Everyday Math: Architecture

Throughout this series, we have discovered mathematics posing as a character on your favorite television show, hiding in your favorite pieces of art, and starring in your favorite big-screen productions. We have even uncovered math in the baseball diamond, at the casino table, and on the soccer pitch.. but we aren't done yet. This Everyday Math blog post goes even further to examine the mathematics behind the very building you are sitting in, and the people who designed it.

That's right– let's take a look at the math behind architecture!

According to the dictionary, architecture is the art or practice of designing or constructing buildings. While this is common knowledge, the close correlation between mathematics and architecture might go unrecognized. The connections between the two disciplines are almost innumerable, especially when you realize that both, in a way, are the study of patterns and systems. Throughout history, some of the greatest architectural feats have been based in the realm of mathematics. This post will further explore some of these buildings, as well as the connection between math and architecture.


The first instance in history that demonstrates a mathematical association to architecture involves, not surprisingly, Pythagoras. Known for being the mind behind the Pythagorean theorem, numbers held a special significance to the Greek mathematician. This significance was mainly geometrical, as Pythagoras spoke of square, oblong, and triangular numbers. He also placed an importance on the aesthetic qualities of numbers and proportions. After Pythagoras's death, his followers– the Pythagoreans– carried on his ideas and utilized them in 447 BC when rebuilding the Parthenon. The ratio 3 : 4 : 5 was used throughout the building of the Parthenon, and later was notated as one of the Pythagorean Triples. Further, the ratio between the height and width (4 : 9), is the same as the ratio between the width and length (4:9). This may not seem significant but Berger, a mathematician, examined that the ratio 4:9 is used in the creation of the columns, as well as the inner area of the temple. Many argue that this consistent ratio accounts for the aesthetic beauty of the temple.

Let's turn the history book ahead a few hundred years. Time– 27 BC. Place– Ancient Rome. In case you dosed off during a few months of History class, there are many consistencies between the societies of Ancient Greece and Ancient Rome. Like Ancient Greece, Rome focused on education, the arts, and beauty. This sets the stage for more architectural accomplishments, and with that more mathematical connections. In fact, in his series of books titled De Architectura, Virtruvius outlines the practical applications of mathematics that are necessary for building design.  The Classical age again mirrored the former great civilizations in Greece and Rome, creating buildings that relied again on the importance of proportions, ratios, and perspective. The Classical age brings us Brunellischi, Alberti, and Leonardo da Vinci– all fascinated with mathematics. 

London City Hall– Created by Foster + Partners
Note the helical staircase inside

After that nice history lesson, you may be wondering.. Why should I care? Throughout history, math was needed to properly structure buildings. As we mentioned, the two areas of study were completely intertwined. Today, technological advancements (like calculators!) help architects with some of the work load. However, in many architectural projects, these technological advancements are still set in place through mathematics. 

One of today's most famous architecture studio is Foster + Partners. The company is famous for constructing enormous structure that dwarf their surrounding buildings. With the added size, comes more of a need for math. The buildings need to be made secure, aesthetically pleasing, comply with building regulations, and maximize a budget. A series of equations and programs help to ensure all of these categories are fulfilled. The Special Modeling Group's (SMG) was created to maximize the efforts of architects. SMG's often build larger shapes from smaller shapes. Makes sense, right? In order to do this, they create equations for various sections of a building. Examples of this are pictured below. 

Has this post convinced you of the connections between mathematics and architecture? 

In case it hasn't, here are some testimonials from architects further explaining their personal opinions.

(Testimonials gathered from http://www.lifeofanarchitect.com/architecture-and-math/)  



Jes Stafford
"Architects should be math ninjas. The aspiring architect should rush headlong into math as if charging into a field of battle. Math is an education in problem solving and of knowing what is asked. There are few stronger parallels to all the the variables in the Builder-Architect-Client dynamic. All math puns intended."

Andrew Hankins
"Math is important to my daily tasks as an Architect. It mostly involves simple calculations, but for me, it is necessary to be able to do them quickly in my head.  And they are mostly simple equations, but it definitely helps if you can do them in your head and on the fly."

Evan Troxel
"That said, it is better if you are decent at math. Here are some examples people usually don’t think of as math, but are things architects use all the time: We are constantly adding and subtracting measurements, thicknesses, volumes and areas. We are responsible for budgets. We work with spreadsheets that tally sizes of spaces and everything has to all add up. We do TONS of geometry, and we love it. Geometry is math, right? Yes it is. Drawing + Math = Awesome. That’s one reason we’re architects and not artists."


Sources: http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Architecture.html
https://plus.maths.org/content/perfect-buildings-maths-modern-architecture
http://www.lifeofanarchitect.com/architecture-and-math/

Everyday Math: Math in the Movies

Coming soon to theaters near you: Mathematics

You may not run to the nearest theater to watch a mathematician prove a theorem, but math may be starring in your favorite movie without your recognition. This installation of Everyday Math features big screen hits that would not be possible without mathematics. Usually a math story line is intertwined with a romance, mystery, or comedy, which can make it easier to miss. Don't worry! We're here to point out our favorite films that carry a heavy dose of real math information. Grab your popcorn and make your next movie night a math-themed experience!

The Classic: A Beautiful Mind

A Beautiful Mind is based loosely on the real life story of John Nash, a Nobel Prize winner. Russell Crowe plays John Nash, a mathematical genius that specialized in game theory, differential geometry, and differential equations. Game theory can be utilized in fields such as economics and political science. In fact, Nash won his Nobel Prize in economics. In a famous scene, the film dramatizes Nash's discovery of the Nash equilibrium, a term used in economics and game theory. The film is said to take artistic interpretation of Nash's real life, but the mathematics in the movie are based on real theorems and theories. The director of A Beautiful Mind enlisted the help of a mathematics consultant, Dave Bayer of Columbia University, to ensure the mathematics were correct throughout the film. The movie contains some math jokes and facts that may only be clear to those well versed in mathematics. For an example, at the end of the film, a student wants to show Nash a proof exploring the idea that "finite Galois extensions are the same as covering spaces", which is actually a true statement. Along with his prowess in mathematics, A Beautiful Mind also demonstrates Nash's story of mental illness and schizophrenia. His schizophrenia initially impacts his career, but he is able to recover and take his place as one of the leading mathematical and economic minds of his time.

Watch the dramatized discovery of Nash's Equilibrium here!

The Romantic Drama: Proof

Proof, starring Gwenyth Paltrow and Anthony Hopkins, ties in universal themes with the backdrop of mathematics. The movie tells the story of 2 mathematicians, father and daughter, fighting mental illness and attempting to prove various mathematical theorems. Before he dies, Robert (the father) makes note of the interesting characteristics of the number 1729. The daughter eventually takes the place of her ailing father and dedicates herself to mathematics, even though she lacks formal training. Critics explain that the film realistically expresses the field of mathematics. It shows the nuances of proving a theorem, as well as the work and studying that are involved. Paltrow's character, Catherine, describes how she feels when she is attempting to solve a difficult problem, comparing "elegant proofs" to music. Proof also makes reference to other real mathematicians that may or may not be known to the public, such as Sophie Germain and Carl Friedrich Gauss. The director of the film consulted heavily with Timothy Gowers, a Fields medalist from Cambridge University.

For the Sport's Enthusiast: Moneyball

Watch a mental math scene here

Moneyball is the perfect flick for sports enthusiasts and statisticians alike. Starring Brad Pitt, Moneyball takes a twist on the average sports film, incorporating mathematics into the strategy of baseball. The movie is adapted from a book of the same name, written by Michael Lewis. The plot is based on the 2002 Oakland A's and General Manager Billy Beane. Billy Beane took a different route when gathering players, in order to deal with the economic impositions placed on the team. He searched for undervalued players, and looked specifically at statistical analysis in order to determine who was worth the cost. Other 'risky' plays like stealing bases and bunting were thrown out the window under Beane's guidance. The heavy dose of mathematics in the movie is centered on the Pythagorean Expectation, which is used to calculate wins based on runs scored and allowed. The sabermetric approach to baseball is placed head to head against more traditional methods, and definitely makes for an entertaining film.

Pythagorean Expectation

The Teen Rom-Com: Mean Girls

This may be an outlier of the group, as the movie is geared towards teenage girls and features all the facets of your typical high-school movie. Mean Girls star Lindsey Lohan plays a homeschooled girl who tries her luck at navigating high school for the first time. She comes across some new friends, and they attempt to steer her in the right direction. Cady Herring, Lohan's character, has an aptitude for math and even joins the math club. Her math teacher, played by Tina Fey, is featured several times in the movie explaining various high-school math concepts. Late in the movie, Cady attends a mathlete competition and is faced with a limits problem. There are also scenes where she is being tutored in calculus, and the math errors showed are typical errors a high school student would make. This movie is perhaps the epitome of teen-movies circa 2004 but the mathematics represented, although correct, definitely correspond with the high school setting.

Click here to watch the scene!

The Story of An Underdog: Good Will Hunting

Known as a classic math movie, Good Will Hunting is a must-see. This underdog tale has a romantic twist and stars both Matt Damon and Robin Williams. Matt Damon's character, Will Hunting is a troubled young adult who's life path weaved in and out of foster homes and trouble with the law. While working as a janitor at MIT, he is able to solve two math problems that were created for graduate students. A professor at MIT took interest in Will, noticing his affinity towards mathematics and genius-level ability. Along his journey, Will faces his inner struggles with the help of a therapist (Robin Williams), and meets a romantic interest who helps to shape his life. The math problem that Will faces on the board is actually a real problem, although not as difficult as it is made out to be. The problem involves a feature of graph theory, homeomorphically irreducible trees. Pictured below is the problem that sent Will into the realm of academia. Interestingly enough, the math brains behind the movie actually appeared on screen as well. Patrick O'Donnell, who had a minor roll in the bar scene, actually ran the math department at University of Toronto at the time. O'Donnell and John Mighton, who plays the professor's assistant, chose the equations and theorems used in the movie. 

See the solution here!

Everyday Math: Your Favorite Hit Series

“When will I even use this stuff?” This question can be heard echoing off the walls of math classrooms everywhere. Okay sure, no one is going to approach you on the street and ask you to calculate the value of x. However, it is impossible to deny that mathematics is everywhere. Here at the Center of Math, we want to make math accessible to mathematicians, but also to people who may not appreciate the roll math plays in their everyday lives. This blog series will highlight unexpected areas where math may be hiding.

Math in your Favorite Hit Series 

Whether its channel surfing in your living room or heading to your local theater, the entertainment industry is a guilty pleasure for almost everyone. With your hands full of popcorn and thoughts submersed in the show, you may not realize how large of a roll math plays in some of your favorite movies and television shows. Here are some television series and movies that include story lines centered on mathematics. 

The Simpsons

Believe it or not, this long-time beloved American cartoon regularly includes a heavy dose of mathematics. In fact, the team of writers includes some paramount mathematicians. Executive producer Al Jean studied mathematics at Harvard, and Jeff Westbrook found himself on the Simpsons team after serving as a senior researcher at Yale. Their love for academia can be found in a wide array of mathematical references throughout the series. Some are very blatant, as full episodes are based on a mathematical storyline. This is the case in "MoneyBart", where Lisa manages Bart's baseball team using her keen sense of math. Other references are well concealed within the comedy and may only be obvious to those well-versed in the area. In "Marge and Homer Turn a Couple Play", three numbers appear on a jumbotron. These numbers seem random to the viewers, but the mathematicians-turned-writers deliberately chose 8128, 8208, and 9181– a perfect number, a narcissistic number, and a Mersenne prime. The most famous Simpsons math moment is when Homer jots an equation on board, in an effort to create an annoying noise, and as a result disproves Fermat's last theorem.

Click here to watch: Simon Singh discusses The Simpsons and Fermat's Last Theorem 

The Big Bang Theory 

The math references in this television show may be more obvious than those in The Simpsons, but the importance of math in the show's dynamic can not be denied. The Big Bang Theory documents the life of a group of 5 'nerdy' physicist friends. The humor that drives the show comes from the witty comedic interactions that occur and the interesting way that the show plays on 'academic' stereotypes. The show has a science and math consultant, Dr. David Saltzberg, who checks the accuracy and relevancy of the jokes on the show. Each episode includes a whiteboard that is crawling with real equations. One of the actresses, Mayim Bialik, even has a Ph. D in neuroscience that backs up her character's role on the show.


Futurama 

Many of the same comedic, mathematical masterminds that are credited with The Simpsons, also write for Futurama. The creator, David Cohen squeezes comedy into every inch of the screen. He makes use of background space, which is where many of the mathematical references can be found. Only a certain niche of people will take note of these jokes, and it makes the show seem custom-tailored to their interests. The writers of Futurama even created a theory specifically for the show. Ken Keeler created the "Futurama Theorem", which was the first mathematical proof designed specifically for entertainment purposes. He was also interested in portraying mathematics as fun and exciting, in order to attract young viewers to the field. 

Click here to watch: Futurama Theorem