## Wednesday, December 16, 2015

### 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.

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.

## Monday, December 7, 2015

### 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 $z_{n+1}=z_n^2+c$.  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"