can be continuously deformed without tearing (McElroy, 2005, p. 210). According to Crilly (2007), Poincaré's conjecture in three dimensions was solved by Grigori Perelman in 2002 (p. 93). According to McElroy (2005), no one has solved the Poincaré conjecture in three dimensions to date (p. 210). Although one book is approximately two years older than the other, the publishing date is well past the 2002 date of when Perelman supposedly solved the conjecture so it could lead one to question the validity of one or both of these authors. Poincaré's work continues to be an influence in modern studies of topology and geometry. Prior to his early age death, he was elected to the Academy of Sciences. He is the only member to have been elected to all five sections-geometry, physics, geography, navigation, and mechanics (McElroy, 2005, p. 211).

Descartes

While several mathematicians contributed directly to topology, there were dozens more that were not even mentioned. There were so many advances in mathematical history that have happened since the prehistoric times to present and it's still evolving. There will always be a 'genius' who can work problems in their head or someone who finds answers an easier way than others. René Descartes was a French mathematician who was said to have created geometry (Bell, 1937, p. 54). He is known for introducing what we know as x, y, and z for unknown quantities as well as the powers of a variable (McElroy, 2005, p. 76). As a young man at Jesuit college, the rector believed he required more rest than the other boys, so Descartes was allowed to sleep in each morning until he was ready to join his peers. This extra time spent in bed is where Descartes believes was the source of his philosophy and mathematics (Bell, 1937, p. 36-37).

Education & Career

Anyone looking for a career in topology can do so by way of focusing on mathematical and/or science-based studies. Careers in engineering, architecture, surveying, astronomy, and navigating are strong mathematical based studies. Areas such as chemistry, physics, economics, and psychology/sociology are strong science-based studies. According to Drucker (2013), It is recommended that one study as many levels of mathematics as possible if considering a career in topology. Upper level algebra will help with some of the language used in the field of topology. A doctoral in mathematics and/or computer science is best if one is looking at a specific career in topology. (Careers and Course Work, para. 1). Another option would be to apply with the National Security Agency (NSA). The NSA employs various types of topologists, but cryptanalysis is possibly the most common. Cryptanalysis is a type of code breaking. The NSA also employs physicists and cosmologists, but they are generally used in an academic setting.

Conclusion

The changes of topology and advances have brought frustration over the years. The beginning said classical math was sufficient; set theory was the result of paradoxes; abstract algebra was discipline without content; and catastrophe theory didn't live up to its name. While the changes may have brought frustration and questions, the brilliant and often misunderstood minds of Euler, Descartes, and Poincaré played a pivotal role in contributing to the mathematics that we use today. Today, various types of topology are in use in universities all over the world.

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