Blogpost #6: Topology, what the heck is that?

It's amazing to me: I am close to graduation with a degree in pure mathematics, and this is the first time (at least that I can recall) that I have heard information about the subject Topology (I am sure that there are many other subjects in math that are typically not discussed/covered in great detail at the undergraduate level). Like many other disciplines in math, Topology (seems to) originate from a mathematician who was trying to solve a particular problem, but maybe did not have the correct tool or approach needed to solve the problem, and while working on the problem, a new subject is created! Here we will take a look at the beginnings of Topology, the two major sub disciplines of Topology, and what types of creations/technologies have been created and might still be used today because of Topology.

Perhaps the first work that is usually considered to be the beginnings of Topology was Euler's work on the "Bridges of Konigsberg" problem. (Here is a link to the specifics of the problem  https://nrich.maths.org/2484 ). Euler wrote about his solution to this problem in his paper that was titled "  The solution of a problem relating to the geometry of position". The geometry of position?!? This was a new way of thinking about geometry: distances and length measurements are irrelevant. Not only did Euler prove that the desired trip in the Konigsberg bridge problem was impossible, he realized what would be necessary in order for this type of trip to be made, in a general case, regardless of how many "bridges" and "land masses", or in modern day math, how many vertices and edges a graph or map has; Euler determined that the only way the Konigsberg path could be done, is if there was exactly two land masses that had an odd number of bridges connecting each land mass, respectively, to other land masses. In modern graph theory, this is stated as the following:

 "A graph has a path traversing each edge exactly once if exactly two vertices have odd degree." 

Euler's creation of this solution is the beginning of Topology.

There are two major sub disciplines of Topology: general, or point-set, topology, and the other is called algebraic topology. 

General topology deals a lot with the congruence of two or more objects to each other. In particular in topology, two or more objects are equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while tearing and gluing together parts is not allowed. This link to an animation shows again what was discussed in my MTH 495 class, showing the equivalence between a doughnut and a coffee cup (now tell me how easy it would be to see this congruence with just basic geometry!)

https://www.britannica.com/topic/topology/images-videos/Because-both-a-doughnut-and-a-coffee-cup-have-one/90829

The coffee cup and doughnut are examples of objects that are simply connected, meaning we can continually shrink and morph any portion of the cup or doughnut into the other without running into any problems. There are other objects, like the torus, that are not simply connected. lets look at the following picture as to why: 

Here we can see, that loop c can be shrunk down to a point without any problems, while loops a and b cannot because they are affected by the torus's hole.

Euler had derived a formula for the polyhedra, that is known as the Euler Characteristic, which states that

v - e + f = 2

where v is the number of vertices, e the number of edges, and f the number of faces. Thisformula was used to prove many theorems dealing with polyhedra's. However, a mathematician in 1813 named Antoine-Jean Lhuilier noticed that Euler's formula did not work for solids that had holes in them. Lhulier's adaptation to the Euler Characteristic was 

v - e + f = 2 - 2g

Where g is the number of holes in the solid (source from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology_in_mathematics.html ). This was the first known result of a topological invariant (meaning the object stays the same no matter if we continuously deform the geometric object) 

The real importance of topology from a mathematical perspective seems to be the consideration of objects in higher-dimensional spaces, or even, in particular abstract objects that are sets of elements. Topology focuses on space and spacial functions, determining of objects are isotopic to a larger space (topological equivalence). This is why Topology is considered one of the abstract fields of mathematics (along with abstract algebra, analysis, and others). 

Algebraic Topology uses tools from abstract algebra to study topological spaces. The goal 

is to find Algebraic invariants that classify topological spaces up to homeomorphism (objects can be deformed into each other by a continuous, invertible mapping), though most classify up to homotopy equivalance (a class below homeomorphism. Does not satisfy invertible mapping). I think of these as classes from abstract algebra; a homeomorphism invariant is a field, and a homotopy equivalance is a integral domain (or less, I am not completely sure) Where if we recall from abstract algebra that integral domains do not satisfy the Multiplicative Inverses Property, while Fields do, which seems to some coincide with the idea of homeomorphism and homotopy invariants. Very interesting stuff!! of course I not even really skimming the surface of Topology, this was just a little bit of insight  I have learned about the subject.

Modern day Topology is used in many fields today, from Cosmology, to DNA research, to Crystallography  (used by chemists) to even Networking (along with Graph Theory, ya know, things like Facebook, twitter, etc) as well as medical imaging software and technology, just to name a few. 

What an interesting topic Topology is!

Blogpost 5: The inception of Graph Theory

The beginnings of Graph Theory is usually traced back to the mathematician Leonhard Euler, who in 1735 solved the famous "Konigsberg bridge problem". The question at hand was " is it possible for the people of Konigsberg to walk around the city, crossing all seven bridges only once, and the person or persons walking do not have to end up from where they started walking from".

(A map of Konigsberg and the seven bridges (in red) connecting the land, picture from www.maa.org )

Euler thought of this problem as trivial, however he stated that
 " This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it" (www.maa.org/leonard-eulers-solution-to-the-konigsberg-problem).
What Euler did from here was one of the many reasons he is considered one of, if not the greatest mathematician of all time.
Euler determined that choosing a route within each land mass was irrelevant. The only important aspect of route choice was the sequence of bridge crossings. This thought process allowed Euler to reformulate the problem in abstract terms (laying the foundations of graph theory), eliminating all features except the list of land masses and the bridges connecting them (Pickover, C, "The Math Book). Euler replaced the map of the city by a much simpilar diagram showing only the main features. In modern graph theory, this diagram is further simplified using "dots", which are referred to as vertices (or nodes), and line elements that are usually referred to as edges. Below is a graph that represents Euler's new viewpoint of the map and a modern graph theory representation.

                                                                         (Euler's representation of Konigsberg city)                                    

                                           (modern graph theory representation.image from  www.kurzweilai.net)   

The relationship between vertices and edges is called a graph. Euler felt that this problem was one that was related to a topic that was once discussed by Gottfried Leibniz called "geometry of position". This so called geometry of position that Euler used is what is referred to today as Graph Theory. Today, graph theory serves many purposes;finding communities in networks (for example, social media networks like Facebook and Twitter), GPS systems in order to find the shortest route home, postal services, and many other purposes including many other applications in computer science.
We cannot discuss the inception of Graph Theory without taking a very simplified look at Euler's proof of  the Konigsberg bridge problem.......

Euler realized that, excluding where the person ends their walk, whenever one enters a vertex by a bridge, one leaves a vertex by a bridge. In graph theory lingo, during any walk in the graph, the number of times one enters a non-terminal vertex equals the number of times one leaves it. Now, since every bridge has been crossed exactly once, for each land mass that was reached (excluding the ones that were chosen for the start and finish of the walk), the number of bridges touching that land mass must be even. However, all four of the land masses in the original problem (see picture above) are touched by an odd number of bridges (one is touched by 5 bridges, and each of the other three is touched by 3). Since, at most, two land masses can serve as the endpoints of a walk, the proposition of a walk crossing each bridge only once leads to a contradiction, and therefore, the walk is not possible.

Euler did figure out what layout would be required in order to complete such a walk for any number of bridges and land masses. Euler figured out that in order to complete a walk of this desired form, a graph must be connected and have EXACTLY zero or two vertices of odd degree, meaning that out of all of the land masses and bridges that connect them, there can only be two land masses or zero land masses that are connected by an odd number of bridges for this walk to be possible. Pretty dang awesome!

Book Review: Journey Through Genius

The book I chose to read for class was titled "Journey Through Genius" written by William Dunham. The first thing that stuck out to me was how closely this book followed our lectures and topics in class. It was not until our group book discussion that I found out that this book is used as the textbook for the other MTH 495 sections.

If you are someone that does not really enjoy reading and working through proofs of theorems and propositions, then I would not recommend this book for said person. However, if you enjoy that aspect of math, this is a great book in my opinion. With that being said, the book is not all "proofs". There is a great amount of historical information and stories that really describe and help distinguish many of the individual thoughts and minds of the greatest mathematicians the world has ever seen, and their thought processes and approaches to creating proofs.

This book flows smoothly in my opinion, and transitions to the next great mathematician/s according to either timelines of existence or particular work that they have done that somehow is introduced or is recognized to have something to do with the work that was described in the previous chapter. For example, the first chapter starts off with Egyptian mathematics ( speculated around 2000 B.C) and Thales (546 B.C) and the last chapter, chapter 12, ends with Cantor (1891), so the book does follow a somewhat chronological order of the history of mathematics dating back to the Egyptians.

Probably the one and only complaint I have with this book, which for me seems to be the same complaint I have for the few other proof-based or math-based books I have read, is that the author, like many others, will refer to a formula or drawing that is on a different page. For example, the section that discussed the Pythagorean Theorem: within the first couple of pages, the author referred to the very famous sketch, but the sketch was not visible until a few pages later. Or in other chapters, when working through some of the proofs, the author will refer to equations and formulas that were either several pages earlier or several pages ahead. I understand that this is probably a pretty common practice, but in my opinion, if you are going to explain something as in depth and as complicated as a mathematical proof, and you are going to right off the bat refer to a picture or formula, it should be visible on that page. Again, just my opinion.

I can also say that reading this book, along with the discussions in class, has really gotten me intrigued about many of these individuals that I might have heard of before in a previous math course, but did not really know anything about. The coolest part is seeing how these individuals thought about problems and their approaches to solving these problems. It is fascinating to realize that there are many areas of mathematics that have literally been created from people trying to solve or prove other propositions or postulates (for example, the creation of non-Euclidean Geometry, which was "invented" when people were attempting to prove Euclid's parallel postulate). If you like proofs, read this book.

What is Math?

Hello everybody! I do apologize for the delay, I am definitely behind on the social media thing. My name is Nick, and I work for a tier 1 automotive supplier as a welding engineer/Technical trainer. I recently married and my wife is pregnant with our first kid (kids, twin boys!). As far as education goes, I have been working on completing my BS in pure mathematics for a LONG time. Over the years my job has required me to travel a lot, so there had been several times that I had to give up school for work. On the positive side, I have been all over the world several times, and have spent extensive time in places like Plsen, Czech Republic, Kunshan, China, Quertero Mexico, Istanbul Turkey, and Nagoya, Japan. My last semester will be this Fall, I have MTH 408 (advanced calc) and then I will be done! I will be enrolling in Western Michigan's MBA program starting the winter semester.


The Syllabus seems pretty straight forward to me, I just might still need some help with the social media stuff :) .


The book I have chosen to read first is titled "Journey through Genius."


The question; what is math? Well, my personal opinion, math, in my mind, is a thought process  that is used to problem solve or innovate. I do not know much about the true history of mathematics, but I think that the "pioneers" of math, did math, in abstraction. I do not think they were solving problems, just thinking about "stuff" critically and in a scientific manner. Then, they learned to apply that logic to their real worlds. Again, just my opinion.


Some of the biggest moments or discoveries or milestones in math, in my opinion, would be things like the Riemann Zeta Function (which is still part of one of the millennium problems the "Riemann Hypothesis"), of course the Pythagorean Theorem, the acceptance of zero as a number,  Circle definitions (circumference, diameter, radius, area, etc), and the Fundamental Theorem of Algebra.


I am truly looking forward to learning the historical perspectives of many of the great mathematicians and possibly tying together many of the disciplines that we have learned!

Working with Cubic Equations

In order for us to trace the history of the solution of the cubic equation, we must look back to ancient Egypt. The first solutions to cubic equations at this time were geometric, which is fitting since ancient Egyptians and Greeks new very little to nothing about algebra. The original cubic equations resulted from problems in land measuring and especially from the duplication of the cube.

Hippocrates of Chios was probably the first person to show that the duplication of a cube could be reduced to finding two mean proportionals between a given line and one twice its length ( Gardner, "A Brief History of Equations"). Hippocrates and others (Menaechmus and Archimedes) were believed to have come close to solving the problem of doubling the cube using intersecting conic sections. Later in the 7th century, it was a Tang dynasty mathematician Wang Xiaotong that solved 25 cubic equations of the form

But Xiaotong could not solve the cubic for the general case, for all values of p and q.

Then, in the 11th century, Persian mathematician Omar Khayyanm made giant strides in solving cubic equations. He discovered that a cubic equation can have more than one solution and stated that it cannot be solved using just a compass and straightedge construction (which was a common methodology used to prove propositions during that time and previously). Omar did find a geometric solution to the cubic equation 

Here is a specific example of Khayyam's geometric solution of a cubic equation.

In the 12th century, another Persian mathematician, Sharaf al-Din al-Tusi wrote the "Treatise on Equations", which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what became known as the "Ruffini-Horner" method to approximate the root of a cubic equation. It was also during this time that the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations was realized by Sharaf.

In the early 16th century, the Italian mathematician Scipione del Ferro (1465 - 1526) found a method for solving a class of cubic equations, those of the form 

.

Unfortunately during this time, negative numbers were not known to del Ferro, or else he probably would have realized that all cubic equations can be reduced to the above form if we were to allow m and n to be negative.

In 1530, Niccolo Tartaglia was able to work out a general method to solve cubics of the above form, and was persuaded by Gerolamo Cardano to reveal his secret method for solving cubic equations. As the story goes, Cardano promised that he would not publish Tartaglia's work. A couple of years later, Cardano was discovered the solution of the general cubic equation

However he was unable to publish his findings because the solution was largely dependent on Tartaglia's solution of the depressed cubic. Lucky for Cardano, he found a solution to the depressed cubic in his pupil's, Ludovico Ferrari, own hand writing, and was then able to publish his work. This is when, in 1545, Cardano published his book "Ars Magna", the "Great Art." In this book he gave credit to del Ferro with the original solution to the depressed cubic.

Even though many today might believe that it was Cardano who, on his own, found the solution to solving the general cubic equation, it is evident that work on cubic equations began hundreds of years before Cardano's time.

The works of others: Archimedes, Brahmagupta, and Omar Khayyam.

This blogpost will skim the surface of three great mathematical minds and some of their works; Archimedes, Brahmagupta, and Omar Khayam.

Archimedes was born in the city of Syracuse on the island of Sicily in 287 BC. As a youth in Syracuse Archimedes had developed a natural curiosity for problem solving. Once Archimedes felt like he learned as much as he could in Syracuse, he went to Egypt to study in Alexandria. Archimedes was undoubtedly influenced by the works of Euclid (Dunham, "Journey Through Genius"). After Archimedes was done studying at Alexandria, he went back to Syracuse and began to problem solve for King Hiero II. One of the problems that Archimedes worked on for King Hiero is sometimes referred to as " The Puzzle of King Hiero's Crown". The King had ordered a crown to be made for him out of solid gold, which he supplied to the blacksmith. Once The king received the crown, he had suspicion that the crown was not pure gold, and that the blacksmith kept most of the gold for himself, and made the crown out of a little bit of the gold and mostly silver. Archimedes was tasked with determining if the crown was solid gold without damaging the crown in the process. Archimedes actually found the solution to this problem while taking a bath. He noticed that the full bath overflowed when he lowered himself into it, and realized that he could measure the crown's volume by the amount of water it displaced. He knew that since he could measure the crown's volume, all he had to do was discover its weight in order to calculate its density and hence its purity.

 

In 1906 unknown mathematical works of Archimedes was discovered, and is referred to as the "Archimedes Palimpsest". Archimedes is considered to have been one of the greatest mathematicians of antiquity. Archimedes produced formulas to calculate the areas of regular shapes, using a revolutionary method of capturing new shapes by using shapes he already understood ( Mastin, "The Story of Mathematics"). This method of using already known shapes in order to approximate the area of a circle is referred to as "Archimede's Method". Archimedes methodology for calculating pi was so powerful that in the 16th century Ludoph can Ceulun used a polygon with an extraordinary 4,611,686,018,427,387,904 sides to arrive at pi correct to 35 digits! Archimedes most impressive use of "Archimede's method" was his proof known as the Quadrature of the Parabola, which states that the area of a parabolic segment is 4/3 that of a certain inscribed triangle. Many also believe that Archimedes had the most prescient view of the concept of infinity of all Greek mathematicians.  

Brahmagupta was (possibly) born in Ujjain, India, in 598. Brahmagupta wrote very important works on both mathematics and astronomy, and became the head of the astronomical observatory at Ujjain in central India. Most of his works are composed in elliptic verse which was a common practice in Indian mathematics at the time (Luke, "Life of Brahmagupta"). Brahmagupta's text "Brahmasphutasidddhanta" is probably the earliest known text to treat zero as a number (class handout, Luke, "Life of Brahmagupta"). Brahmagupta also had alot of work on arithmetic, explaining how to find the cube-root of an integer and gave rules facilitating the computation of the squares and square roots (Mason, "Indian Mathematics"). He also did alot of work with fractions. Brahmagupta gave the sum of the squares of the first n natural numbers and the sum of the cubes of the first n natural numbers. Brahmagupta established the basic mathematical rules for dealing with zero (1+0=1; 1-0=1; and 1 x 0 = 0). Brahmagupta even wrote down abstract concepts using the initials of the names of colors to represent unknowns in his equations, which is one earliest signs of what we now know as algebra. Brahmagupta had many major discoveries; he was the first to discover the formula for solving quadratic equations. He wrote that the length of a year was 365 days 6 hours 12 minutes 9 seconds, and many other properties dealing with both positive and negative numbers.

Omar Khayam was a Persian mathematician, philosopher, poet and astronomer born in 1048 in Nishpur (modern day Iran). Omar started his career by teaching algebra and geometry. One of Khayam's most famous works include a treatise called "Treatise on Demonstration of Problems of Algebra". He also laid the foundation of Pascal's triangles (even though Pascal's Triangle was discussed by al-Karaji, before Omar's time) with his work on triangular array of binomial coefficients. Another major work was titled "Explanations of the Difficulties in the Postulates of  Euclid" It is believe that Omar Khayyam was trying to prove Euclid's parallel postulate when ended up proving the properties of figures in the non-euclidean geometry.

All three men, Archimedes, Brahmagrupta. and Omar Khayam, were revolutionaries of their time, and helped pave the way to modern day mathematics. 

Euclid's proof process

Out of all of the many different disciplines of the sciences, mathematics is the only science that uses the rigorous proof concept. Steven G, Krantz, author of "The History and Concept of Mathematical Proof", said that "There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics." Even though there were several well known mathematicians that used some sort of proof concept in their work and findings, for his time, no one quite solidified the bluepring of mathematical proof as fluently as "the Father of Geometry",  Euclid.

It was Euclid of Alexandria (325 BC - 265 BC) who first truly formalized the way that we now think about mathematics. Unlike his predecessors like  Thales (640 B.C.E - 546 B.C.E) who did actually prove some of his theorems in geometry, Euclid was the first person to have definitions, axioms and then theorems, in that particular order. This proving schematic that Euclid followed is mathematics done right. Euclid's process was so powerful that 2300 years later, it is still the way in which many practice mathematics, today.

Euclid's first task was to create definitions that could be used in the proof process. Euclid would use words outside of mathematics to explain particular mathematical "things" that could later on be used in other definitions. For example, Euclid defined a "point" to be "that which has no part"(Dunham, "Journey through Genius"). Since "point" has no been defined, Euclid could now use the term "point" in the process of defining other mathematical "things".

The next step for Euclid was axioms. An axiom is a proposition regarded as self-evident true without proof. "axiom" is a slightly archaic synonym for postulate (mathworld.wolfram.com/Axiom.html). Euclid would formulate axioms using previously created definitions. Once Euclid had a solid foundation of axioms and definitions, he then would start constructing proofs for his theorems.

Euclid's method of proving theorems was always approached from a geometrical perspective, which is only one of many tools used today would working towards proving theorems. Even though the modern day mathematical proof still tends to the traditional form of Euclid that goes back 2300 years, technology and science has advanced so much that there are many, many different methods that can be used today in order to proof a mathematical theorem. For example, while Euclid only used a compass and straightedge as his tools for proving theorems (Dunham, "Journey through Genius"), today, someone might prove a theorem that would consist of a computer calculation, or it could even consist of the construction of a physical model. Furthermore, one's modern day proof method might consist of using a computer simulation or model, or a computer algebra computation using software like Mathematica, Maple, or MatLab. 

Even with all of our technological advances that did not exist during the time of Euclid, no writings or books have had an impact on mathematical thought and exact sciences as has Euclid's Elements. the only other book that has been revered more so then the elements is The Bible, which to me, is astonishing, considering it was written so long ago. Indeed, I would agree, that Euclid is in fact the "Father of Geometry."