Intense math behind

Architecture BY: Shaylee Renner

Lydia Christian

Katie Guido Roman Architects implemented the Golden Ratio and the Golden Rectangle when they built the Colosseum. Egypt Sydney, Australia -The golden ratio is a standard ratio for length to width of rectangles used in a structure.

-The Golden Ratio is represented by the symbol Phi and is equal to 1.618033. A pyramid is a polyhedron whose base is a polygon and its sides are triangles that have a common vertex. Pi was inadvertently applied

by the Egyptians in creating the pyramids. Rome, Italy - Divide the perimeter of the pyramid by its height = 2 pi.

-Slope of each face = 4/pi The Colosseum Agra, India Australian builders were inspired by segments of an orange when designing the Sydney Opera House.

The shells used in construction were all taken from the same sphere.

Each shell was made up of a series of ribs formed by parallel lines. Egyptian Pyramids Sydney Opera House Taj Mahal Reflective Symmetry -The formal gardens are divided into four sections.

-The four minarets are 137 feet high and are cylindrical columns with beveled angles. Located at each of the corners of the raised marble plinth the minarets repeat the right angles. Seattle, Washington Space Needle An important concept in building the Space Needle is angular velocity. Leaning Tower of Pisa Piazza, Italy Will the tower ever fall over?

5.5 degree angle

-theta>= arcsin(D/2//H)

-D represents the width of the base

-H is the height of the center of gravity. Euler's Rotation Theorem states for a rotating frame there exists an instantaneous axis of rotation in any instant. In other words, Angular velocity is the angular speed at which something rotates- instantaneous rate of change. Fibonacci Sequence: The first two numbers are 0 and 1, and each following number is the sum of the previous two. (0,1,1,2,3,5,8...)The formula is Fn=Fn-1 + Fn-2 The Golden Rectangle The Golden Rectangle occurs when a square section is removed and a golden rectangle is left over. It is repeated indefinitely. Accidental Pi Relationship The Egyptians did not have to calculate pi to use it. By using a wheel with a diameter of 2 units (cubits) they rolled out the perimeter exactly as many revolutions as the pyramid was units tall. Geometry Angular Velocity of Space Needle: PEACE OUT In cubits, the pyramid's perimeter = 365.24 - the number of days in a yr

The Sagrada Familia also features a Magic Square within the Passion façade – an arrangement where the numbers in all columns, rows and diagonals add up to the same sum: in this case, 33. The Magic Constant, or M is the constant sum in every row, column and diagonal and can be represented by the following formula M = n (n^2 +1)/2. Antoni Gaudi, the famous architect who designed the church, used hyperbolic paraboloid structures (a quadric surface, in this case a saddle-shaped doubly-ruled surface, that can be represented by the equation z = x2/a2 – y2/b2), which can be seen within particular façades. Remember the Golden Ratio? The Fibonacci sequence goes hand in hand. As you take the numbers of the sequence (...2, 3, 5, 8, 13, 31, 34, 55, 89...) and divide them by the previous term (2/1=2, 3/2=1.5, 5/3=1.67, 8/5=1.6, 13/8=1.625, 21/13=1.615, 34/21=1.619, 55/34=1.618, 89/55=1.618...) as the sequence gets larger the ratios will eventually converge to the same number : 1.618. That is the same number that the Golden Ratio is rounded to. Relationship The Gherkin’s unusual design features – the round building, bulge in the middle, the narrow taper at the top and spiraling design – create a huge impact on the structure itself. The cylindrical shape minimizes whirlwinds that can form at the base of large buildings, something that can be predicted by computer modeling using the math of turbulence.

What’s more, the bulging middle and tapered top give the illusion of a shorter building that wouldn't block out sunlight, helping to maximize natural ventilation and save on air conditioning, as well as lighting and heating bills. Built with the help of CAD (Computer Aided Design) and parametric modeling, the Gherkin is now an important part of London's skyline. The Church of the Sagrada Familia Magic Square The Gherkin CAD software for mechanical design uses vector based graphics to depict the objects of traditional drafting. However, it involves more than just shapes. The output of CAD must convey information that includes materials, processes, dimensions, and tolerances.

CAD may be used to design curves and figures in two-dimensional (2D) space; or curves, surfaces, and solids in three-dimensional (3D) space.[5]

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