This week I spent hours watching the Youtube videos from Vihart and playing around with the ideas discussed. Several of the doodling patterns and connections shown blew my mind. In one video, the Hilbert Curve is mentioned and the first few iterations of it are drawn. I attempted to recreate them several times and ended up frustrated. What I eventually did realize is that each iteration of the Hilbert curve takes the previous image, shrinks it, and places identical copies of it into its four "corners" (see below).
There is obviously more to it than just sticking the shapes into the corners since each iteration is connected and becomes more tightly woven as it progresses. One thought that I had right as I realized how the Hilbert Curve was created was, "This is like Sierpinski's Triangle!" I had looked into Sierpinski's Triangle for one of my daily works. You can notice in the picture below that its premise is very similar to the Hilbert Curve.
Both of these patterns begin with one simple shape and build from that shape in order to create a complex, continuous design. I researched and came to find out that both Sierpinski's Triangle and the Hilbert Curve are fractals. I then looked into fractals and discovered that they contain self-similarity on all scales.
Some clear differences between these two fractals are their base shapes (one is a triangle, the other an open square) and the orientation of their images. While Sierpinski's Triangle solely shrinks and shifts its components, the Hilbert Curve also involves rotations and extra "links" to connect them.
Fractals are prevalent in the natural world. It's almost unreal how perfectly arranged some objects are. Fractals can be found in:
And those are just a few! Math always amazes me, but this stuff really got me excited. I spent more time than planned (which may not be evident in my writing, sorry..) researching and just examining pictures of natural fractals.
This week I decided to work on the "Doing Math" category of weekly work. After learning about Desmos and Daily Desmos in class, I started to play around on the site. This was time well spent, since I have not done legitimate graphing of equations in years. I also came to realize that graphic results of equations never really stuck with me (i.e. the effects of the type of equations, shifting/rotating of graphs, etc.) when I learned about them previously, so I found myself learning several new things and even having fun while doing it. This post will outline the process that I followed in order to create the graph of a familiar symbol and lessons that I learned (and re-learned) along the way.
When I opened Desmos and began to explore, one that came to my mind was, "I wonder if I can make a smiley face graph." It seemed simple enough to just put together circles and arcs. I remembered that the equation for a circle was x^2+y^2=b, so I started there. Using that equation and a bit of shifting and value changing, the head and eyes were a breeze to create. In order to create the left eye, I subtracted from x and shifted it horizontally. I added to the x value in order to create the left eye. Moving the eyes toward the top of the face was done by subtracting from the y value that was being squared. Using the same numerical values for the eyes (and only changing a sign) assured their symmetry.
To make the face more detailed, I decided to add "pupils". In order to do so, I used equations identical to those for the eyes but made the b value much smaller. This resulted in the "pupils" having the same center as the eyes but being placed inside of them.
-left eye: (x+2)^2+(y-2)^2=1
-right eye: (x-2)^2+(y-2)^2=1
-left pupil: (x+2)^2+(y-2)^2=1/6
-right pupil: (x-2)^2+(y-2)^2=1/6
Next I thought about making the nose and mouth. I knew that I wanted to form arcs. I naively tried a few equations, including 1/2 times the equaton for a circle. No luck with that or the others. I ended up looking online to find out what type of equation would make an arc. I found that equations of the form sqrt(b-x^2) would work. So I felt my way around in Desmos and eventually formulated appropriate equations to fit into my smiley face as its nose and mouth. The only main difference between these features (besides size, which depended on b) is their orientation, which I set with a simple sign change at the beginning.
-nose: sqrt(1/2 - x^2) - 1/3
The step-by-step pictures of my creation as well as the final result are shown below.
Response to The Math Book:
What an awesome read! Here is a description of pros and a con of the book, in my opinion.
When I bought the book and flipped through it, my first impression of it was very positive. The way that it is organized as a timeline with just one event (plus an eye-catching illustration) on each pair of pages is incredibly appealing. Each idea was introduced, described, and related to recent events in just one page. The following page would move on to follow the same summarizing process for a completely different concept. The sequential order of the book helped me to connect and compare historical events and discoveries and arrange their progression in my head.
I expected The Math Book to be like a traditional textbook, full of plain and confusing text. In reality, the book was quite appealing and accessible to me. I believe that the majority of it would be accessible to a variety of readers. The concepts discussed throughout the book were introduced in an informal and relatable way.
The book's introduction states that it outlines 250 of the greatest math events in history. I would definitely have to agree! Although the book resembles an encyclopedia in size, once I began to read it I couldn't put it down. This is again due to the brief nature of each mini-history on each page. I didn't become bored or overwhelmed by what was being discussed because the descriptions rarely went into deep detail. Clifford Pickover really did a great job exploring and summing up each idea with just a few paragraphs.
Besides the way that the content was communicated, the actual content was informative and intriguing. My personal favorite parts of the book include those highlighting women mathematicians, historic remnants of math tools and writings, and math in nature. Pickover's appreciation of math was obvious through his writing. He shared such eye-opening and mind-boggling facts that his belief in math's beauty became contagious.
Although content was one of the best features of The Math Book, it also plays a part in the one con that I came across. About halfway through the 516-paged monster, the topics discussed shifted from interesting and somewhat familiar to hard to grasp and over my head. I am not a master mathematician and had trouble wrapping my mind around certain ideas presented. Some of these are related to calculus and abstract math. Even after reading and re-reading a page several times, I found myself saying, "What?!?" With more education in higher-level math, I would probably be able to translate the information so that it is more meaningful to me. However, reading through some of math's history, I felt like the guy that misses the punchline of a joke and has to awkwardly move on with a blank look on his face.
When we explored Fibonacci's Liber Abaci on Thursday, I was intrigued by chapter 13: On the Method Elchataym and How with It Nearly All Problems of Mathematics Are Solved. What a title! I couldn't stop wondering about this mysterious method of elchataym and how it could possibly be so universal. I read through a translated snippet of the chapter online and also studied some articles which discuss elchataym in modern math terms.
Elchataym comes from the Arabic "al-khata'ayn", meaning "the two errors". In Liber Abaci, Fibonacci introduces elchataym as "the method of double false position". This method was first used before his time by Chinese and Arab mathematicians. Fibonacci's description of the method went right over my head, each of the eight times that I read it. I decided to start with the basics and find out what double false position actually means.
It seems that elchataym is a form of "guess and check". An article titled "False Position in Leonardo of Pisa's Liber Abbaci" by John Hannah (University of Canterbury, New Zealand) says that to use double false position, "Leonardo uses two guesses at the value of the unknown, and then compares them to see how much closer he is to the target value" (Hannah 11). Fibonacci seems to approximate his answers by adjusting the values for x and y until he reaches the number he is looking for (or one very close to it).
A quote from the translation of chapter 13 of Liber Abaci shows Fibonacci's thought process regarding elchataym: “For the one pound that I increased in the second position, I approached more closely to the true value by 3 ounces of silver. How much should I increase the second position so that I approach more closely by another 3 ounces?”
Multiple sources defined double false position as a method that we now call linear interpolation. This term was also new to me, so I researched it. Wikipedia defines linear interpolation as "a method of curve fitting using linear polynomials." This definition opened my eyes a bit. I read on to find out that for two known points with coordinates (x0,y0) and (x1,y1), the formula for linear interpolation is
An axiom is a statement that is accepted as true without proof. Axioms form the basis of mathematical proofs that are written in order to establish theorems. In order to formally prove conjectures, we must start with given information. Axioms often supply us with this assumed information.
There are two major roles that axioms play in mathematics.
1. They describe undefined terms (line, point, etc.).
2. They provide us with a starting point to prove conjectures.
The Greeks (and majorly Euclid) are known for first using mathematical axioms over 2000 years ago.
Their axiomatic approach involved undefined objects such as points and lines. These axioms are simple forms of statements which cannot be further broken down without philosophical considerations.
The axioms that I am most familiar with are those within Euclidean geometry. Euclid’s famous book The Elements contains his geometric axioms. Axioms are sometimes organized into systems, as Euclid's were. An axiomatic system is a collection of axioms such that each axiom is independent from the others – that is, no axiom can be proven from any other axiom in the set.
In Euclidean geometry there is a system of five essential postulates (which are synonymous to axioms) called Euclid's Postulates. These postulates are the building blocks which are used to prove every theorem of Euclidean geometry.
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.
Euclid also wrote a set of "Common Notions", which are axioms that go beyond geometry and refer to general logic. These notions are:
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Euclid's axiomatic writing was widely accepted and stable for thousands of years. Then, in the 19th century, mathematicians Ivanovitch Lobachevski, Karl Gauss, and János Bolyai extensively studied Euclid's Postulates and attempted to find contradictions within them. Instead, these men ended up creating non-Euclidean, hyperbolic geometry. In the 20th century, Albert Einstein referenced hyberbolic geometry to write The General Theory of Relativity.
It is evident that axioms (especially Euclid's) are the foundation of mathematical theory and have remained meaningful for over 2,000 years of mathematical discovery. Without axioms, math would not have developed into the beautiful science that it is today!
We commonly see it as a representation of "none" or a placeholder within a number. Where did zero come from anyway?
Was it always considered a number or did someone specifically discover it?
I decided to do some digging.
I read article after article and struggled to organize all of the information in a sequential, meaningful way. So here is a rough attempt to trace the history of zero.
According to Yale Global Online, the earliest form of zero was used by the Babylonians, around 2000 B.C., who used it as a placeholder in their numbers. They first represented an “empty” position with a space between their numbers. Eventually, they marked a place-holding zero with two slanted arrow-like symbols.
Thousands of years later, around 900 A.D., the Indians, chiefly a mathematician named Brahmagupta, acknowledged zero as a number rather than solely a placeholder. He introduced zero to the number system by placing dots under numerical symbols. He also wrote rules for doing math with zero. These rules, with the exception of dividing by zero, still hold true today. The Indians were the first to denote zero with the oval shape (0) that we now use in our number system.
The Indian number system and Brahmagupta’s work with zero was eventually introduced to Arabian mathematician Mohammed ibn-Musa al-Khowarizmi. He conducted early forms of algebraic equations equaling zero as well as created algorithms for multiplication and division.
In 1202 the well-known Italian Fibonacci continued al-Khowarizmi’s work with zero, exposing more businessmen to it and its real-life usefulness. Later on, Rene Descartes made use of zero in his Cartesian coordinates, using (0,0) as the origin of his Cartesian plane.
Although this is a very general and unelaborated piece of writing, I did put in time and learn a lot of by researching this topic and plan to expand on the concept of zero in future work.Sources: