495 Final Project

This project was really fun for me. I started with the idea to recreate George Hart's tunnel cube puzzle, but with the added idea of having the different cards making patterns within the shape. Each step had a different challenge to it. First putting it together was the challenge and figuring out which cuts should connect to make the right solids. After I got the general shape I tried to make patterns on each of the faces. This was a bit frustrating because I was trying to move the least amount of cards since they were hard to move, but it wasn't at all obvious which cards would be best to move. As I was writing about the different symmetries I thought maybe it could become a puzzle in which the player is given the symmetries and is asked to recreate the solid. This could also just be interesting as an exercise to see the different combinations that could be made from the same clues.

These are the various observations I made about my card creations:

For every solid, each card has four sets of two cuts. No two cards can be connected at more than one place. Every corner cut is matched to a side cut.

My smallest shape made of 6 cards most closely resembles a Triangular Pyramid. This was the least amount of cards needed to make a shape that left no open cuts. Each face has one card from each suit. Four bigger triangles for each vertex and four smaller triangles in the middle of each face. There are four tunnels in which a small triangle matches with a big triangle. One face has all twos, but no face has all threes. The three threes are connected only with a vertex hole. Every vertex besides the one with all threes has two twos and one three. Each vertex and face has one of each suit. Each face besides the one with the three threes has two threes and a two.

The nine card solid most closely represents a Triangular Prism. I made this one on accident. I started trying to fit each cut together without realizing I had three cards left on the table. Six total faces, three square faces and two triangular faces on the ends. Six large triangular holes, three square holes, two small triangular holes. Each square face has two of the same number on one “diagonal” and two of the same suit on the other “diagonal.” Each triangular face has one of each suit and one of each number, these only came together once I figured out how to do the square face symmetries. On each vertex, two cards have the same number and one of those cards shares its suit with the other card. Looking at the top of each long edge, there's a diagonal of all one suit, and the other diagonal is all one number.

The shape that looks kind of like a cube has 12 cards. Possibly a cuboctohedron. Six faces, eight vertices. Six square holes, eight large triangular holes, no smaller triangular holes, which means there are 14 tunnels to the center. The top and bottom have all one color, one is red the other black. Every other face has both colors. Each face has two different suites and two different numbers, one number of each of the two suits. No number touches another card of its own number. Each of the eight vertices has a 9, 10 and Q. Four of the vertices have all the same suit and four have three different suits. There are six square symmetry axes, one from each face. There are six triangle axes from the vertices. I believe the two card symmetry axes connect the corners. So each number is in a two card symmetry with its number in the other color. Like the 10 of spades and the 10 of hearts. The exception is the 9s which match up with the 9 of the same color. This makes me wonder if somehow I could change the order of the 9s to make those two card symmetries alike with all of the ranks.

I also attempted to make an object with a pentagonal face, trying to use 15 cards. The pentagonal face did not work out though. The cards were too slippery to stay together. I also suspect the angle wasn't quite right to do more than four cards on one face.

Finally I made an object, trying to use 15 cards, but ended up using 12. It is sort of the reverse of the cube. It has 6 large square holes from the numbered corners coming together (like vertices). Then it has 8 small triangle holes from the sides coming together. I formed it just by following those two rules. Looking at a square hole it looks like a cube, but then looking at a triangle face it looks like a rhombus. I have no idea what kind of shape it actually is. I didn't attempt making patterns, but I'm sure it's probably possible. I think if I did clubs for example 4-7 on one square face, and on the opposite square face 4-7 in spades and 4-7 diamonds around the “circumference” through the middle, then hopefully make those middle squares have 4-7 on them. But then the triangles would have two suits and three different numbers around them. It wouldn't make as nice of a pattern for the triangle faces. I bet I could mimick the pattern from the cube, but with that one I used three cards of all four suits, and this shape has four numbers from three suits so I'm not exactly sure how it would work. Unfortunately I think the cards are a bit too worn out from trying to do a pentagonal face to take it apart and make a pattern right now.  

 

Overview

I've decided for my last blog post I will review MTH 495.  It's titled History of Math and I feel like we definitely covered that.  We started at the beginning to learn various milestones in math which seem so natural now.  We talked about the development of the number system and additions to this, from zero to irrational numbers and even infinity.  I found it very interesting at how deeply these concepts were debated at the time of their discovery/invention.  Something that stuck with me throughout the course was history repeating itself in the sense that every new idea had some adjustment time where it was debated then accepted.  It made me curious what parts of math we have yet to discover or invent and debate about.

The class also helped me discover the creative and artistic side of math.  Of course there was always graphs and some geometric drawings in math class, but we talked about new things I hadn't thought of as math.  Talking about tesselations was one of my favorite classes.  I also really enjoyed when we talked about topology because of the emphasis in visualizing how different shapes can mold into others.

I also had lots of fun with my individual project which led me on a journey trying to put together this puzzle by George Hart.  (You can see my whole project in my other post here.) Overall I really enjoyed this class as my final math class.  It gave a nice background to tie together all that we had learned over the past four years.

Fractals

Communicating Math

From IBM's tumblr

Do you know about Benoit B. Mandelbrot?  If you don't, you should!  Dr. Mandelbrot was born in Poland, but had citizenship in France and the US.  He worked for IBM for 35 years and dabbled in such vast areas as finance, aeronautics, physics, and of course math.  Oh yeah, and he invented a thing called fractal geometry.

I may have undersold it.  Fractal geometry is a big deal.  It can be applied to coastlines, bronchi in lungs, clusters of galaxies, and financial price increments, to name a few.  The basic idea of fractal geometry is that as you zoom in on a picture, it looks pretty much the same as when you were zoomed out.  Also, while zooming in, the curve doesn't get simpler.  For example, zooming in on a section of a parabola makes it eventually look like a line segment (Jordan Ellenberg actually discusses this in his book "How Not to be Wrong: The Power of Mathematical Thinking" when introducing the logic behind calculus).  But when you zoom in on a section of a fractal it continues to be as detailed as it was before, if not more so.  Benoit referred to this detail as "roughness" which he would calculate.  He discusses an instant in this TED talk of a colleague giving him a picture of a curve and asking him to estimate the roughness.  His estimate was off by .02.  This idea of roughness is interesting though because it can describe so many things from mountains to cauliflower.

Anyway, fractals have a little wiggle room in their definition.  How else could these both be fractals?

Sierpinski Carpet

Mandelbrot Set

Both of these have some sort of self-similarity--that idea of the picture being about the same at different "zooms."  Plus, fractals have "fine or detailed structure at arbitrarily small scales. irregularity locally and globally that is not easily described in
traditional Euclidean geometric language, and simple and 'perhaps recursive' definitions" (https://en.wikipedia.org/wiki/Fractal).

The Mandelbrot set is a bit "complex" ;) but can be written as a simple recursive rule.  The Mandelbrot set is the set of complex numbers c for which the sequence (c, c² + c, (c²+c)² + c, ((c²+c)²+c)² +c, (((c²+c)²+c)²+c)² + c, …) does not approach infinity.  It's defined recursively as .  Some numbers like c=1 do not belong to the Mandebrot set because the rule would go to infinity.  However, c= -1 is in the set because that sequence oscillates between 0 and -1 so it does not approach infinity. (https://en.wikipedia.org/wiki/Mandelbrot_set)

Anyway, fractals are fun to look at and play with (check out John Golden's Geogebra file) and Mandelbrot was a very interesting man.  He was highly revered for this discovery that is so applicable to nature.  Plus he has some great quotes:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
-The Fractal Geometry of Nature

Bottomless wonders spring from simple rules, which are repeated without end.

-TED talk "Fractals and the art of roughness"

Euler's Square

Nature of Mathematics

This week we learned about the incredible Euler.  One of the activities we could do was to make an Euler Square of 16 playing cards. Elaine Young describes it in a MAA post, "In 1694, Jacques Ozanam posed the problem of arranging 16 playing cards in a 4 x 4 array such that no row or column contained more than one card of each suit and each rank. The solution forms an Euler square."  It's an Euler square because it combines two different Latin rectangles.  A Latin rectangle is like Sudoku, no number (or rank and suit in this case) is in the same row or column.  

Below are my two solutions to Euler's Square.  The first one is interesting because it has symmetry along the diagonals with the ranks of the cards.  It was really hard to get started on this one.  I started off by trying to make diagonals with one kind of number.  That didn't really work except for the Aces on the big diagonal.  From there I just tried to randomly move around cards to get something started.  The breakthrough was when I not only had the Aces on the diagonal, but also the Jacks on the other diagonal.  The second picture shows that each row, column, diagonal, and 2x2 square satisfies the requirements of no repeating rank or suit--including the middle 2x2 square.  I struggled with this one a bit too at first.  I knew I had to change up the diagonals so I attempted that first.  Then I tried to place each group of numbers as in a sudoku, with one number in each row or column and just filled in the gaps with the other numbers.  This worked a little, but then I had to modify and switch some based on suit.  I found that moving one usually results in changing three more.  But looking at it like a sudoku and filling in the blanks was helpful.

What I think is more interesting than just forming these Euler squares is thinking about if doing this is really math.  On one hand it's hardly more than arranging the cards in a puzzle type of way.  I certainly wasn't using any mathematical formula to figure out the arrangement.  However, there are many patterns to be seen in these squares and I believe finding patterns is a large part of mathematical principles.  For instance, if these squares had missing cards, you should be able to complete them based on the patterns within them.  Incidentally my Honors thesis is on rook polynomials which have similar properties to Latin rectangles, so I know there are ways to calculate the number of combinations of the cards.  There is a whole section of math dealing with combinatorics, which is counting combinations of elements.  So these squares can use math to describe and generalize them.  

Next question, if I think these are a part of math, was I doing math while I was making them?  This is a tough question.  I admitted I didn't use any of those formulas that count how many arrangements can be made.  But perhaps recognizing the patterns and following those pattern rules to fill out my square was doing math.  

It's hard to nail down exactly what is math and what is considered "doing math."  Instead of determining if this exercise was math, one could consider if this exercise is math, then what is math in general.  In that case I would definitely consider strategic pattern finding as a major part of math, in addition to the ability to numerically find conclusions that would be hard to find without math.  

Since I've decided to consider this math, what similar things could possibly be considered math?  Sudoku?  Jigsaw puzzles?  What examples can you think of?

Review: The Math Book

For class we each read a book of our choosing.  I chose the Math Book by Clifford Pickover because it looked like an interesting quick read.  The front states "250 Milestones in the History of Mathematics" and that's exactly what it is.  It's written in chronological order all the way from circa 150 Million BCE to 2007, two years before it was published.  Every page is a different milestone with an accompanying picture.  

Not only does it discuss the famous mathematicians that we've all heard of like Pythagoras, Euler, and Fibonacci, but it also discusses people, discoveries, and inventions that I've never talked about in math class.  My favorite pages were ones that talked about different puzzles and games like Rope around the Earth Puzzle (1702), and the board game Go (548 BCE) .  So although not every page is a gigantic breakthrough that everyone has heard about, they're all important in some way, even just for entertainment.  Most of the pages discuss some history which I found really interesting because not often do we put these discoveries in context of the times.   One that I found interesting that ties in the puzzles and also history was the entry on Hex from 1942.  It discussed the game of hex and how to play and then also tells about how it was manufactured by Parker Brothers.  It's inventor, Piet Hein had to go into hiding in 1940 because of WWII.  It really makes you think about what could have been discovered and invented by some people if their society had allowed them to keep doing math.

Overall I really enjoyed the book.  It really shows how vast math is and showed how interesting it could be.  It's written at a level that is easy to read and understand.  I believe the general public would enjoy this book, but it might help if that have a slight interest in math to start with.  No deep knowledge of math is needed to read and understand this book.  For the first half of the book I read straight through in chronological order, but the second half I skipped around a bit, so if the reader decides to read in chronological order or not, either way is interesting.  My only complaint is that sometimes I wish the pages were a little more in depth instead of just introducing the topic and then moving on to the next one.  Perhaps if I had had more time to get through this book I could have looked up the ones I was really interested in online instead of immediately moving to the next one.  I definitely liked this book though and would recommend it to anyone with the slightest inclination towards math and its history.  It was an easy read and entertaining.

The Tale of the Cubic

Once upon a time there was no uniform way to solve all cubic functions--how sad.  It was a problem that puzzled mathematicians up until 1535.  What happened in 1535 you ask?  Well there was a math competition in Italy that a guy named Niccolo Fontana attended.  You might know him by the nickname Tartaglia which means "the stutterer."  Or you might not.  Anyway this guy was an engineer and amateur mathematician--because who doesn't want to do math in their free time?  So he came to this competition and SURPRISE he won it by solving a cubic function using his general formula!  Everyone was shocked because they had thought it to be impossible.  

Since it was such a coveted formula, Tartaglia wanted to keep it to himself, even hiding it by encoding it in a poem.  Here's a picture of the formula from this Vanderbilt website. Can you imagine fitting that whole thing secretly into a poem? 

So Tartaglia wanted all the glory for himself and kept it a secret until the smooth talking Gerolamo Cardano came along and got the formula from Tartaglia which he subsequently published in his book "Ars Magna" in 1545.  Needless to say Tartaglia was not pleased that Cardano broke his promise of keeping it a secret, but got his revenge by helping Cardano get arrested for heresy after Cardano made a horoscope for Jesus.  

When we talked about the cubic formula in class I attempted to use the formula to solve the following cubic.  The (-3,0), (2,0), and (5,0) are the solutions I found from graphing.  My incomplete attempt gave me imaginary numbers after taking the first square root.  I may have messed up a negative somewhere...  But take a look and then take a moment to appreciate graphing calculators, the factor theorem, and long division of polynomials that can find the solutions a whole lot faster!

It makes me wonder what kind of applications they had that needed the solution of a cubic function.  As I thought back on my schooling to try to remember an application we learned that used cubics I couldn't think of a single one.  I think we were always just given a graph and/or the equation and had to find the solutions from there without giving an adequate reason why we should even solve it.  So I went to google and found this page where several people came up with applications of cubics.  Most of them seem to be far more advanced than what we would learn about when first seeing cubic formulas.  I thought the most interesting one was the claim that the typeset letters are formed from cubic functions.  So thank you cubics for helping me type this blog!  Do you know of any other applications of cubic functions?  Any that people would have needed in 1535?

Hey look!  If you want more information about the characters in this tale click here for a more in depth story courtesy of Luke Mastin! 

Qiandu, Yangma, and the Bienao

We talked about ancient Chinese mathematics in class the other day.  Among those we discussed was Liu Hui who lived in the third century and made numerous contributions to mathematics.  Our main topic of conversation was the cube he made from three different parts that he called the qiandu, yangma and bienao.  The qiandu is half of the volume of a cube, yangma is 1/3 and bienao is 1/6, so together they make up the whole volume of a cube.  So with the finished product in front of us (but without tracing or measuring from them!), we were told to make the nets of the shapes.  A net of a three dimensional object is a flat drawing of all the faces of the shape correctly positioned so that when you cut it out you can fold it into the 3D shape.  Before I started the nets I drew the above sketch to get an idea of how they would fit together.  I decided to make my final cube 2 inches on each side which made the diagonal of each face the square root of 8 (approximately 2.8) and the diagonal of the cube the square root of 12 (approximately 3.4).






After analyzing which sides would need to fold together I made my nets.  Here's a rough sketch of each shape with the measurements.  Here's a link to Jennifer's ready made nets (with a better picture).








So with my sketches, a ruler, scissors, tape and coral paper I made my nets and then folded them up to make my three shapes!  Not the prettiest, but they still make a cube!

Left to Right: Bienao (1/6), yangma (1/3), and qiandu (1/2).

As I played around with them I decided I didn't want to use the qiandu anymore, but I wanted to make two new shapes.  I decided I didn't want the yangma and bienao to form another qiandu like in the original, so I positioned them kind of diagonally (pictured).  

Once I got started on the nets for the two new pieces, I realized they were the exact same as the yangma and bienao.  I still made them though and played around with the ways that I could put them together.  This was interesting to me because I realized I couldn't put two complete faces of the same shape together.  In other words I had to make two qiandus out of a yangma and bienao.  I couldn't make any other formation from them that would still form a complete cube.  

Overall I think it's interesting that all the way back in the third century Liu Hui was thinking about volume and how to divide it in such a way that gave a unique combination of the shapes.  (You can check out a proof and more pictures here from Sherlock Holmes in Babylon: And Other Tales of Mathematical History.)  Plus, I very much appreciate the kind of puzzle approach to putting it together.  It makes me want to make more cubes made up of more complicated pieces, although I'm not sure if I would be able to manage making shapes that have such nice divisions of the volume like he did!