Problem of the Week: Finding the sides of a Square [Geometry]

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Solution below.

Problem of the Week: 11-08-18: Prove Pasch’s Postulate as a Theorem [Geometry]

Happy 175th birthday Moritz Pasch! The German mathematician was born on November 8, 1843. Pasch specialized in the foundations of geometry. He is best remembered for Pasch's axiom, which he discovered in 1882. 


Although axioms do not require proofs, in this week's problem we celebrate Pasch's birthday by- you guessed it- proving Pasch's axiom. Why? Because proofs are FUN.     

Solution:

Video:

Valentine's Day Theorem

Here is a neat little theorem referred to as the "Happy ending problem" by Paul Erdős, due to it resulting in the marriage of two mathematicians. Happy Valentines Day!

Flag Day and Mathematics

Flag Day celebrates the adoption of the United States' flag on this day in 1777. The symbol for unity, spread over thirteen stripes and fifty stars, stands tall as a momentous proclamation of the United States' values. Over the years, with the growth of our country, the flag of the U.S. has also changed, from thirteen stars to fifty. With each revision of the flag's design, a great deal of thought goes into the arrangement of our star spangled banner; and while mathematics is not always considered in this process, we know math is capable of bringing to our attention beauty, so today we will consider how math could play into our flag.

Read more after the break. 

Problem of the Week: 4-18-17

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Solution below the break.

Problem of the Week: 4-11-17

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Solution below the break.

Problem of the Week: 2-21-17

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Solution below the break.

Problem of the Week 2-14-17

Here is this week's Problem of the Week! Let us know how you did in the comments below or on social media!

Solution below the break.

Problem of the Week

Check out this week's Problem of the Week! Let us know how you did in the comments!

Solution below the break.

Problem of the Week

Check out this week's Problem of the Week! Let us know how you did in the comments!

Solution below the break.

Problem of the Week

Check out this week's Problem of the Week. Try the interactive Desmos graph below the break, and let us know how you did in the comments!

Solution transcript and interactive below the break.

Advanced Knowledge Problem of the Week

Check out this week's Advanced Knowledge Problem of the Week. Let us know how you did in the comments!

Solution below the break.

Problem of the Week

See if you can come up with a cool angle for this week's geometric Problem of the Week!

Solution below the break.

Problem of the Week

This week's Problem of the Week asks you to come up with a geometric argument for equality of the area of four triangles.

Solution below the break.

Everyday Math: Sports

Game Time: Math Style

Known for underdog upsets and nail-biting game winners, the NCAA Basketball Tournament takes the cake for the most exciting annual college sports event. Nicknamed March Madness, the tournament begins with 64 teams, but after 6 rounds of games and plenty of Cinderella stories, the field is narrowed to only one National Champion. March Madness rallies the obvious sports fanatic, but also makes it easy for anyone to stay informed and enjoy the thrill that is collegiate sports. You may be thinking, "Okay.. but what does this have to do with math?". Staying with the trend of Everyday Math, the tournament inspired us to look into some key aspects of mathematics that are incorporated into various sporting events.


We'll start by taking a look at our inspiration- March Madness.

College Basketball

     Before the tournament tips off, it is estimated that more than 40 million Americans flock to ESPN or other sports-based websites to fill out their own tournament bracket. In all, it is estimated that about 70 million brackets will be completed. Bracket-filling strategy is a personal preference. Some stay loyal to their favorite college, selecting them to win-it-all, even though it isn't probable. Others hope to shock their competitors and pick underdogs, crossing their fingers that a rare bracket will do the trick. On the other hand, some people pick top seeded teams, leaning towards favorites. 

According to USA Today, the odds that you randomly select a perfect bracket– that's guessing every game correctly– are 1 to 9.2 quintillion. 

       More serious betters may look at the mathematics behind the tournament. Experts broke down mathematical models that can be used to simulate the tournament. The tournament can be thought of as a series of coin flips rather than games. However, instead of there being a 50/50 chance of flipping heads or tails, the odds are stacked differently depending on the teams playing. The NCAA has given each team a seed and has published data regarding the results of each of these seed numbers in the past. For an example, when #5 seeds and #12 seeds play against each other, the #5 seed wins 65% of the time. This data can be used to manipulate your bracket choices. This isn't the only way to mathematically simulate the games. Some use a point spread given by Vegas betting odds and convert these point spreads into winning percentage. The Bradley-Terry model converts computer rankings into probabilities. Of course, no method can perfectly account  for the twists and turns that occur in the tournament. Every year there are upsets that result in ruined brackets and one lucky guy who decided to take a chance on the underdog. The favorite this year is Kansas, and CBS Sports calculated the chance of a Kansas victory to be 16%.

Baseball

    Busy swinging the bat and keeping their eye on the ball, baseball players aren't consciously thinking about the mathematics that make their sp ort possible. However, hitting a home run or even making it to first base is centered heavily on angles, velocity, and energy. The speed of a pitch or of a ball after it's hit can be found using a specific equation. H_o is the height from which the ball is thrown, α is the angle at which the ball is thrown, v_o is the speed at which the ball is thrown, and x is the distance that the ball travels. From the graph below, it can be seen that hitting the ball at a 45 degree angle will cause the ball to travel the farthest. 

Distance baseball will travel

Graph showing range with different α's.
Black graph α = 30^o, blue graph when α = 45^o, red graph when α = 60^o

Projectile motion of a baseball

It can be argued that any sport involves similar physics. So why was baseball one of the sports chosen to examine with a mathematical lens? Mathematical approaches to managing baseball teams have surfaced throughout the sport. In fact, the term sabermetrics specifically describes the way in which statistical analysis is applied to baseball records. The term was coined from the acronym SABR (Society for American Baseball Research), and is used to evaluate and compare baseball players. Sabermetrics takes an emotionless, objective approach to baseball. It aims to answer only questions that can be proven with facts. Perhaps one of the most celebrated proponent of sabermetrics is Billy Beane, who inspired the movie Moneyball. Beane was the General Manager of the Oakland Athletics and used statistical analysis to lead the A's to a winning season. He looked at the risk factor of each player he brought into the program, and examined who was worth the money. He looked for players who may not have carried a famous name, but could still contribute to the team. In other words, he stretched the dollar and looked for economically smart choices. This is one example of how sabermetrics and analyzing stats is prominent in the sport of baseball.

Soccer

     Soccer is the world's most popular sport, and every day millions of people around the world take the field– from professionals to small children. However, I doubt any of these players take the time to examine the geometry that makes a soccer team successful. The basic shape of soccer is a triangle. the players on the field are connected by imaginary triangles, that build upon each other to diamonds and other shapes. Why a triangle? It allows for the most passing lanes and provides an option in every direction. The best teams in the world are known for being able to move the ball around the field in these triangles. 

     Free kicks provide another use for the application of geometry. Defenders line up in a wall, in hopes to impeded a direct path to the goal. The wall is set up at a specific angle and the person kicking the ball tries to bend the ball behind the ball. However, the amount of people that stand in the wall is dependent on where around the goal the ball lies. While the goalie isn't thinking about math when he/she sets up the wall, the logic behind it is definitely mathematical. The chart below shows that as the ball moves away from the center of the goal, less people are needed to defend the wall. When the ball is in the middle of the goal, the shooter has a larger angle, so the width of the wall needs to be greater. 

Problem of the Week

Give this Problem of the Week a tri-angle, and let us know what you came up with! There are a lot of great possibilities for this week's problem.

Solution(s) below the break!

Everyday Math: Mathematics in Art

Everyday Math: The Art of Mathematics

Students and teachers alike often place art and mathematics on opposite sides of the academic spectrum. They consider there to be little overlap between the logic that blooms in a math class and the creativity that flourishes is an art room. Rather than running beside each other as parallel lines, mathematics and art intertwine in a way that creates some of the world's greatest masterpieces. The mathematical techniques used by renowned artists proof that math is everywhere, seeping into every aspect of academia and entertainment. 

The Golden Ratio

The Golden Ratio is explained algebraically as .
This ratio, also known as the Divine Proportion, is used by artists on a geometric scale. For an example, the Golden Rectangle is a specific rectangle that follows the Golden Ratio in its side lengths. The Golden Ratio also manifests itself in nature and the human body, making it useful for artists.



Da Vinci

Leonardo Da Vinci is credited with some of the world's most famous paintings, and also is one of the prominent artists to utilize the Golden Ratio in his work. The most acclaimed painting showing clear examples of the Golden Ratio is Da Vinci's "The Last Supper". From the architecture of the background to the minuscule details on the shield, implications of the Golden Ratio are evident throughout. Other, more controversial examples of the Golden Ratio is Da Vinci's work can be found in the "Mona Lisa" and "The Annunciation". Da Vinci's fascination with ratios can be seen in "The Vitruvian Man". Here, Da Vinci measured the human body using specific ratios drawing ideal proportions according to the work of Vitruvian.


Raphael

Working in the shadow of Da Vinci, Raphael also used the Golden Ratio to create visual beauty and harmony in his art. In his piece, "The School of Athens", a perfect Golden Rectangle can be seen painted in the forefront. Many scholars say that this was Raphael's way of noting his recognition of the Golden Ratio, and prompting others to seek them in his work. Like Da Vinci, the placement of subjects as well as the architecture in the background is painted with the Golden ratio in mind.

Geometric Patterns

Many artists create pieces that involve only simple shapes, yet are situated such that a beautiful design is created. Using only circles, triangles, and squares, many artists have created masterpieces that are still viewed today.

Flower of Life

The Flower of Life can be traced back 5,000 years to Egypt. It is made by overlapping circles at specific points. By starting the next row of circles on the circumference of the row before, a certain pattern in created that stays constant throughout. In the religion of Islam, animals and people are not typically depicted on Mosques. Because of this, geometric shapes like the Flower of Life are painted on Mosques around the world. It also serves as a symbol of the divine order of the Universe. 

Modern Tessellations

Originally, tessellations were large images created by placing smaller tiles in a certain pattern. However, a modern interpretation of a tessellation is a design that incorporates non-square tiles and completely fills a space without gaps or overlaps. Tessellations can be found in nature, but are commonly created in art. M.C Escher is known as the father of modern tessellations and is infamous for his passion for creating complex tiled designs.