That's All Folks

For the past 8 weeks, I have spent two days a week traveling through the history of mathematics.

Sounds...boring right? Honestly, it was anything but. I've never been one for history or facts, but learning about my area of concentration deeper, was the furthest thing from boring.

The course started with our definition of math. We needed to get an understanding of what we already knew before we could develop our thoughts.

Next, we moved onto the Greek where we learned about Thales, Pythagoras, and Euclid. We learned not only about the contributions of these mathematicians but facts about the people themselves. We researched weird facts about Pythagoras which, I never would've thought to do on my own. He was apparently suuuuper weird. Some of Thales' foundational results include:

• A circle is bisected by any of its diameters 
• The angles at the base of an isosceles triangle are equal
• When two straight lines cut each other, the vertically opposite angles are equal
• Two triangles are equal in all respects if they have two angles and one side receptively equal

After, we learned about Archimedes. His claim to fame was seeing if the King got jipped on his gold crown. Archimedes did this by simply using water displacement. He simply found the crown's volume by immersing the crown and exact measurement of pure gold in a tub filled with water to the brim. He measured the spillage and compared the volumes of spillage. If the crown was made of pure gold, the volumes would be the same...but they weren't. Thank you, Archimedes. Archimedes also has a puzzle, that was a pain to complete but a sense of accomplish came shortly after finally figuring it out. Try it here.

Then we had tessellations! Everyone's favorite. I think I'll always be excited about tessellations. Along with tessellations we learned about Al Khwarizmi, who was an Islamic mathematician who wrote on Hindu-Arabic numerals and was among the first to use zero as a place holder.

Skipping ahead, we arrive at Galileo where we learned about his though experiments and conducted one of our own. Which would be awesome to somehow use in my classroom one day. On this day we collected data using what we had, the predict what we would need, very similar to what Galileo would do.

Day 12 brought us the topic of women in mathematics. Weird, I know, but women can do math too. Here we learned about Sophie Germain and her proof. Astounding the work that this female did, even making connections with Gauss, which was another mathematician we learned about. Sophie proved that women can do math just as well as men.

We then met Georg Cantor, the Infinity Man. This man brought to light set theory. Cantor revolutionized the foundation of mathematics with set theory. Set theory is now considered so fundamental that it seems to border on the obvious but at its introduction it was controversial and revolutionary. The controversial element centered around the problem of whether infinity was a potentiality or could be achieved. You go, Georg Cantor.

Our mathematical journey ended with John Conway, who created life, one of the earliest studied and most well known examples of cellular automaton.

In 8 weeks we covered the highlights and sequence that math was explored in. This class has brought light to several topics that I may have not been able to explore in my career. It was fascinating to see to development of mathematics through contributions and discoveries. Quite interesting. It's interesting to look back to that first day, not knowing what to expect and being asked if math was discovered or invented. I won't tell my opinion, because I don't want yours to be altered but I will say that after this course, I have a better understanding as to what math actually is.

The Math Book

Picking this book up I thought it was quite intimidating, however skimming through it, that's not the right word to describe this book. This book is thick but full of 250 of the most intriguing mathematical milestones. I really like how the book travels through math history, much like we have done in class. Each page is dedicated to a single topic and accompanied with a related picture. The readings were simple and easy to follow and understand. At the bottom of some of the pages is a "see also" section that provides other pages within this book that are related. I found this interesting because math is all about connections and I like that this book recognized that.

Some highlights that I found while skimming though include:

= Ant Odometer (c. 150 Million B.C.): This first milestone in this books discusses how Swiss and German scientists discovered that ants "count" steps to judge distances. Once the ants had reached their destiniations, the scientists would add stilts or shorten the ant's legs to see how the ants traveled back to their original destination.

= Rubik's Cube (1974): This gives a summary of the Rubik's Cube and informs you that to solve the Rubik's Cube, only 20 moves are required. One day, maybe I'll be able to figure out the pattern but it was really interesting to read about the history of the Rubik's Cube because it's probably not something that you think about much, the Cube has always just been around for me.

Overall, I received a good impression from this book. It had a good set up and felt like it wouldn't be cumbersome to read through. Hopefully someday soon I will be able to return to this book and give it the attention it deserves.

Can Women Do Math?

History proves they can.

Some of the great women mathematicians include Hypatia, Agnesi, and Noether; each contributing their own piece into the world of mathematics.

Hypatia, more often remembered for her violent death, was one of the last great thinkers of ancient Alexandria and one of the first women to study and teach mathematics, astronomy and philosophy (source). Hypatia's father, Theon, one of the most educated men in Alexandria (source), taught mathematics and astronomy to Hypatia. She even worked with her father on some of his commentaries. Some think that Book III of Theon's version of Ptolemy's Almagest was actually the work of Hypatia (source). It is also thought that Hypatia helped her father in producing a new version of Euclid's Elements (source).

When searching for information on Maria Gaetana Agnesi, the first few things that appear are related to the "Witch of Agnesi" which is a curve that she studied in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana, which is also the first surviving mathematical work written by a woman (source). In 1738 she published a collection of complex essays on natural science and philosophy called Propositiones Philosophicae, based on the discussions of the intellectuals who gathered at her father's home. In many of these essays, she expressed her conviction that women should be educated (source). Agnesi began working on her most important work at the age of twenty where she started dealing with differential and integral calculus in Analytical Instituations. The publication of her work in 1748 caused a sensation in the academic world, becoming a model of clarity that was widely translated and used as a textbook (source).

The search for Emmy Amalie Noether brought up many statements such as "Most Significant Mathematician You've Never Heard Of" and "Creative Mathematical Genius" both of which sparked my interested, so I dug deeper. Noether invented a theorem that united with magisterial concision two conceptual pillars of physics: symmetry in nature and the universal laws of conservation (source). So what is this theorem? "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time" (source). In the 1920s, Noether did foundational work on abstract algebra, work in group theory, ring theory, group representations, and number theory (source).

Not only can women do mathematics but they can do it well. Small contributions that helped develop further understandings later on. There's obvious proof that women can indeed do math, so why is this topic even an issue?

Gender stereotype. That's why. As a society, for some reason, we are bred to believe that one gender does certain things better than others. This, however, is not correct. There are several women in math who accomplish great things. Three of them are listed above. And while these three may have made their contributions years in the past, there are several women today that are accomplishing great things. There are several women in the math department at Grand Valley. If women couldn't do math, wouldn't the department be solely men? They're obviously doing something right.

Women themselves need to believe that they are capable of doing math, as well. If you're smart, embrace it, don't shy away just because it's not "accepted" in society. Gender stereotypes will (hopefully) change and you're going to regret staying in your comfort bubble when you could be out doing something great. Maybe use a pseudo name like Ms. Sophie Germain did to get started. She created this and she was a woman. Who's to say you couldn't do the same?

The Joy of X

"Did O.J. do it? How should you flip your mattress to get the maximum wear out of it? How does Google search the Internet? How many people should you date before settling down? Believe it or not, math plays a crucial role in answering all of these questions and more." (Goodreads summary)

When researching which book to pick for this project, this is one of the things that caught my attention. Math really is all around us and The Joy of X draws this out. 

The book has six parts, each presenting certain elements of mathematics: Numbers, Relationships, Shapes, Change, Data, and Frontiers. Sounds intimidating but these sections represent a tour through the history and development of mathematics, including the practical applications. Never again will I fall into the trap of bungling the answer to the classic "If three men paint three fences in three hours, how long will it take for one man to paint one fence?" (answer: 3 hours). Now I understand why a piece of paper can't be folded in half more than 7-8 times, and how a high school junior was able to beat the record using a monstrously long roll of... toilet paper! I know how Luke could guarantee himself a win over Darth Vader in a game of laser tag (hint: it involves a conic section). For young lovers, mathematics could help in finding the perfect mate (if you make a few simplistic assumptions, that is). And if the prosecution in the O.J. Simpson murder trial had understood probability and statistics, could they have gotten a conviction?

The Joy of X covers a lot of math. This is a book that progresses from simple number theory, e.g., what do we mean when we say we have six of something? to basic arithmetical operations (adding, subtracting, multiplication, division) to discussions of fractions, and percentages. There are chapters explaining basic algebra, and how often we use the theory of solving for X without even realizing it. Next up is Geometry, the theory of infinity, negative numbers--Strogatz covers them all and then marches through integral calculus (here it started to become a bit more difficult ), differential equations, and vector analysis.


As enjoyable as the first five sections of the book were, my favorite section was the last, "Frontiers," where the author covered topics including prime numbers, where I learned that no one has ever found an exact formula to find primes; group theory, which bridges the arts and sciences; topology; spherical geometry; and infinite series. This section presented some fascinating ideas. For example, group theory suggests how to get the most even wear from a mattress and confirms the old mnemonic "spin in the spring, flip in the fall." For topology, the famous Möbius strip is examined. I thought I understood the properties of a Möbius strip, but they're actually more remarkable than I would have guessed. And the most mind-blowing concept was that some infinities are larger than others. This finding, which was bitterly contested at the time, is brilliantly demonstrated with a parable named the Hilbert Hotel.Sounds intimidating but the readings were so interesting that it never felt cumbersome. The author, Steven Strogatz, writes in a way that is almost engaging and no where near dry, like you think a math book would be.

The later chapters are definitely more advanced, but if you manage to stick it out that far, you'll be rewarded with an esoteric but lilting discussion of number theory.

The book, however, is simple. Think of dipping your toes in the water versus plunging into it. It brings to light some concepts that you might have always wondered about but it never really dives deep into any one topic, really explaining in depth the concepts. This is good, in my opinion though, because for those reading, math might not be that fun for them. Too much explanation could turn them off. The reader is able to get a general idea, maybe an 'ah-ha' moment or two, move on to another topic or explore more using the notes section Strogatz provides at the end of the book.

I would recommend this book to a wide range of people. Just because the book is about math doesn't mean you have to be interested in math to read it. The real world tie ins are appeasing to all, which is probably one of my favorite qualities of this book.

I would also like to somehow incorporate some of his explainations into my own explanations when teaching. Strogatz is able to explain in a simple matter that may be useful for others.



Check out the book trailer to get a look at the book and the author!

Elephants Can't Bungee Jump

Oh, but they can.

It all started with Galileo's Thought Experiment. Several of Galileo's most important work was in the form of imaginary experiments. One of his most famous experiments was dropping two different weighted balls, one large and heavy, and one small and light, from the same height and seeing which hit the ground first. Galileo was thinking of one of Aristotle's laws:

If a certain weight moves a particular distance in a particular time, a greater weight will move the same distance in a shorter time, and whatever is the proportion which the weights bear one to the other, so too the times will have to each other. For example if the half as heavy weight covers the distance in a certain time, a weight that is twice as heavy will cover the distance in one half the time. (Source)

So, following in the footsteps of Galileo, we experimented.

In a real bungee jump event, you have one chance. One jump. No do overs. So the tension of the rope needs to be perfect enough that you come close to the ground but not slam into it. These were also our requirements. We were allowed one jump. To make sure our jump was as accurate as possible, we started gathering data.

We selected our object, Mr. Elephant, and began to take several things into consideration to make our jump the best it could be. We needed to observe  how far one rubber band would stretch, how the weight of our object affected the pull of a rubber band, what would happen when another rubber band is added, etc.

To begin, we started with a single rubber band, measured the length at resting, then added another rubber band and observed how the length changed. We continued to do this, recording our data as we went, to generate a ratios with the number of rubber bands used. We took the distance and created a ratio with the number of rubber bands used.

We found that as the number of rubber bands increased, our decimal result grew larger and larger. To make a sound prediction for our one and only jump, we stayed on the safe side and used our smallest decimal result. Through our data we concluded that we would need 16 rubber bands to bungee our elephant 505 cm.

We were wrong.

Our first attempt landed us about 1 meter shy of even touching the ground. Our elephant was safe but we knew we could give him a better bungee experience.

Second time around we used more of a trial and error method. We added two more rubber bands, for a total of 18, and dropped again. Elephant got much closer to the ground. But what would adding one more band do?

Splat.

Too much. Elephant got a trunk full of concrete. Eighteen was the golden number for Mr. Elephant and a successful jump.

Some noticeable moments from this experiment include:

• It is difficult to prediction how much each rubber band is going to stretch and which rubber bands will stretch the most. Maybe all of the rubber bands we used didn't hold the same tension. It would also be interesting to test this...do all rubber bands from the same bag with the same size hold the same tension?
• Trial and error, as cumbersome as it may be, used to be helpful with out second jump. Had it been a real bungee experience we wouldn't have done this, however, it was quite interesting to see how the addition of one rubber band can alter your results.
•  The weight of an object has a huge affect on the number of rubber bands needed. Mr. Elephant was one of the heavier objects. We started at 16 rubber bands for our jump whereas Ms. Barbie's group started with 25. 

Numbers and Their BFFs

Amicable numbers. First off, what are they? Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number (thank you Wiki). But what does that exactly mean? Let's look at an example:

To show that two numbers are amicable you must first find their proper divisors. So first you need to know what a proper divisor is: a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3 (again, thank you Wiki). Let's work with 220 and 284.

Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
Proper divisors of 284: 1, 2, 4, 71, 142

If we add together all the proper divisors of 220 we obtain 284.
If we add together all the proper divisors of 284 we obtain 220.

Weeeeeird. In short, that is what amicable numbers, or more commonly amicable pairs, are.

This site gives a nice little brief history on amicable numbers. Briefly discuss famous mathematicians who found amicable pairs, Pythagoras being the first to find 220 and 284 as an amicable pair, and several other devoting their time to finding others. The page even offers a way to discover other amicable pairs. Fancy.

Who knew that numbers had BFFs. I guess everyone deserves them.


FUN FACTS

• There are some amicable pairs (m, n), in which the sum of digits of m and n is equal [4]. For example, consider amicable pair ( 69615, 87633), 
• Sum of digits of 69615 = 6+9+6+1+5 = 27 
• Sum of digits of 87633 = 8+7+6+3+3 = 27 
• There is no amicable pair in which one of the two numbers is a square

(http://www.shyamsundergupta.com/amicable.htm)

Tessellations!

Growing up my sister and I would always color in these books with our colored pencils at my grandma's house. Little did I know that what I was actually coloring were tessellations. I wish I could remember the names of the books but apparently I have been a fan of tessellations for quite some time now.

In MTH 221 we also work with tessellations but to a different extent. In MTH 221, we took a square, cut piece(s) out of it, then reattached them somewhere else on the square. This topic was covering the different types of tessellations so we created two different shapes. My first shape was created using slides. From this, my shape resembled the Teenage Mutant Ninja Turtles. My second shape was created using rotations. This shape resembled distorted bats, so I colored them Halloween colors. I wish I had my past school stuff with me so I could post a picture, hopefully I come back to this with an update.

For MTH 495 I first started using the pattern blocks to create tessellations. I later moved to the isometric dot paper. I think overall I probably spent 3-4 drawing the page. Obviously there were distractions but I watched two movies while doing the drawing. Below of pictures of the beginning and the in progress. I'll updated with the completed colored version when it's finished.

I also really enjoyed this website that allows you to create patterns online. I wouldn't be able to spend as long on a site versus doing a tessellation with pen and pencil but it's still an excellent resource and a good time killer.