The Golden Ratio

When you think of the golden ratio, what do you think of? I usually think of sunflowers, but why? That is because we see the golden ratio in nature very often, and it is present in the middle of a flower..

That really pretty spiral that we see in the middle of a huge flower is known as the golden ratio, or a number that is approximately equal to 1.618. It is only approximately this number but the digits do keep on going without a repeating pattern, isn’t that weird?! This number does not just appear in nature, it also appears in geometry, art, and architecture. It is said that the golden ratio makes the most pleasing and beautiful shape to the human eye. Many buildings and artworks actually contain the golden ratio within them, like the Parthenon in Greece,
but it is not known if it was intentionally designed to be this way or not.

How do we find this golden ratio? We divide a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. Kind of confusing so here’s a picture from mathisfun.com
The funny looking symbol that you see at the end is the symbol that we use to represent the golden ratio, it’s easier than writing out all those decimals. But what’s interesting is that this golden ratio also has a connection with the Fibonacci sequence, there is a special relationship between the two. We know the sequence to be
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …. and so on and we find the next number in the sequence by adding the two previous terms. Now the golden ratio relates to the Fibonacci sequence when we take any two successive Fibonacci numbers, the ratio of those two numbers is very close to the value of the golden ratio. It seems as though the bigger the pair of numbers, the closer the approximation is to the ratio, that’s pretty cool!

Another interesting aspect of this ratio is that it is said to be the most irrational number. A reason why is because it can be defined in different terms, like this:  and this can even be expanded into a more complicated fraction!
This fraction appears to continue on forever and it is very complicated to understand but for these reasons alone, the golden ratio is thought of as the most irrational number.

Surprisingly, there is not much history out there on the golden ratio but it is said to have first been spotted in Euclid’s book of Elements where Euclid is giving applications such as the construction of a regular pentagon, an icosahedron, and a dodecahedron. I just think it’s really cool that it has made its presence in nature and in architecture! Check it out!

 

 

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html
https://www.mathsisfun.com/numbers/golden-ratio.html
https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

Wallpaper Groups

For this post, I am going to be completing the ‘doing math’ requirement.

I completed the 17 wallpaper groups as part of my project for this course and decided to complete a blog post on the work that I completed! The 17 wallpaper groups represent the seventeen different ways to cover a two-dimensional plane if one only uses symmetries. Each individual group is a collection of these symmetries that are used in unique ways, making each group different from one another. First, I wanted to define the different types of symmetries used to tile the plane.
Rotation – a transformation in which an object is rotated about a specific point, typically rotated in degrees.
Reflection – a transformation in which a mirror image is created around an axis of reflection.
Translation – a transformation that moved every point in an object the same amount of distance.
Glide Reflection – a transformation that is a combination of a reflection and a translation.

All of these symmetries are important for understanding how these wallpaper groups have been created and how they move around the plane. Another important definition that is used to describe the movement in the plane is a lattice. This term describes all the centers of rotation and the axes of reflection for the objects in the plane. Now, we can continue on to show each of the seventeen wallpaper groups that my group and I have created for our project.

Symmetry Group 1 (p1):
This group is the simplest of all the symmetry groups, it consists only of translations throughout the plane. It does not contain reflections, rotations, or glide reflections and its lattice is parallelogrammatic.

Symmetry Group 2 (p2):
This group consists of translations and 180 degree rotations within the plane. It does not contain reflections or glide reflections. Also, the two translation axes may be inclined at any angle to each other. The lattice for this symmetry group is parallelogrammatic.

Symmetry Group 3 (pm):
This group consists of reflections and translations. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. There are no rotations or glide reflections and the lattice is rectangular.

Symmetry Group 4 (pg):
This symmetry group contains glide reflections and translations. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. There are no rotations or regular reflections and the lattice is rectangular.

Symmetry Group 5 (cm):
This group contains reflections, glide reflections, and translations.

The reflections and glide reflections have parallel axes. There are no rotations in this group and the lattice is rhombic. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes. This group applies for symmetrically staggered rows of identical objects, which have a symmetry axis perpendicular to the rows.

Symmetry Group 6 (pmm):
This group contains reflections and rotations whose axes are perpendicular. There are no glide reflections and the only rotations are half-turns whose fixed points lie at intersections of axes of reflection. The lattice for this symmetry group is rectangular.

Symmetry Group 7 (pmg):
This symmetry group contains reflections and 180 degree rotations. It does not contain translations or glide reflections. The fixed points of the half turns do not lie on the axes of reflection. The lattice for this group is rectangular.

Symmetry Group 8 (pgg):

This group contains translations, glide reflections, and 180 degree rotations but it does not contain regular reflections. There are perpendicular axes for the glide reflections, and the fixed points of the 180 degree rotations do not lie on these axes. The lattice for this group is rectangular.

Symmetry Group 9 (cmm):
This group contains reflections and glide reflections. This group also contains half turns (rotations). This group has perpendicular reflection axes and has half turns. The lattice for this specific group is rhombic.

Symmetry Group 10 (p4):
This group contains a 90 degree rotation, a 180 degree rotation, and translations. The 90 degree rotation is an order 4 rotation, and the half turn is a order 2 rotation. The centers of the half turns are midway between the centers of the order 4 rotations. There are not reflections or glide reflections and the lattice is square.

Symmetry Group 11 (p4m):
This group contains reflections, 180 degree rotations, and 90 degree rotations. The axes of reflection are inclined to each other by 45 degrees so that four axes of reflection pass through the centers of the order 4 rotations. All rotation centers lie on the reflection aces. The lattice is square for this symmetry group.

Symmetry Group 12 (p4g):
This group contains translations, reflections, glide reflections, 90 degree rotations, and 180 degree rotations. Different from symmetry group 11, the axes of reflection are perpendicular and none of the centers of the 90 degree rotation lie on the reflection axes. The lattice for this group is square.

Symmetry Group 13 (p3):
This group contains a 120 degree rotation.
This rotation is of order 3. The lattice for this group is hexagonal. There are no reflections or glide reflections. The rotation centers can be found at the corners of these triangles and at the centers of them.

Symmetry Group 14 (p31m):

This group contains reflections and rotations of order 3 (120 degrees). Some of the centers of rotation lie on the reflection axes, and some do not. The lattice is hexagonal.

 

 

Symmetry Group 15 (p3m1):

This group contains translations, 120 degree rotations, glide reflections, and regular reflections. The axes of reflection are inclined at 60 degree to one another, but all centers of rotation lie on reflection axes. The lattice is hexagonal. The glide reflection is given after the translation and reflection have been done.

Symmetry Group 16 (p6):
This group contains 60 degree rotations (order 6), 180 degree rotations, and 120 degree rotations. This group does not contain reflections or glide reflections. The lattice is hexagonal. A fundamental region for the symmetry group is one-sixth of an equilateral triangle of the lattice.

Symmetry Group 17 (p6m):
This is the most complicated group consists of translations, reflections, glide reflections, 60 degree rotations, 120 degree rotations, and 180 degree rotations. The axes of reflection meet at all the centers of rotation. The lattice is hexagonal. The glide reflection is given after the translation and reflection have been applied.

Shown with examples that my group and I created, these are the 17 wallpaper groups that exist in the two-dimensional plane.

Georg Friedrich Bernhard Riemann

Georg Reimann had a huge influence on mathematics and the way that we look at the mathematical world today. He started by taking mathematics courses from Moritz Stern and Gauss, learning from the best from the start of his mathematical career. In the spring of 1847, he moved to Berlin University to study under Steiner, Jacobi, Dirichlet, and Eisenstein. Focusing on elliptic function theory, Eisenstein and Riemann worked together to discuss theories and advances in the subject. But during his time at Berlin University, the man who influenced him the most was Dirichlet. This was the time that Riemann worked out his general theory of complex variables which stemmed off into some of his most important works. Riemann returned to Göttingen and began to work on his Ph.D. under Gauss, he officially finished and submitted his research in 1851. Riemann was influenced by many important mathematicians throughout his career, Weber and Listing were among some of the others that helped push him further into his research. Through working with these two men, Riemann gained a strong background in theoretical physics and important ideas of topology. He continued his work and started his thesis where he began researching the theory of complex variables and Riemann surfaces, he introduced topological methods into complex function theory, examined geometric properties of analytic functions, and completed research on conformal mappings and the connectivity of surfaces.

Riemann contributed many things to the mathematical world, but a very interesting part of his research consisted of Riemann spaces. While lecturing in Göttingen, he focused his lectures on geometry and defining a n-dimensional space as well as the definition of the curvature tensor. Now we ask the question, what is a Riemann space and why is it important? Before Riemann, there were not very many mathematicians that dove deep into the idea of n-dimensions but Riemann was determined to look deeper into what was really going on. He started thinking about geometry and how the geometry of space is not just one with straight lines or corners on a zero curvature plane. When we think of this statement, we typically think of Euclid and his development of geometry but Riemann wanted more from this. Shapes and planes more than two dimensional are more difficult for any person to picture, for example, can you picture what n-dimensions would look like? We see things like a table or a chair and just think, it has curves and the chair has a hole in the middle of it, but so what? We don’t think that a chair or a table can have positive or negative curvature like the other types of geometry, but we see things like this all the time. From this insight, Riemann developed a new geometrical concept called the Riemann space where force simply can be understood by geometry.

Riemann apparently borrowed from Gauss’s idea of a bookworm living on a two-dimensional piece of paper. Riemann took the idea that you place

these worms on a crumpled sheet of paper, what would this world look like from the worms two-dimensional view? What is interesting is that
the Riemann crumpled ball has all three types of
geometric curvature in a plane (sheet of paper), it contains zero curvature, positive curvature, and negative curvature. What do you think the world would look like to a book worm? According to Riemann, the world would look flat and undistorted because these creatures can only perceive two dimensions, they would not be able to detect that their bodies are crumpling with the paper in the higher dimensions as they move across their new world. Well why can’t the bookworm just move on top of the ball in a straight line? Riemann found that the “force” that would keep the bookworm from moving in a straight line is a result from the unseen warping from the third dimension (Cool, right?). Does this crumpled ball only have three dimensions? What if we were able to find more? Riemann didn’t stop with just the idea of the bookworm living in this new world, he cut the crumpled ball in half and found that on the inside, there are more dimensions to discover!
The approach to this new math opened new ways of mathematics, adding new dimensions to the way that we look at geometry and planes and thus proving why his research and discoveries were extremely important to our mathematical world.

Coming to the end of this post, you might still be asking why his discovery was so important? Well being in the field of mathematics and other fields like science, it is obvious that this discovery leads to the knowledge that other dimensions do exist in our world. It led to the discovery of math that was able to help calculate these dimensions and figure out just how many there were, which seems to be infinite! It was a milestone in math that helped develop where we are today, we are able to elaborate on the ideas of positive and negative curvature and go beyond just what we can see.

 

Citations:
http://www.mu6.com/riemann_space.html

 

 

The Number System

Today, our number system is known as the Hindu-Arabic Numeral System consisting of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This number system is a place-value system developed by Indian Mathematicians and is designed for positional notation in a decimal system. It is known that Indian Mathematicians founded the number system but down the road, Al-Khwarizmi and others accepted it and adopted this numeral system as their own, contributing to the spread and acceptance of the system. The spread of the numeral system was not fully in effect until these numbers reached Europe and became integrated into the normal practice of mathematics. The nine digits that we use today originally evolved from the Brahmi numerals, a Indian numeral system from the third century B.C. Buddhist inscriptions.

Before this numeral system was the most accepted, there were many others that had been developed. It all started after language was developed, at which point in time we started to produce counting methods through creating marks. The next known system was created by Egyptians, known as the Egyptian numbers created around 3000-1600 B.C.. Through surviving records, it has been discovered that 1 is a vertical line ‘|’ and ten is ‘^’ and they write their numbers from left to right. Babylonian mathematicians developed a numeral system with 60 as its base towards the middle of the second century B.C..But this system proved to be difficult to use since 60 was the systems base where numbers below 60 were represented through clusters of ten. As of today, this system is still surviving through 60 seconds, 60 minutes, 180 degrees of a triangle, and 360 degrees of a circle. Through the development of this system, we were introduced to the place-value concept which is very important for our number system today. Around 300 B.C., the breakthrough for the numbers came with the creation of zero, decimal system, and the Arabic numerals. The introduction of this system came with the new idea that every number in the system has their own symbol, making it different from all the rest created before. Another system of counting that was created was the abacus, created in the first millennium B.C. and is sometimes still used today. This creation had the idea of zero and it was in it’s place before it was put into written systems. Being able to use this form through counting and drawing in the dirt, it was used as early as 1,000 B.C.. All of these different forms of counting contributed to the development of the Hindu-Arabic Numeral System over the years.

The introduction of the Hindu-Arabic Numeral System took time to become integrated into the world of mathematics. The idea of this system was created in India and traveled to the Arabic/Islamic peoples and from there traveled on to Europe where it became the accepted form of representing numbers and counting. It is known that Persian and Arab mathematicians in India commonly used the numbers but the spread of this system into further western regions occurred before this system was taken to Europe. Before the adoption of these numbers in Europe, it first was used by the Arabic peoples. Although the book is lost, between 825 and 830 B.C., Al-Khwarizmi and Al-Kindi each wrote books on the principles of using these Arabic numbers which eventually lead to the adoption of these numbers into the middle east and parts of the west. In the tenth century, middle Eastern scholars used the numbers for the development and creation of fractions and percentages. Within that same century, Sind ibn Ali introduced the decimal point which introduced a new way of writing numbers called “sand-table”. This new way of writing encompassed the numbers of the written form that we still use today.

Where would we be without zero?

To this day, the concept of zero continues to be a struggle for students. It is difficult to think about the fact that there could just be nothing, just an empty space. Or that someone could have “zero” of something, meaning that they really just have nothing. Before we talk about zero and the discovery of it, we should talk about who truly discovered zero.

The Ancient Indian astronomer and mathematician, Brahmagupta, is given the credit for discovering the number zero. Born in the city of Bhinmal in Northwest India, Brahmagupta is the only scientist that we are able to thank for discovering the concept of zero. If he hadn’t discovered it, where would we be? The lack of zero in our number system would mean that calculus, algebra, quadratics, negative numbers, geometry, almost every aspect of advanced mathematics would not exist in our world today. Along with the discovery of zero, Brahmagupta had many other successes during his lifetime career of being an astronomer and mathematician. Some of these are the rules and properties of zero, discovered the formula for solving quadratic equations, indicated that Earth was closer to the moon than the sun, calculated Earth’s circumference, and established the rules for working with both positive and negative numbers. As we can see, Brahmagupta was thoroughly involved with mathematics and made impressive discoveries along the way. One interesting fact about this man, is that most of his mathematical discoveries were actually documented in the form of poetry. I thought that was pretty cool! Now, let’s talk about nothing.

Where did zero come from? How did it all of a sudden show up into our number system? Well to begin with, the Sumerians were the first to develop a counting system to keep track of all their goods. They used a positional system, where the value of a symbol is dependent on its position. After they developed their system, it was commonly used and was eventually passed down to Akkadians around 2500 BC and eventually to the Babylonians in 2000 BC. It is told that the Babylonians were the first to use a ‘mark’ to symbolize that a number was absent from a column in this positional system. This may have been the start of the concept of zero. But who really took it one step further and investigated the mysterious number? It was the Indians who began to understand zero both as a symbol and as a concept in the numerical system. This is where Brahmagupta comes into the picture, determined to understand the meaning of zero. Around 650 AD, Brahmagupta was the first to articulate the formal operations using zero, his form of ‘zero’ was using dots underneath numbers to indicate the zero. These dots were sometimes referred to as ‘sunny’, meaning empty, or ‘kha’, meaning place. After Brahmagupta formalized his theories for the discovery of zero, he wrote a series of rules for obtaining zero through addition and subtraction. Along with these rules, he also found the results of the operations with zero.

The only thing that Brahmagupta has seemed to get incorrect with these rules is the fact that a number divided by zero is actually undefined or infinite rather than zero. This has sparked many arguments throughout the mathematics world, and even in my capstone. The general argument is that a room full of math majors could not decided whether 0 divided by 0 was going to be 0, 1, undefined, or infinite. After doing a good amount of research, it seems as though a number divided by zero is simply undefined, there is no answer. As told in history, the division by zero would have to wait until Newton and Leibniz invented calculus. Some parts of the world would also have to wait for the concept of zero to come along to expand their knowledge on their number systems since it still took a few centuries before the concept of zero reached Europe. It is told that first, Arabian voyagers would bring the texts of Brahmagupta and he colleagues back from India along with other exotic items, like spices. By 773 AD, the concept of zero had reached Baghdad and would eventually be developed into the middle east by Arabian mathematicians who worked through the Indian number system.

By 879 AD, many years after the concept of zero had reached Baghdad, zero was written as how we know it today! Along with this progression of the concept, zero had finally reached Europe around this same time thanks to the conquest of Spain by the Moors. Since zero had been discovered by all at this point in time, it started to be put to use in many different ways. Fibonacci, for example, used zero through his work on algorithms in the book Liber Abaci in 1202, Rene Descartes founded the cartesian coordinate plane, with the origin of (0, 0), and towards the 1600’s, Newton and Leibniz worked with numbers as they approached zero, leading to the discovery of limits and calculus. As we can see, we would not be as advanced as we are today in the mathematical world if zero had never been discovered. It is a magical number that is necessary for our number system to contain. If zero didn’t exist, what would we use to represent nothing? How would we subtract 4 from 4? We could not do simple addition or subtraction without zero, we would be just as confused as those before us.

http://yaleglobal.yale.edu/about/zero.jsp

http://www.smithsonianmag.com/history/origin-number-zero-180953392/?no-ist

http://www.famousscientists.org/brahmagupta/

Is math a science?

According to the Merriam-Webster Encyclopedia, science is “knowledge about or study of the natural world based on facts learned through experiments and observations.” When we think of science, we think of the study of the human body, medicine, nature, or how the systems within the world work like gravity or travel. All of these things are part of the study of science. When you ask whether or not math is a science, you have to think about what goes into the different aspects of math. Are we experimenting? Can we observe math in its natural form? To me, the answer would be yes, that math is a science. I base my opinion off of the fact that in almost all aspects of science, math is included in the observations and experiments. Many of the facts that they obtain from these experiments would not be arrived upon if math was not part of the scientific world. Also, I think that math is observed and experimented upon. When we are solving for a system of equations, for example, we are observing the scenario, first attempting to figure out what we want to solve for, or what our end goal is. After we have figured this out, we can solve for a variable through experimentation. This is justification enough for me to believe that math is a science.

There are arguments that sciences study the natural world but mathematics does not. If you take apart what goes into the natural world, like gravity or growth of plants/animals, how do you think that we know the rate of gravity or the way that things grow? Mathematics does study the natural world, it tells scientists and the general public the way that things work with numbers instead of just observing.

How Not to Be Wrong

I recently read the book titled How Not to Be Wrong, The Power of Mathematical Thinking by Jordan Ellenberg. This book is composed of many examples of different types of mathematics, how that specific topic was used in a context, and basically how they were either wrong or right and how they may fix this issue. Ellenberg really puts the misuse of mathematics into perspective as you go through all of the examples provided in the book. Besides putting the misuses into perspective, I also learned a lot from this book about mathematical ideas and history that I had not previously known before. For example, I did not know that the man, who discovered the square root of 2, named Hippasus, was thrown into the sea to his death for this discovery. The book progresses through a series of realistic, real-life situations that involve mathematics, making it easier to comprehend the complexity of the situations. But I really liked how this book showed how these situations went wrong in the world of mathematics, how they went right, or how mathematics can be applied to other areas of our lives like science, politics, conspiracy theories, and the probability of completely random events.

One chapter that I really enjoyed reading, chapter 15, titled Galton’s Ellipse, really looks into regression and the patterns that you are able to see when you plot data on to a graph. Galton created our idea of a scatterplot by comparing different factors of heredity. Galton compared a father’s height and son’s height from many different families and decided that the best way to arrange the data would be in a graph, therefore creating the scatterplot. This scatterplot made the relation between the to factors easier to see, explained in the book on page 313, “The coxcomb and scatterplot play to our cognitive strengths: our brains are sort of bad at looking at columns of numbers, but absolutely ace at locating patterns and information in a two-dimensional field of vision.” After reading this, I could not agree more. We can be right in mathematics by visualizing things, the book even states that we are more likely to see a pattern when we are seeing something visually rather than seeing a bunch of numbers.

What I took away from this book was not how to be right, even though this is the title that the author had picked. Even in the last pages of the book, Ellenberg states that there is no way to always be right in mathematics, but it’s more about how we learn this mathematics and how we teach it to our future students. “We have to teach a mathematics that values precise answers but also intelligent approximation, that demands the ability to deploy existing algorithms fluently but also the horse sense to work things out on the fly, that mixes rigidity with a sense of play” (Ellenburg pg. 58). There is an entire section dedicated to how you should and should not teach algebra or calculus. Mathematicians and teachers are afraid of creating students who are purely computation based, but they are also afraid of creating students who are only conceptual thinkers, not able to do simple computations. This book is able to give suggestions of where to meet in the middle. Between the different views of how we should or should not teach math, I found it interesting that they were suggesting that we throw out all of the algorithms and procedural connections to math in order to have students stop relying on them. I agree with Ellenburg, that these are necessary but students should also not be dependent on them.

Overall, I thought that this was a really great book. I learned an awesome amount of history from the book and I also learned how to be right and how to be wrong. You have to be very careful when you’re completing computations and different environments bring in different factors that you need to be aware of. I think that all of the ideas really brought light to my eyes about what direction I should head into with mathematics and how to become a better teacher. I think that I would recommend this book because I think it’s filled with a lot of great real-life examples. It is difficult to follow sometimes because it jumps around a lot from topic to topic and it also tends to talk in circles, but I think that he makes great points at the end of each chapter. This book would have been better if it had an overhanging theme, tying every single topic back to this theme since it was easy to get lost in the computations and the long explanations. With a title like How Not to Be Wrong: The Power of Mathematical Thinking, you would think that it would be more about how not to be wrong, instead of these lengthy examples about history and abstract concepts. But still, I would recommend this book to anyone who is interested in mathematics.

Some quotes that I really enjoyed:

“Pythagoras was said to have the ability to talk to cattle and one of the few ancient Greeks to wear pants.” (pg. 34)

“Rule of mathematical life: if the universe hands you a hard problem, try to solve and easier one instead, and hope the simpler problem is close enough to the original problem that the universe doesn’t object.” (pg.35)

“In the mathematical context, the good choices are the ones that settle unnecessary perplexities without creating new ones.” (pg. 47)

“An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something’s wrong with your method.” (pg. 64)

“Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.” (pg. 85)

The Proof

For this blog, I am going to complete the ‘doing math’ requirement.

Euclid didn’t just develop the concepts of proofs, he also developed a set of conjectures and propositions that connected all proofs together and allowed them to make sense while looking at them piece by piece. Euclid used the most general concepts to create these proofs that explain larger theories, in an attempt to prove the proposition or conjecture correct. In the process of explaining these theories, we started with proving theorems. Theories consist of a set of ideas that are used in order to explain why something is true, while a theorem is a result that can be proven to be true using axioms or postulates. We have proven our theories and shown that they are true through proving theorems and conjectures, showing that the invention of the proof is more important than we think. In the process, he uses complicated wording and bases most of his geometric proofs off of his five postulates.

Euclid’s five postulate, representations shown above, consist of: “1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The fifth postulate is known as the parallel postulate ” (Wolfram Alpha). Euclid did not just use these five postulates while creating the idea of the proof and proving different theories, he also had a set of common notions.

These common notions were used to guide the rules of completing proofs and what should commonly be followed by others when attempting to do so. Since Euclid has created the concept of a proof, things have changed. Euclid brought the idea of proofs into the world of mathematics, showing that there is a strategic and more sophisticated way of proving an idea. Euclid set the groundwork for bigger and better things that occurred later on in the mathematical world.

Since Euclid has created proofs using conjectures, postulates, and theorems, proofs and mathematics have come a long way. Since math has continued to develop, the list of ways to prove things has advanced. We have gone beyond the idea that geometry is the only thing that can be proven using axioms and postulates. We have advanced into more complicated areas of mathematics, like proving that an equation can be solved or proving that something equals something else. Euclid primarily focused on geometry, leaving so much more room for improvement. For example, the creation of the proof and Euclidean geometry lead to the creation of Hyperbolic and Elliptic geometry.

Although he only focused on geometry, he did get a lot of things right. The idea that there needed to be a basic list of axioms or postulates in order to start, set the framework for the concept of writing proofs and was the biggest improvement in the world of mathematics. Euclid even set up the perfect outline for writing proofs, starting with what you know and ending with what you wanted to prove. To this day, this is still the general outline for how proofs should be written. The idea of combining multiple conjectures in order to create one proof through valid reasoning is another thing that Euclid did right, the need for explanations is essential while writing a proof. If the reasoning is correct, there is no room for argument, therefore showing that the reasoning about why the conjecture is correct serves as a valid proof.

Now that all I’ve done is talk about Euclid and his works, now we should check out just how some of it works. Many of the conjectures and theorems could be proved just using the five postulates or the axioms, which was very interesting! I am going to prove a theorem doing exactly that:

Consider the following incidence geometry defined by these three axioms:

A1. There are exactly four lines in the geometry.
A2. Given any two distinct lines, there exists exactly one point that lies on both of them.
A3. Every point is on exactly two lines.

Theorem: Each line in this geometry contains exactly three points.

Proof. 

According to the first axiom there are exactly for lines in this geometry. Call these four lines l, m, n, and o. Take line l. By the second axiom, lines m, n, and o must each have a point in common with line l. 

The intersection of these three lines (m, n, and o) with l, create three distinct points at their intersections. The reason that these points are distinct is because if they were not distinct they would violate the third axiom, creating a point on three lines.

Line l therefore contains three distinct points at the intersection with line m, n, and o. 

Suppose we have a fourth point P on line l. P cannot be on line m, n, or o because doing so would violate the second axiom. But every point is on exactly two lines, thus we would need a fifth line in order for point P to exist. But this violates the first axiom that only allows for four lines in the geometry. Thus we cannot add a fourth point to line l. 

Since we cannot add another point to line l  and it must have at least three points at the intersections with m, n, and o, l must have exactly three points on it.

Similar arguments can be made for lines m, n, and o. Therefore, each line in this geometry contains exactly three points.                                           QED.

Euclid also was able to prove the pythagorean theorem in many different ways using the proof! I am going to take the pythagorean theorem and prove it only one way but there are many!

Theorem: If triangle ABC is a right triangle, then the measure of angle c will equal 90º, b = |AC|, c = |AB|, and a = |CB|, then a² + b² = c².

One of the many ways of proving the pythagorean theorem includes a square inscribed within a square. We can prove this algebraically, which makes it much easier to see. First, we can separate this into the square contained on the inside, whose area is c². Next, we have 4 triangles located on the outside of the square, all with area ¹⁄2ab. Lastly, we have the area of the big square, which is (a+b)². Now, the are of the large square must equal the area of the small triangles and the smaller square contained within so we can set these numbers equal to each other…

c² + 4(1/2 ab) = (a+b)²
c² = (a+b)² – 4(1/2 ab)
c² = a² + 2ab + b² – 2ab
c² = a² + b²

This allows us to see why the pythagorean theorem works and it proves the pythagorean theorem at the same time!

I believed that proof mattered to Euclid because it was a solid base for the studies and work that he created for Euclidean geometry. He was working with geometry, attempting to expand on what he was finding. In order to expand, there needed to be a base, something to work up from. Proofs were important to Euclid because it shows that his work was expanding, and it shows that his work is valid. He created the framework for proofs, and this became more and more important to him as he began to make more and more discoveries about how complicated geometry is. But when you think about the history of math and the advancements that were made, you can ask yourself, why did Euclid create the proof? In my opinion, I think that he created the proof because there was no other formal way of completing mathematics. I think that Euclid wanted a concrete format for proving theorems. If these theorems were proved, they would be able to log them and build the world of mathematics on top of them instead of continuing to work with ideas that may or may not be true. To this day, proofs are the most important framework for figuring out whether ideas are true or not. Mathematicians and even scientists work to prove theories and new ideas through already proven ideas, if the framework was not set through these proofs then there would not be the advancements that we have today.

 

What is Math?

When thinking about math and how all the equations and concepts were discovered, it makes one wonder how all of it came together to form the subject of math. When I think of math, I think of numbers. This is one of the greatest discoveries in the history of math, the creation of numbers and the application of these numbers into other things. For example, using these numbers to add or subtract. Using these numbers for counting, measurements, anything that was possible to use numbers for lead to the biggest discovery in the history of mathematics. Numbers are the foundation of math and the ability to extend math into other areas. Next, I would think that the biggest discovery is measurement. The ability to take these numbers and apply them to measuring certain objects, angles, lengths, what have you, would be another huge discovery in the area of mathematics. This is huge because it was the foundation for geometry, the equations involved in geometry, measuring simple lengths, constructing buildings, cooking, chemistry, measurement and numbers take the cake for the greatest top two discoveries in the history of mathematics.

Other than numbers and measurement, the creation of geometry was a milestone for the history of mathematics. They had all the components of geometry but putting it all together was the biggest part of it all. After the creation of geometry and fully understanding how geometry worked, it lead into other aspects of geometry other than just Euclidean. For example, it lead into Hyperbolic geometry and Elliptic. Both being very abstract forms of geometry that couldn’t have been discovered without the creation of Euclidean geometry.

From these discoveries, other types of abstract math started branching off of each other. Math started introducing topics like statistics, modern algebra, discrete mathematics, but they were all introduced based on the common idea of numbers and measurements. How the different types of numbers like, rational or irrational, produce different answers and how they work in different planes or environments. The idea that math can go into different dimensions changed the world of mathematics because now mathematicians don’t just work in 2-D, but 3-D or 4-D. Without the simple discovery of numbers or measurement, we might not be as advanced as we are now in the world of mathematics or technology.

Lesson on Chance

For this lesson, students will be introduced to the idea of chance. The teacher will set up five stations of chance, each with a game to play for students to interact with. First, the teacher will talk about probability and explain the formula for finding probability. The teacher will go through examples of probability and explain what different fractions would mean in terms of the chance situation. Students will go through experiments in order to gain a conceptual understanding of chance and probability. For example, they will find the chances of flipping heads, getting a 4 on the die, picking a red card, picking a diamond, and picking the 5 of diamonds. Through experimentation, students will understand what probability is and how to compute the probable outcome. In addition to understanding probability, the students will understand the difference between unlikely, most likely, and very likely when determining their probabilities. A link to this lesson is posted below:

http://illuminations.nctm.org/Lesson.aspx?id=2895

This lesson aligns with the CCSSM standards for grades 6-8 because it is an introduction into probability and what it means. It allows the students to gain a greater understanding of how probable an outcome is and what the chances of it actually happening are based on experimentation. They can draw conclusions about the probability of events, overall I would give the lesson a rating of 5.