Computer Science and Checkers

                For my final blog post I am going to talk about computer science.  To help me explain such a broad topic, I am going to use a program I wrote a few months ago that simulates a game of checkers when you run it and, due to a limited amount of space, I can only briefly talk about the concepts behind this program. Since Computer Science and the program I'm using as an example are both very complex topics, I'm going to primarily focus on the way a computer thinks when it compiles code. Before I can go on to explain checkers, I would like to talk about how computers think and function very briefly. A computer can't look at a situation and assess it like we can as humans.  A computer can only recognize two things: whether a condition is true or false. To best show what this means, you can look at binary. Our number system is called decimal.  It has ten digits and is easy for us to understand because we can count to 10 on our fingers and it's a fairly simple system. Binary is different in the sense that it only has two digits--0 and 1, 1 being true and 0 being false. At the end of this post I've linked a video that explains how to convert from decimal to binary for those who are interested in the topic.

                 Since a computer can only recognize things as true or false, you  have to keep this in mind when you construct the game board. To do this, I used something called a two-dimensional array or a matrix. A matrix is like a grid where it has two values (an x and a y) and it uses those two values to find the place in the matrix you're looking for, then it returns that information.  I've posted a picture of the matrix bellow.

                             

      As you can see, there are four different values on the board: an X which represents the spaces you can't "jump" your pieces to, an empty space which represents the empty places on the board, a capital O which represents one player's pieces, and a lowercase o which represents the other player's pieces. Now that I have a board created, the next step I had to take in writing my program was taking this matrix and making a visual representation of it as a game board. To do this, I used text and something called a loop. A loop is a type of function that will repeat itself until its condition is false.

     The image above is a picture of the two loops I used. While it may look complicated at first glance, when you break it down it is relatively easy to understand.  First, I needed to create two integer values, which are values that can only be whole numbers.  I named them count and countTwo. Then to make it easy to understand the grid and select pieces I include the line System.out.println("      0.   1.   2.   3.   4.   .5   .6   7.").  The function of this is that it prints out whatever is inside of the quotation marks so, in this case it is  0.-7.. The next line says while(count<8). This line creates the loop that I mentioned earlier.  What it is saying is that it is going to re-run the lines of code inside of its curly brackets until count is no longer less than 8, or in other words, until the condition is false. The next line is another print line; however, the only difference between this one and the previous one is that it says print instead of println. What this difference does is that the println automatically creates a new line after it prints out its statement and the one that only says print does not. The next line is another loop that will run until countTwo is no longer less than 8. Then inside of that loop is another print statement that prints (board[count][countTwo ] + " | "). This print statement is going to grab the value at board[count][countTwo] which is like a co-ordinate plane at the point (x,y). It then separates that with a line from the next value it prints. The second loop also contains the line countTwo++ which adds 1 to countTwo.  The first time it runs, it will grab the value from board[0][0] which is X. Then  the second loop will run again printing out the value of board[0][1] which is an o. It repeats this process 8 times.  Then because countTwo is no longer less than 8, it leaves the second loop. The next couple of lines are count++ and countTwo=0.  Count++ adds one on to count, countTwo = 0 resets countTwo so that it no longer equals 8 but 0 again. Then the first loop repeats. Since countTwo was reset to equal 0, the second loop also repeats again, going through another 8 times.  This time it is printing out the values at board[1][CountTwo] instead of board[0][countTwo]. The entire process repeats until count equals 8 and it prints out the entire board, which ends up looking like the following picture. 

 

     Unfortunately due to a lack of space I could only briefly describe to concepts of this program. Other parts of this program include: selecting a piece to move, returning move options for the selected piece, identifying when a player can jump another piece, identifying the possibility of multiple jumps, and also implementing the king piece.

https://www.youtube.com/watch?v=Q2UgMYwWiO4 - David Williams: Base 10 to Base 2 (Decimal to Binary) Conversion

Blackjack

One of the most popular card gambling games in the world in blackjack. It is played in mostly every casino, all over the world, every day. People make thousands and thousands of dollars playing blackjack, but also lose the same amount, if not more. As with every casino game, the "house" (the casino) has the odds in it's favour. This is set in order to ensure that the casino will make more money than it loses in the long run, after all it is all a game of percentages, but this will become much clearer later on when I discuss basic game strategy. If following basic strategy, the house still has an advantage of about 0.5 percent, which may not seem like a lot, but imagine playing $100 per hand and losing 50 cents a hand. After playing hundreds of hands, the 50 cents per hand adds up. Knowing basic strategy is one way to improve your odds of winning. Another, more frowned upon, way is counting cards. Both ways give the player a higher success rate of coming away making a profit.

Basic strategy is a very common way for players to improve their blackjack game and odds of winning money. Here is a chart of what the basic strategy includes.

Now looking at this chart may seem very confusing if you don't know the rules of black jack. So here are the rules of how to play. First, you must place your bet. Once everyone sitting at the table's bet has been placed, cards will be dealt out. The number of decks a casino uses varies, but normally they use between 4-6 decks per table. Therefore, it is harder to anticipate what cards will be coming out next. Everyone sitting at the table will first get one card faced up along with the dealer receiving the same. Next, another faced up card will be dealt to every player at the table, but the dealer's second card will be faced down. The objective of the game is each participant attempts to beat the dealer by getting a count as close to 21, without going over 21. What is a 21 count you may ask? Well, each card in the deck is given a value. Cards 2,3,4,5,6,7,8,9,10 all represent their numerical value, while every "face card" (jack, queen, king) also represent 10. Lastly, aces can represent either 1 or 11 depending on the better value for your hand. Once you have determined your hand's value the next step is to either "hit"(take another card) or "stay"(chose not to receive anymore cards).The order of who is to make this decisions runs from the left of the dealer to the right, in order. For example, lets say you are the second person to make a decision at a table. After the cards are dealt, you have a value of 8. In this position, the best play is to ALWAYS hit. This is because as the objective states, the goal is to try get the closest value to 21 without going over. With the value 8, no card in the deck will cause you to "bust". Busting is when you or the dealer goes over a numerical value of 21. If this occurs to you, you automatically lose and if this happens to the dealer, you automatically win (if you have not already busted). So with the value 8, you hit and receive the next card. Depending on the value of that card, you will either decide to hit or stay again. In this example, we will say you received a card with the value 10. You now have a value of 18. In basic strategy, once you have a value of 17 and above (not higher than 21 obviously) you are supposed to stay. So you would indicate to the dealer you would like to stay at 18. Each player takes their turn. Now it is the dealer's turn. They first must flip their faced down card to indicate their two card score. In this example we will say the dealer was showing a 7 (dealer's faced up card was a 7). In basic strategy you must always assume the next card you and the dealer will receive when "hitting" is a 10 because the majority of cards in the decks value is 10 (all 10s, jacks, queens, kings). The dealer flips their faced down card and as we assumed, the dealers faced down card has a value of 10. The dealer now has a value of 17 and the rules indicate that the dealer must stay on a value of 17 and above. The dealer is essentially playing the basic strategy. Your numerical value of 18 is closer to 21 than the dealers value of 17, therefore you win the hand and receive the amount you bet plus your original bet back. There are two other options to use when it is your turn. They are called "doubling down" and "splitting". In these cases you are doubling your original bet because you believe you have a better chance of winning. Doubling down usually occurs when you have a starting value of 10 or 11 and hope to receive a 10 or 11 on your next card. Splitting occurs when you have two of the same cards and want to split them into two different hands. Lastly, you get "blackjack when your first two cards add up to a value of 21. Blackjack pays 3 to 2. The chart shows the basic strategy and when do utilize your options best. 

You are also able to play blackjack online. This blackjack uses a random number generator algorithm to be able to randomly deal cards to its users. Many questions have been risen on how random this algorithm really is.  "The program is designed to provide a sequence of numbers that lacks a discernible pattern. The sequence of numbers is so large, often in the billions, which ensures a pattern could never be found without using sophisticated technology and equipment operated by a genius"(legalonlineblackjack.net). The fear is that these online sites manipulate this algorithm in order to ensure they make much more money than they lose. This is evident in the movie "Runner, Runner", which follows a college student to Costa Rica in order to prove he had been cheated by an online gambling site. The difference between online blackjack and casino blackjack is one that has always been debated due to the scepticism of online blackjacks randomness. Both online and casino blackjack are massive gambling outlets and are played by millions of people everyday. 

Finally, counting cards has been an issue many casino owners have faced throughout the years. This is a mathematical strategy few people are able to use. It involves mathematical skill and quickness. Each card is given a value of +1, -1 or 0. Depending on the cards that come out, the count will either be positive, or negative. If the count is high (+10), then more face cards will b coming out. If the count is love (-10), low number cards will come out. This is used to determine which cards will be coming out of the deck and can be used to predict your next card as well as the dealers. The movie "21" is a good way to learn about counting cards. 

Blackjack is a fun game to play, learn about, and even helps you with quick adding!

Bibliography:

1."How a Basic Strategy Player Can Beat Blackjack - BlackjackInfo.com."BlackjackInfocom. N.p., 30 Oct. 2012. Web. 17 Nov. 2015.

2. "Blackjack – Card Game Rules." Bicycle Playing Cards. N.p., n.d. Web. 17 Nov. 2015.

3. Random Number Generators In Blackjack | How Random Are They?"Random Number Generators In Blackjack | How Random Are They? N.p., n.d. Web. 17 Nov. 2015. 

What are Fractals?

Fractals were technically discovered by Benoît Mandelbrot in 1975. I say technically because you can’t really discover something if you are making it up. Mandelbrot basically put fractals into words. He explained them as being geometric shapes that when divided into parts, each part would be a smaller replica of the whole shape. Benoit came up with the word "fractal" as the new scientific term for these patterns. The word comes from the Latin word “fractus”. The latin word fractus means "broken" or “fractured”. Fractals have been naturally occurring for as long as we can imagine, but no one realized it until Mandelbrot.

           Mandelbrot began his research when he found fractals in nature. However, they are also found in mathematical equations and within modern industries such as cell phones and medicine. The main attribute of a fractal is its repeating pattern at any scale (zoom level). Additionally, fractals are known for expanding and evolving symmetry. They can be called a self similar pattern if the replication is the same at every level. In my opinion, the easiest fractal to understand is the Sierpinski Triangle fractal (1st picture). It is created by drawing a triangle, then separating it into 4 triangles, and then removing the middle one. You’re probably wondering why fractals are important and why they are worth knowing. Now that you know the most basic one, we can go deeper into fractals. Many people would find no interest in fractals and do not know that they are naturally occurring in day-to-day life. We find fractals in 3 main areas: Nature, Geometry, and Algebra.

First, we will look at naturally occurring fractals. Fractals in nature often range in scale. There are fractals inside the human body such as blood vessels and within neurons, in the branching of trees and sprouting of plants, and also in weather such as lighting, hurricanes, and rivers. A large portion of fractals in nature are in the spiral pattern, which appears more often than other patterns in nature. The spiral is found inside of tree’s trunks, hurricanes (3rd picture), and agave cactus’ (2nd picture).
Second, there are an infinite amount of fractals in geometry, you can even make your own. The most popular one is the Sierpinski Triangle (1st picture). This triangle is simply created by removing the middle triangle from the triangle(s) from the prior generation. Every generation increases the amount of triangles by a factor of 3: 1,3,9,27,81,243,729 to infinity. Another popular geometric fractal is the koch curve (4th picture). You start with the first bold line, which has 4 line segments. For the next generation of the pattern, you simply replace each of the 4 line segments with the original 4 line segment. You can keep doing this forever. Geometric fractals are never ending because we can perfect them, unlike fractals in nature.

The third main fractal is the algebraic fractal, which are the hardest to understand in my opinion. The most popular algebraic fractal is the Mandelbrot set founded by Mandelbrot in 1980. The equation is found below: you plug in a constant for C and any number you want for Z old. When you get Z new, you plug it back into the equation and you repeat this step. We are interested in C and how different values of C affect the equation. Eventually, you will see one of two patterns. Either the answers will be going towards infinity or the value of the equation will stay between a fixed set of numbers. Values of C that are found in the black cause the equation to stay finite whereas all values of C outside the black cause the equation to go towards infinity. This was hard for me to comprehend at first but Mandelbrot simply found every value of C that makes the equation repeatedly stay finite instead of going to infinity. At the end of this he got a fractal. At first, I was spectacle to how this was a fractal, but as you see in the zoomed in image, it begins to repeat itself.

So, like me, you are probably still wondering what the point of fractals are. Until recently fractals were never used in human engineering, but now they are used for cooling circuits in computer chips, antennas in cellular devices/ radios, and doctors are now using fractal analysis for blood vessels and the lungs to find illnesses. Right now the most common illnesses assessed with abnormal blood vessels is cancerous tumors (shown to the left).

What's the Best Monopoly Strategy?

TABS Monopoly is the type of board game that brings families together for a night of fun, only to have everyone in that family hating each other because Susie won with all four railroads. I have played this game countless times with my family, and almost every time I end up losing to one of my older brothers because they threatened to beat me up if I didn’t give them Boardwalk. However, throughout the years I have always wondered what the best strategy is for beating my family, and it turns out that going for the most expensive properties doesn’t give you the best chance at winning.

TABS The first Monopoly ever created started production in the early 20th century, and has gone through a lot of remodels to get to be where it is today. The earliest strategies of Monopoly didn’t have the different rules that we have today, so original Monopoly was believed to be fair and equal. Ian Stewart, who has written multiple papers on this probability wrote about the fairness of Monopoly back in 1996. His conclusion was that it was equally proportional, “So, with a little help from Markov, we can prove that a game as complicated as Monopoly is fair, in the sense that—in the long run—no particular square is more or less likely to be landed on.” With evolutions in computer programming, and a better understanding of the game and its rules, we can now show that there is an unequal probability to which squares get landed on more.

TABS The most popular spaces to land on are, in no particular order: GO, JAIL, Illinois Ave., Pennsylvania Railroad, and B&O Railroad. This wild phenomenon of the probabilities is explained using something called the Markov Chain. Ian Stewart uses the Markov chain theory to explain, “The theory of Markov chains tells us that the evolution of this probability distribution is given by the sequence of vectors v, Mv, M2V, Mlv and so on: each throw of the dice corresponds to the matrix M operating on the vector v.” He basically goes on to say that you need a basic computer to compute the 1,600 transition probabilities that stretch across a Monopoly board.

TABS Stewart uses some basic rules when determining the probabilities for each of the squares. First off, the double dice rule. Each person on their turn throws two dice, and if they get double numbers, meaning two 5’s or two 3’s, they get to roll again. However if they get a double on their third roll, they go to jail. This determines that the greatest distance possible in one turn is 35, but the most likely distance to be rolled is 7.

TABS The M vector described earlier gets extremely complicated when the “Jail” space is added in. Jail is the most landed on space on any monopoly board, because there are so many ways to get sent there. For instance, one player can visit jail, land on the “Go to Jail” square, draw a Chance or Community Chest card sending them to jail, or roll doubles on dice. He didn’t take the different ways to end up in jail during his first conclusion.

TABS The most shocking thing I learned while researching this topic is that jail is almost twice as likely to be landed on as any other square. Jail has a 5.89% chance of being landed on, whereas the next highest probability of 3.18% belongs to Illinois Avenue. B&O Railroad has a staggering 3.06% chance of being landed on, Reading Railroad has a very close 2.99% chance, and Pennsylvania Railroad has 2.91%. Boardwalk, the square with the greatest rent value, has a landed on probability of 2.63%, and Park Place only has a 2.19%.

TABS Now where do we implement the business side to Monopoly? Because after all the math is done, the object of the game isn’t to have the most properties, but instead to bankrupt the other opponents. Friddell dove deep into the property market behind Monopoly. He outlines when the best times to buy houses and upgrade to hotels. His strategy assumes that houses can be bought from the start, but in most house rules you need all three properties before you can build even one house. Friddell describes that even though it might be more difficult to buy houses early, the point in which you will start to make money will be achieved sooner. Likewise, with only two houses on your properties it will take roughly 20 turns for you to reach even. Now, if you can add another house to make three, the time it takes to reach even and start making money drastically shortens. He also points out that between Go and Indiana Avenue, breaking-even can be reached on spot New York Avenue in only 10 turns.

TABS Tim Darling outlines the four best ways to achieve total monopoly. The first of the four positions is to own all four railroads if at all possible. It is the best way to stack up on cash, and there is no need to pour money into the railroads to add houses. The next three strategies all include 3 houses on the following colors: orange, light blue, and light purple/magenta. “I would trade Boardwalk to get St. James Place. Sure, you don’t want your opponent to have the chance to build houses and hotels on Boardwalk, but they’ll never be able to do it unless they have houses on cheaper properties generating cash. Meanwhile, you’ll be able to throw 3 houses on each of the orange.”

TABS The next time you are playing one of the 600 versions of Monopoly, try one of these strategies out and see how it fares with you. You might be able to bankrupt your older brothers, and the rest of your family.

Stewart, Ian. "How Fair Is Monopoly?" Scientific American (1996): 104-05. Print.

"Monopoly Revisited by Ian Stewart." Mathematical Recreations. Web. 16 Nov. 2015.

Friddell. "Markov and Mr. Monopoly Make Millions." 2010. Web. 16 Nov. 2015.

"How to Win at Monopoly ® - a Surefire Strategy." How to Win at Monopoly ® - a Surefire TABSStrategy. 1 Nov. 2007. Web. 16 Nov. 2015.

Symmetry in Nature

   I have always preferred being outside than being inside. There was always a pull for me to go camping, or backpacking, or rock climbing. I’ve gone on many adventures all over the U.S, and one thing remains the same across the board. Nature is beautiful. Leaves and flowers are so simple in their construction, yet so pleasing to look at. Spider webs, while not pleasant to get on you, are lovely to look at all covered in dew in the early morning. Starfish and anemone stand out in tide pools. Butterflies are vibrant and you can’t help but watch them if they choose to flutter by you. But why is this? Why is nature so aesthetically pleasing more often than not?

   The answer to this question is symmetry. Symmetry is defined as, “the correspondence in size, form, and arrangement of parts on opposite sides of a plane, line, or point; regularity of form or arrangement in terms of like, reciprocal, or corresponding parts.” This is the technical definition, but it can also be defined as, “beauty based on or characterized by such excellence of proportion.” The Greeks based their ideas on symmetry on regular polyhedrons. Each regular polyhedral was also associated with a natural element. Fire was the tetrahedron. Air was an octahedron. Earth was a cube. Water was an icosahedron. The universe itself was a penta-dodecahedron. The Greeks tried to explain biological symmetry in terms of geometry. They had little success determining the underlying cause, but they opened the door for further exploration of the root cause for symmetry.

    In nature, there are many types of symmetry. Each has a specific axis for reflection. In organisms, the type of symmetry it exhibits is an adaptation that the organism has evolved to in order to survive.

The first type of symmetry is called linear, or bilateral symmetry.

                                                                                        

 This means that you can make an axis of symmetry straight down the center. One could also think of it as a reflection along the line that you draw. Humans display this type of symmetry, although most people are not perfectly symmetrical.

   Another type of symmetry is radial symmetry. Radial symmetry is what Starfish have.  This type of symmetry means that the parts of an organism are all arranged around one main axis. Flowers most often exhibit this type of symmetry, but when other organism manage to come together in this configuration it is stunning. There is research being done currently trying to link the Fibonacci sequence to radial symmetry. I found this very interesting due to the fact that we have discussed the Fibonacci sequence so often in class.

   The third type of symmetry often found in nature is called Strip Pattern symmetry. This is most often found with striping patterns on snakes, and the spacing of animal tracks and tree branches. There are seven main patterns within this category. They are Translation symmetry, Horizontal mirror symmetry, Vertical mirror symmetry, Rotational symmetry, or Glide reflection symmetry. In the case of tree branches, scientists can use Strip Pattern in order to tell when a tree was having a hard year for growth. If the space between two supporting weight branches varies significantly, scientist can tell something was wrong. They can use a similar strategy for animal tracks. Any deviations from the symmetrical pattern can indicate an injury or disease in the animal that is affecting its strides.

  

  The last type of symmetry commonly found in nature, and my personal favorite, is Wallpaper symmetry. This type of symmetry is the most complex and is based on tessellation. The most common example of this type of symmetry is honeycomb. The hexagons are all identical. There is also a very interesting place in Ireland where the stones form hexagonal prism. It is called the Giant’s Causeway. These patterns are caused by the repetition of a certain pattern or shape, that isn’t symmetrical on its own, over and over until it completes a cycle. One way to achieve this type of symmetry artificially is to create a vibration diagram using a Goold elliptic pendulum.

   While most of my post has talked about symmetry in the realm of biology, many time organic compounds are symmetrical as well. For instance, a water molecule is symmetrical. When frozen, water remains symmetrical. Many people are aware of this phenomenon. Snowflakes exhibit radial symmetry. Although snowflakes may all be unique, I think that it is interesting to note that all snowflakes are radially symmetrical.

   It is fascinating how so many things are symmetrical. Is there a cause for this? According to researchers at the University of San Paulo in Brazil, the answer is yes. Using ALICE (a large ion collision experiment), they found that ions exhibit a fundamental symmetry due to their charges.

  Symmetry has been a topic of interest for very long time. As some of the drawings that Monica posted by Leonardo DaVinci, it has been a subject who’s meaning has been debated and questioned time and time again. Personally, I am glad for symmetry. It makes nature beautiful, and provides a bit of mystery for each of us.

Iachello, Francesco. "Beauty in Nature: Symmetry." AIP Conference Proceedings 1488.1 (2012): 401-12. Print.

Jaeger, F. M. b. 1877. (1917). Lectures on the principle of symmetry and its applications in all natural sciences. Amsterdam: Publishing company "Elsevier,”.

Macnab, Maggie. (2012) Design by nature :using universal forms and principles in design Berkeley, CA : New Riders

Abney, Angel, Andy Tyminski, and Pawel Nazarewicz. "Symmetry in Nature." UGA Student. University of Georgia Press, n.d. Web.

Leonardo da Vinci and The Golden Ratio

            What is the Golden Ratio? As with any ratio this term can, in simple terms, be described as a proportion. Originally, the Golden Ratio was discovered by Euclid of Alexandria, who used the example of a line broken into three segments to derive the concept of an “extreme” and “mean” ratio. Mario Livio writes, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser”, when a segment is broken up into segments A, B, and C as pictured below.

Though Euclid derived this ratio solely for geometry, it was discovered to play a part in a myriad of things ranging from nature to physics to astronomy, and even in the realm of arts and aesthetics. Similar to pi (), the Golden Ratio is one followed by a never-repeating, infinite decimal number. Conversely to pi, the Golden Ratio cannot be expressed as a fraction – it is both irrational and incommensurable, which occurs when a line has two lengths with no common measure. Originally the symbol for the Golden Ratio was tau (τ), but at the beginning of the twentieth century, an American mathematician gave the ratio the name of phi (ϕ). The first six numbers of its value are 1.61803, continuing infinitely.

We as somewhat simple human beings do not usually seek or recognize mathematics in nature - or even anywhere outside of the classroom for that matter. However, the Golden Ratio has been discovered in many facets of the natural world ranging from the reproduction of rabbits, to the arrangement of leaves around branches, to the optics of light rays through glass plates. One form in particular that I find very interesting which incorporates the Golden Ratio is the Nautilus pompilius, or chambered nautilus, which is the shell of a mollusk. In pure mathematical construction, the nautilus is composed of "whirling squares" (Livio, The Golden Ratio, p.119) which are formed by cutting off squares from a Golden Rectangle, connecting the squares, and dividing the sides by the Golden Ratio. This creates an image of a logarithmic spiral curling inwards. It is also possible to devise a chambered nautilus from a Golden Triangle. 

Leonardo Fibonacci, discoverer of the famous Fibonacci sequence, played an important role in the development and application of the Golden Ratio in a variety of different areas including geometry, and nature as well. It was he who studied the reproduction of rabbits and how they regenerate in the way of the Fibonacci sequence. What he discovered was that when two rabbits of opposite genders are paired together, they will produce one rabbit, the new sum of all of the rabbits being three (0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3…). While this sequence of rabbits may seem unrelated to the Golden Ratio itself, Fibonacci experiment with the ratio by dividing the first number of the sequence by the second, the third number by the second, the fourth number by the third, and so on (1/1, 2/1, 3/2, 5/3, 8/5…987/610 = 1.618033…). Eventually by going through these prime numbers and dividing the latter by the former, one will reach the approximate value of the Golden Ratio.

So how is the ever-intriguing Golden Ratio and the absolutely enthralling Nautilus pompilius, among other things, relevant to Leonardo da Vinci? In many of his paintings, da Vinci focuses heavily on symmetry and geometrical balance. For example, his painting “Leda and the Swan” portrays the chambered nautilus in the arrangement of the woman’s hair. Da Vinci’s painting “Deluge” also imposes these logarithmic spirals even further as he describes the scene:

The sudden waters rush into the pond that contains them, striking the various obstacles with swirling eddies…The momentum of the circular waves flying from the point of impact hurls them in the way of other circular waves moving in the opposite direction.

The photo below of “Deluge” shows us even more clearly the coils of the winds and floods of the storm. It’s incredible to me how an artist, da Vinci in particular, plans their work of seemingly un-math-related artistry, of nature or of humans for example, with such intricate underlying sequences that most people would never be able to decipher without prior knowledge of the patterns.  

I, for one, would not have been able to tell whether there was a method to the madness of this storm or not. Now that I have learned about the concept of “whirling squares” and the connection of the vertices of the Golden Triangles, I can more easily sort out the disparity of the spiraling waters. Because of his prevalent use of the Golden Ratio, da Vinci has been attributed by several authors the invention of the name “the Divine proportion”. If it weren’t for the ratio, his well-known works “St. Jerome”, “Madonna on the Rocks”, and the most famous “Mona Lisa” (to name a few), Leonardo da Vinci’s paintings would not be the same. Thanks Euclid!!

Sources:

Leonardo, McCurdy, Edward. (1906) Leonardo da Vinci's Note-books London, Duckworth & Co.

Livio, Mario. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway, 2002. Print.

Müntz, Eugène. (2006) Leonardo da Vinci: Artist, Thinker, and Man of Science. New York : Parkstone Press

"The Golden Ratio: Phi, 1.618 - Golden Ratio, Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face." PhiPoint Solutions, LLC, n.d. Web. 15 Nov. 2015.

Sports Analytics

Everyone knows how important statistics are for an athlete when it comes to points scored or mistakes made. It could make or break an athletes career. But they are also used for baselines and studies on injuries, bets and financing an athlete, and performance predictions. Statistics are used not only in young athletes but also while recruiting and can be a large determinate in scholarships or salaries as well.

Lately, there has been a lot of talk about how the “game” has changed for many young athletes. The number of injuries has increased some being more serious than others. According to an article published by USA Today called, 1.35 Million Youths a Year Have Serious Sports Injuries. They state that the most common injuries occur in football with 394,350 and basketball with 389,610 injuries among athletes under the age of 19. The U.S. Consumer Product Safety Commission conducted a report on 46.5 million child athletes that found that in 2012, 12% of the sample involved a concussion and that 47% of that were 12 to 15 years old. Those findings don’t seem to be too troubling at first until the article states that it takes a longer time to heal than those who are older. A childs brain is still developing at that time and having that many concussions during that process could hinder it. A second concussion could cause even more issues. Part of that same study found that girls soccer and basketball players have been diagnosed with a concussion rather than boys although a link in that has not been determined. Lastly, the study found that there was a significant number of overuse injuries and 25% of those ended up being rather serious. Some common areas of overuse that they found were in tendons, bones, and joints. The article concludes with that rest/recovery and preventative measures such as strength and stretching exercises is recommended following an injury or to prevent one. They encourage kids to speak up about an injury so that it does not worsen overtime and hinder their athletic career.

Most people have heard of the movie Moneyball starring Brad Pitt. In the film, the Oakland team scouts high school players since there are slim pickings in the major leagues. They use analytics to recruit the players and determine their salaries in order to gain a competitive strategy on the field. This is a very common strategy for any organization. They can type in a certain height, weight, and batting average into a system and individuals fitting that criteria will come up. The only issue with is that there could be other individuals that meet this criteria not in the system they have used. While the recruitment process is going on, usually some finances are worked out as well. Teams use statistics of a player to determine what they need to meet performance wise in order to retain a certain salary or payment that the player wants. This whole process about finding players and determining salaries and so on was found in an article on Analytics Magazine’s website. The article is titled Beyond Moneyball: The Rapidly Evolving World of Sports Analytics. In this article, the author highlights how the process of “decision making” comes into play with this diagram. The diagram sums up the determinants of recruitment and eventually salaries as well.

Finally, statistics can be used as predictors as to what a player could do in the future. Teams take past statistics and identify strengths and weakness and what could be improved by the player. And they use that to figure out if this player were to improve, how their improvements could play out in future games. But they use this process with other teams as well. Having another teams statistics could help pin point the best strategy in order to compete and win. Again this process in mentioned in the Moneyball article but also in an article about the Boston Celtics on boston.com. The article is Numbers Don’t Lie, Teams Use Advanced Analytics. The article talks about how the team uses previously obtained statistics to determine starting lineups, who will sub in and out for players, and the best strategy for competing against opponents. The Celtics have this individual named Michael Zarren who processes all the statistics of their own players and opponents. He is quoted saying, “If you're a Celtics fan, we’re doing everything we can to make the team as good as we can. if you're not a Celtics fan, maybe this is an interesting part of other sports that you don’t know about.” He goes on to say “It’s not stuff that everyone will be interesting in, but like every other field that is becoming more scientific, this stuff is not going away.” What he is saying is that all other teams use analytics and its becoming more of a need in order to win and improve. It may not be more interesting than the game itself but the whole game rests on analytics and predictions.

Statistics are more commonly known for involving probability. Probability is used for determining injuries and future performance, but there’s a lot more that goes into analytics with sports. The statistics on injuries could help prevent severe issues in the future and eventually hurting young athletes and older athletes careers. Most teams use them and those who don’t will begin to just because of how important it is for the game and for the basics of a team such as lineups or salaries.  It may not be the biggest and most exciting part of the game, but it ends up being one of the most important.