**Calculus in Architecture**

Background info

Historically, architecture has been a part of mathematics, so much so the two disciplines were indistinguishable. Ancient architects were mathematicians, as well. Throughout time, architects have created mathematically amazing structures such as; temples, pyramids, ziggurats, and integration projects. Architects could use simple geometry if buildings were simple squares and triangles, but a vast majority of buildings involve curves and strange angles- this is where calculus comes into play.

When Calculus is neglected

A well-known example of a design plan gone wrong is the Leaning Tower of Pisa. The building was originally intended to stand upright, but the weight was not accounted for and early on in it's construction began leaning over. Wet soil and decay have caused the structure to sink downward and tilt. The tower is just one example of a result of neglecting the math that goes along with the architecture process.

Applications of Calculus

In addition to accurately depicting the layout of the building, calculus also is used to find the volume and surface area of buildings for their building requisites, as well as to find the specifications for acoustics within a building.

Surface area/Volume

Calculus is used to find the area under a curve,or the area of two intersecting curves, as seen in the Athenuem. This is important because with the area and surface volume architects will then calculate the amount of materials needed, as well as the cost of the project.

Acoustics

Calculus is used for acoustics to better define curves of a structure, in order to produce the right reverberations within the building. The curves of the Jubilee church were designed using calculus to achieve a strong echo.

Varied shapes

A parabola, or an archway, has been used in the past for buildings such as churches. They are efficient in supporting structures because they displace the force exerted down on them. Calculus comes into play because these shapes often vary and allow architects to design such buildings that have repetitive structures, but with modifications that are aesthetically pleasing and useful in supporting the building.

Architectural design optimization (ADO) is a subfield of engineering that uses the method of optimization to aid and solve architectural design problems. Optimization would also be useful in minimizing the amount of materials needed for buildings that are non uniform shapes, such as curved domes, and help to find the weight of those shapes/determine the necessary support structures.

Architectural design optimization

Derivatives in use

Although the tower looks complex, surprisingly, it can be graphed with two simple equations;

y= e^-x

y= -e^-x

As the tower was being built, wind became a major concern for the architects. But by utilizing calculus they found the tangent lines of the towers main curves (their derivatives). The tangent lines were used to locate the center of the tower or where its mass is focused. By knowing this center the architects balanced the tall structure by using wind as an ally instead of an obstacle. Thus, the tower was molded by wind.

Integral Calculus

Many factors come into play when an architect designs a building. Ascetics, lighting, size, and even heating are taken into consideration. If an architect is focusing on the heating of his/her structure and wants to figure out how much heat is being lost based on temperature variations throughout the day, they can utilize calculus. They can graph heat loss vs. time and find the area under the curve. This area can be found by evaluating a definite integral and finding the area between the curve and the x-axis.

Applications of Calculus

Calculus can be utilized by architects to express design plans through graphs or drawings. They can mathematically describe surfaces for the adaptation of drawings to computer software. This can be done through various differential equations. Calculus was used in the designing and construction of the mathematically sound Eiffel Tower.