Mathematics and architecture are related. In the past, architecture was a part of mathematics and architects who constructed those architectures we still marvel at today were also mathematicians. Furthermore, lots of magnificent buildings in the past contained or was build based on mathematical models. Understanding how mathematical knowledge has helped the construction of buildings in life since many yeas ago can help people be aware of the importance of mathematics and motivate their passion towards mathematics. The approach of this research paper is to come up with findings on importance of mathematics in ancient architectures and magnificent bridges, as in geometry, from very important site analysis to final design as well as giving readers, especially people outside the construction industry a simple idea of how mathematical knowledge have been related to architectural design and construction.
Keywords: Architects, Geometry, History, Design, Construction, Ancient Architecture, Bridges.
The basic study of forms, shapes and spaces contributes to the process of designing and building architectures.
Composition in architecture begins with space developing and their relations. Geometry and its study make an important input to this process by dealing with studies of geometric figures, shapes and forms as elements and at the same place proportions, differences, angles positions and transformations as relations between them. The foundation of composition is built by structures. Mathematics and crucially geometry, can be seen as a specific study of structures by considering collective sets of architectural elements and their relations as well as operations. In the following paper we will analyse the role of geometry in the sequential architectural design processes through several examples of ancient architecture and magnificent bridges.
In ancient times, people always favoured some special types of construction.
The golden section (see figure 1) shows the coherence of composition and geometry. This idea steps long time through history of architecture. Hippasos of Metapont (450 B.C.) found it in his research about the pentagon and the relation of its edge length and the diagonal. Euclid (325-270 B.C.) was the first person who described the golden section precisely also as a continuous division. In the following time golden section was seen as the ideal proportion and the epitome2 of aesthetics and harmony. Especially in the renaissance, harmonic proportions were based on the geometric relations according the golden section in art, architecture as well as in music. Filippo Brunelleschi built Santa Maria del Fiore (See figure 2) in Florence 1296 based on the golden section and the Fibonacci Numbers. Golden Rectangle, Golden Triangle, Golden Spiral, Penrose Tiling, Pentagon and Pentagram are some kind of Golden geometry based on golden ratio. (Ashish Choudhary, Nitesh Dogne, Shubhanshu Maheshwari)
The mathematical history of fractals began with mathematician Karl Weierstrass in 1872 who introduced a Weierstrass function which is continuous everywhere but differentiable nowhere. In 1904 Helge von Koch refined the definition of the Weierstrass function and gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1975, Mandelbrot brought all previous years’ work together and named it ‘fractal’. Fractals can be constructed through limits of iterative schemes involving generators of iterative functions on metric spaces Iterated Function System (IFS) is the most common, general and powerful mathematical tool that can be used to generate fractals. The iteration procedure must converge to get the fractal set. Therefore, the iterated functions are limited to strict contractions with the Banach fixed-point property. Cantor set, Sierpinski Triangle, Menger sponge, Dragon curve, Space filling curve, Mandelbrot set are some of best examples of fractal geometry.
Fractal geometry plays an important role, from Hindu temples, where the self repeating and self-similar components are supposed to reflect the idea that every part of cosmos contain all information about the whole cosmos, to gothic architecture, with a high degree of self similarity and complex detailing.
Clearly, pyramids are built based on the model of what people call triangular pyramids nowadays. The pyramids of Ancient Egypt are tombs constructed with deliberately chosen proportions, but which these were has been debated. The face angle is about 51°85’, and the ratio of the slant height to half the base length is 1.619, less than 1% from the golden ratio. This design would imply the use of Kepler’s triangle (face angle 51°49’) in the pyramids. However, it is more likely that the pyramids’ slope was chosen from the 3-4-5 triangle (face angle 53°8’), known from the Rhind Mathematical Papyrus (1650 – 1550 BC); or from the triangle with base to hypotenuse ratio 1:4/π (face angle 51°50’). (Gazale, Midhat. 1999).
The possible use of the 3-4-5 triangle to lay out right angles and the knowledge of Pythagoras theorem which that would imply, has been much asserted. (Cooke, Roger L. 2011) It was first conjectured by the historian Moritz Cantor in 1882. It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use knotted cords for measurement; that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3-4-5 triangle; and that the Berlin Papyrus 6619 from the Middle Kingdom (before 1700 BC) stated that ‘the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other.'(Gillings, Richard J. 1982) The historian of mathematics Roger L. Cooke observes that ‘It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem.’ Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle’s sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor’s conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that ‘there is no evidence that they used it to construct right angles’. (Cooke, Roger L. 2011)
Mughal architecture, as seen in the abandoned imperial city of Fatehpur Sikri and the Taj Mahal complex (Rai, Jaswant. 1993), has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony (Michell, George. 2011). Generally, it’s built on the models of cylinder and polygon.
The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. The traditional accounts give the width of the complex as 374 gaz, the main area being three 374-gaz squares. These were divided in areas like the bazaar and caravanserai into 17-gaz modules; the garden and terraces are in modules of 23 gaz, and are 368 gaz wide (16 x 23). The mausoleum, mosque and guest house are laid out on a grid of 7 gaz.
Byzantine architecture. The Byzantine architecture includes a nave crowned by a circular dome and two half-domes (Fazio, Michael. 2009), all of the same diameter (31 metres (102 ft), with a further five smaller half-domes forming an apse and four rounded corners of a vast rectangular interior. This was interpreted by mediaeval architects as representing the mundane below (the square base) and the divine heavens above (the soaring spherical dome). (Gamwell, Lynn. 2015)
Sagrada Família, Barcelona. Antoni Gaudí used a wide variety of geometric structures, some being minimal surfaces, in the Sagrada Família, Barcelona, started in 1882 (and not completed as of 2015). These include hyperbolic paraboloids and hyperboloids of revolution, (Burry, M.C., J.R. Burry, G.M. Dunlop and A. Maher 2001) tessellations3, catenary arches, catenoids, helicoids, and ruled surfaces. This varied mix of geometries is creatively combined in different ways around the church. For example, in the Passion Façade of Sagrada Família, Gaudí assembled stone ‘branches’ in the form of hyperbolic paraboloids, which overlap at their tops (directrices) without, therefore, meeting at a point. In contrast, in the colonnade there are hyperbolic paraboloidal surfaces that smoothly join other structures to form unbounded surfaces. (Usvat, Liliana)
Bridges are built in a variety of different forms including simple beams, trusses, arches, suspension, and cable-stayed structures. The choice of material and structural form depends on a wide range of factors such as the load to be carried including the weight of vehicles, the span length of the bridge, the construction and maintenance costs, the visual impact and so on. (R.A.E)
Suspension Bridge. In a suspension bridge, the roadway is actually hanging from large cables(mathcentral). The cables run over the top of two large towers (which are rooted deep into the earth) and connect to anchorages at each end of the bridge. When constructing a bridge, architects must consider the compression and tension forces that the bridge is going to have to withstand – compression and tension. The cables and the towers of the suspension bridge are designed to deal with the weight of traffic. The towers are dug deep into the earth for stability and strength. Tension is combated by the cables, which are stretched over the towers and held by the anchorages at each end of the bridge. Wind can be detrimental to a bridge.
For that reason, a deck truss is often placed under the roadway of the bridge. This provides additional stability for the bridge. The parabolic shape of the suspension bridge is also interesting. At first glance, the curve may be described as a catenary. A catenary is a curve created by gravity, like holding the end of a skipping rope in each hand and letting it dangle. However, because the curve on a suspension bridge is not created by gravity alone (the forces of compression and tension are acting on it) it cannot be considered a catenary, but rather a parabola. The parabolic shape allows for the forces of compression to be transferred to the towers, which upholds the weight of the traffic. The parabolic shape can also be proved mathematically, using formula comparisons.
Steel Truss Road Bridge. A truss bridge is a bridge whose load-bearing superstructure is composed of a truss, a structure of connected elements usually forming triangular units. The connected elements (typically straight) may be stressed from tension, compression, or sometimes both in response to dynamic loads. Truss bridges are one of the oldest types of bridges. The basic types of truss bridges shown in this article have simple designs which could be easily analyzed by 19th and early 20th-century engineers. A truss bridge is economical to construct because it uses materials efficiently.
In analysing the structure of Steel Truss Road Bridge above a number of simplifying assumptions are made. Firstly, the three dimensional bridge structure is idealised as a two dimensional structural model as shown in figure 10 below. Secondly, the structure is formed from a series of braced triangular frames. Thirdly, all loads (weight of traffic etc) are transferred to the structure through the nodes or joints. Then, the only forces in the members are either axial tension forces (putting a member into a state of tension) or axial compression forces (putting a member into a state of compression). (RAE. The Mathematics of Framed Bridge Structures)
The approach to analysing the structure is to consider the equilibrium of each joint of the structure as show in figure 10(b) above. The general engineering principle to be applied is that at each joint equilibrium is assured if the components of all forces acting in (a) the horizontal direction and (b) the vertical direction summate to zero. In mathematical terms this can be expressed as:
The application of these expressions will give two equations which can be solved for two unknown forces. Hence, if at any joint there are no more than two unknown member forces then there will be two simultaneous equations that can be solved for the two unknowns. For example, figure 12(a) below shows a joint connecting two members inclined at 60o and 45o respectively with an external applied force of 20kN acting in a downwards direction and 10kN acting to the right. The unknown forces in the two members are shown as F1 and F2 and are shown as acting away from the joint which is a convention that assumes that the members are acting in tension. If subsequent calculation indicates that the value of force in either of these members is negative then this indicates that the members are in compression.
Above all, we can see that mathematics including geometry plays an indispensable in ancient architectures and bridges. We could say that without mathematics, people are unbale to design those beautiful, magnificent, practical and high-quality architectures.
Without any experiment, this study is just objectively set to discern the relationship between buildings and mathematics, and how the mathematics modeling affected changes of the building components. Today the various sciences and arts are in most cases strongly separated. Therefore there is the risk that the powerful relationship between geometry and architecture gets lost. Specialization segregates the fields nowadyas; yawning gaps prohibit potential cross-fertilization.” By remembering the historical relations between geometry and architectural design we help to keep the background of our culture but also to understand the fruitful combination between geometrical thinking and architectural designing.
In conclusion, things in the paper remind people that while admiring those magnificent architects the ancients built, there are more tasks necessary in the future needed to be work out, both historically and theoretically, from an architectural and a geometrical point of view as well as to experience and apply it in the practice of architectural design.