Trigonometry in Architecture

Trigonometric identities are used heavily in architecture. All of the six different identities come in hand when finding the length of the sides of a wall, or at the angle the material must be placed at to receive the desired outcome for the given location. Before architecture became primarily digital, architects had to be very good at math. Blueprints of a given structure always involve trigonometry, because something must be built perfectly, and a structure simply will not hold if the walls are not made to match up with the ceiling properly. Knowing the sin and cosine of an angle between two walls, allows the architect to evaluate the amount of material that will be necessary to complete the project. Trigonometry allows one to be as accurate as possible when determining the correct sizes of geometric structures. Intricate bridges, benches, and buildings that have curves can use trigonometry by mimicking the unit circle and following the rules of this genre of mathematics accordingly. It is much easier to build a structure when positive of all of the measurements. Even Vectors, which have a starting point, magnitude and direction -- allows one the ability to define the forces and loads that a given structure can support. Trigonometry is obviously based upon the principles of the triangle, and this shape is a main component in architecture.  By understanding the key concepts of trig, one can obtain all information regarding angle measurements and lengths, which is a necessity when building a strong, aesthetically pleasing structure.

Sin

        Dr. Stefano Bertozzi's speech given at the 2014 convocation for Fountain Valley School of Colorado evoked many solutions as to fixing our world.   I thought he did an excellent job of conveying his message through metaphors and real life experiences.   The main concept outlined throughout his "send off" to the graduating class was to give to this world as much as you take.  Of course, as he noted, as a teenager one takes substantially more than giving, but that is only because they have yet to begin their actual life.
       Out of all of the nations in this world, Americans are specifically notorious for being consumers.  We take your food, we take your jobs, we take your home, we take everything we can if it is lucrative and beneficial to us.  I perceived Bertozzi's message as moving and motivational.  He highlighted the atrocities in third world nations and how they could use assistance desperately, while making his audience question their greed.  By listening to this I had doubts about my own morals and I began to think of the good I have contributed to this world-- i already know I am "that person" who literally eats more than she brings!  In order to move forward, we need to be more conscientious  of our surroundings and stop being so selfish, and I think carrying on that advice was Bertozzi's principle purpose of addressing high school students.

The Quadratic Function

Not only is the quadratic extremely helpful and used in abundance in math class, but it is pertinent in daily life as well- especially in sports. Any ball thrown up into the air will follow the trajectory of a parabola, which of course is the parent function of x squared. In order to find the height, time, or speed of a throw, you must use the quadratic function. This can be used in any sport such as tennis, soccer, football, etc, and a lot in architecture.  In addition to these uses, the quadratic function can be applied when calculating the area of an object or finding the profits a company makes, like in this example:

• Unit Sales = 70,000 - 200P
• Sales in Dollars = Units × Price = (70,000 - 200P) × P = 70,000P - 200P²
• Costs = 700,000 + 110 x (70,000 - 200P) = 700,000 + 7,700,000 - 22,000P = 8,400,000 - 22,000P
• Profit = Sales-Costs = 70,000P - 200P² - (8,400,000 - 22,000P) = -200P² + 92,000P - 8,400,000

 We say all the time in class, "when am I ever going to use this?!" but in reality, you most likely will use it during sometime in your career. There are so many forms of a quadratic, (such as vertex form), but one thing that's always true is the parabolic shape that it graphs. Knowing how to graph is extremely important because it brings an equation to life and it becomes visually apparent how it would look if it were a ball thrown in the air.

Mapping the trajectory of a ball.

https://www.youtube.com/watch?v=ReHwNtoRMrY
A video bringing the quadratic formula to life.

Tesselations

There are only three regular polygons that tessellate on the Euclidean plane.  These polygons include triangles, squares and hexagons. I recently found out what it even meant for something to "tessellate".  I did an artistic project involving the, a while ago, but I have long forgotten the actual definition. One with no knowledge on this matter, in other words, me prior to some research, would think that "to tessellate" is to decorate a surface with mosaics, but in the math world, a tessellation is what's created when a single shape, is copied and repeated throughout a whole plane, covering every area and leaving no gaps. Regular tessellations are made up of congruent regular polygons.  It makes sense for a hexagon to be capable of regular tessellation considering 6 triangles make up a hexagon, but the reason that triangles, squares, and hexagons are the only regular polygons that tessellate, is because the interior angles of these three shapes, (60, 90, and 120 degrees), are exact divisors of 360. It is possible to create tessellations using a variety of different polygons in a repeated and uniform pattern, but that is not considered a regular tessellation.  Tessellations are not only mind boggling to grasp the concept of, but when you put color and flare onto them, they are truly beautiful.  Having mathematical knowledge about them is not only useful in the world of numbers and shapes, but in architecture and decorating as well.

http://mathforum.org/sum95/suzanne/whattess.html

Pascal's Triangle

      In the 10th century, hundreds of years before the birth of Blaise Pascal, Indian mathematicians utilized the triangle in addition to Middle Eastern.  Following the Iranian mathematician, Omar Khayyam, a Chinese mathematician used it and discovered how the triangle provided coefficients for expanding (a+b) to a certain degree which is an imperative fact to be aware of when in high-school math class.

       There is a plethora of different patterns which can be made by shading in specific numbers on Pascal's Triangle.  Whether you shade in odd numbers, even numbers, multiples of 6, or multiples of 11, you are bound to see  a trend within the patterns. Blaise Pascal, who was a French mathematician, was a sickly child and lived from 1623-1662, and obviously, the triangle was named after him.  Even though Pascal receives the majority of credit for this mathematical gem (for it is named after him), other people from other nations such as China, India, Western countries, and Iran discovered it long before.

     Aside from Pascal's triangle, Blaise discovered the first digital calculator, the Pascaline.  His calculator was not very popular amongst the people, for it could only perform addition.  He created this device in order to make his father's job involving finances to be easier. His father was a major influence on his life considering he was home-schooled by him and grew up with no mother.  Not only was Pascal a math guru, but he was a philosopher as well and he laid the foundations for theories of probability.

"Small minds are concerned with the extraordinary, great minds the ordinary"- B.P.