# The Center of Math Blog

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## Thursday, April 14, 2016

### 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 + PartnersNote 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/

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