Signifigance of Mathematics in Archiecture

 

By: Katharine Charlotte Meinhover

Presented to Mrs. Kristina Rogale Plazonic- the most wonderful calculus professora

In this web page, I will be delighted in demonstrating the importance of mathematics in the construction of three different structures in three different parts of the world.  In each of these cases, the structures have become symbols for the brilliance, creativity and ingenious tactics derived from the world of mathematics to further build upon theories of structural analysis. 

  • The St. Louis Arch depicts the concept of an inverted centenary curve.
  • The whispering room utilizes the geometric components of the ellipse and the reflective property of ellipsoids.
  • The spiral staircases of Blois demonstrate the circular helix.

 
 
 

 

 

 

 

 

 

 

 

 

 

The St. Louis Arch was erected in 1947 as part of the project for the Jefferson National Expansion Memorial Association.  The project was sited to be built at the side of the great Mississippi River facing westward to commemorate westward expansion, a defining period in the development of the United States.  However the initial idea of the project is based on an interesting story.  The original organizers of the Jefferson Memorial posted a worldwide competition in search for a design for the new memorial.  A man by the name of Eero Saairen submitted his memorial design which made use of the new concepts of stress analysis and structural design.  He proposed the inverted centenary curve to define the project’s form.   

 
 


St. Louis Arch                                             

The arch is an inverted centenary curve. This is the

curve that would result by hanging a heavy chain freely between

two fixed points. The arch is 630 feet high. Each leg is an equilateral triangle; 54 feet at the ground level (base) which

 lengthens and tapers to 17 feet at the top. Concrete acts as the main structural component past 330 feet off the ground.

 The arch doesn’t have a structural skeleton though it is a composite structure of triangular pieces 8-12 feet high stacked on top of each other to form the inner walls.  Its inner and outer steel skins, joined to form a composite structure, give it its strength and permanence.                                                                                                                                                                                 

The mathematical concept behind the structure of the arch is fairly complicated.

 The equations are listed below:

 

The Whispering Room found in the Museum of Science and Industry in Chicago, Il is a great example of sound wave communication.  Two parabolic structures along with a uniquely shaped room, in the form of an ellipsoid, allow quiet messages to be sent and received between two people, or foci.

 

The line of sound traveling from one focus reflects directly to the other focus. This can be deduced by understanding the Reflective property of ellipsoids which states: “All waves (of light, sound, etc.) emanating from one focus of an ellipsoid at the same time will reflect off the surface of the ellipsoid and intersect at the other focus of the ellipsoid at the same time.

 

 
 
The Whispering Room

 

 

 

 

 

 

 

 

 

 

 

 

 

                                   

           

 

 

 

 

Text Box: The concept behind the property of ellipses can be traced to theories from Kepler and Newton, who investigated that the gravitational force between the Sun and the planets is central and lies along a line of a plane. Newton said that the ellipse is separated by a place into conic sections of equal size, like a reflection; a focus point, like where the sun, also has a reflected focus on the other side of the intersection of the conic sections. When a plane cuts at the base of a cone, it is a circle- cut at an angle it is an ellipse. If the cone is cut straight on the side, it is a hyperbola and at an angle, a parabola.

A crucial concept of the elliptical orbit is that object, wave or anything traveling on the orbit is always equidistant from the focus when the Kepler’s second law that the radius vector from the sun to a planet sweeps out equal areas in equal times. 

 

 
 
                                         

 

 

There are two point A, B, a line L and a moving point Q on L.
When AQ + BQ = minimum at Q = P,
angles made by AP, BP and L are equal to each other.
 

Point P is a point on the ellipse.
L is a tangent line of the ellipse at point P.
Point Q moves on L.
If Q = P , AQ + BQ = minimum.
 

A. If the interior of ellipse is silvered to produce a mirror, rays originated at ellipse's focus are reflected to the other focus of the ellipse.
 
 

 

 

 

 

 

 

 

 

 


                        A.                                                        B                                                         C

          



Some Math to consider: 

Major axis length = 2a 

Minor axis length = 2b

 

2a>2b => b2 = a2 – c2 => b2<a2 => b2<a2<0 => (b-a)(b+a)<0 => b-a<0 => b<a =>2b<2a => minor axis < major axis

 

  • X and –X; Y and –Y because of equation create

symetrical graphs with respect to origin.

  • Use Pythagorean theorem to understand relationship of minor and major axis in relation to the foci P(x,y).

d + d = 2a   2d = 2a d = a   Because P(x,y) is a point on the ellipse d(P,F’) + d(P,F) = 2a;

The square root of {[(x+c)2 +(y-0)2 ]+ [(x-c) 2 + (y-0) 2]} = 2a

(a2 - c2 )x2 + a2 y2 = a2 (a2 - c2 ) => x2 / a2 + y2 / a2 - c2 = 1

When a>c; a2 > c2 ; a2 - c2 >0, so b2 = a2 - c2 when b>0.

 
 
Foci y-axis F(o,c) F’(0,-c)

X^2 + y^2 = 1  a>b>0  b^2 = a^2 – c^2

d1 + d2 = constant c>0
 

 

The circular helix in spiral staircases

Spiral staircases have been used in architecture for hundreds of years. The phenomena behind the double helical structure, or the helicoid, are the relation of minimal surface area of the disks within the structure.  In the nineteenth century, a man named Joseph Plateau used soap bubbles and films to understand minimum energy and surface areas. In the case of spiral staircase, the shape is taken by a straight line on a plane when the plane is wrapped around a cylinder. In the circular helix, there is a ratio between a point of a plane and a translation perpendicular to the plane.

The drawing below is a simple example of its structure.  

 
 

Later, Leonardo de Vinci put these theories to test in the design structure in double spiral staircases.  In these designs, one person could descend while one could ascend the case and not meet one another.  The cases spiraled together to be very space efficient.
 
 

 


 

 

 

 

The staircase travels on a point of path much like a screw would twist about an origin of a straight line.  It moves along the surface of the cylinder of radius a, a distance of z, which is proportional to the angle of the twist. Since we are dealing with a cylinder coordinates (r, z, θ), the equations of the helix are: r = a, z = cθ, where x = rcosθ, y=rsinθ, z – which are the same as Cartesian coordinates.
 
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