Abstract
In this paper, we will discuss and describe the application of mathematics in architecture. Starting from the explanation of the meaning of architecture, we will consider the relationship between architecture and mathematics since ancient times. One of the most significant contribution of mathematics in architecture is strengthening the building constructions. This property can be traced in many ancient structures (for example, the Parthenon, the Colosseum, Stonehenge). The evolution of the types of construction design was tightly related to mathematics. In our research, we will discuss various designs, such as pyramidal, post-and-lintel and carcass construction systems.
Special attention will be paid to symmetry as to one of the bases of aesthetics in architecture. Different kinds of symmetry (such as mirror symmetry and rotation symmetry) will be discussed. Another fundamental contribution of mathematics in architecture is Golden Ratio. The history of this concept and the application of its use in famous architectural masterpieces will be provided. Finally, the paper is summed up in theconclusion section.
Introduction
The term "architecture" has several meanings. Architecture is the most ancient sphere of human activity and its results. In the wide sense, architecture means a set of buildings and structures for various purposes. It is a space created by a human, used for his life and work. Architecture was born with mankind, accompanied it during the historical development. It reflects the human’s worldview, values, knowledge of people living in different historical epochs. It focused particularly on the culture of different nationalities. Architectural monuments have come down to us from immemorial time. They help us understand the purposes, attitudes, thoughts, traditions and habits, perceptions of beauty, knowledge of people who once lived on Earth. Why the architectural structures were erected? First of all, they were built for the convenience of human life and activity. They were to serve human’s favor: to protect him from heat and cold, rain and sun. They had to create a comfortable environment for various human activities - to provide adequate lighting, provide sound insulation or good distribution of indoor sound. The constructions should be safe and long to serve. But human nature also relates to the pursuit of beauty, so whatever he does, he tries to make nice.
The close relationship of architecture and mathematics has long been known. In Ancient Greece, geometry was considered as one of the sections of the architecture. Modern architect must be familiar with the various ratios of rhythmic series, which make the object the most harmonious and expressive. In addition, he must know analytical geometry and mathematical analysis, the foundations of algebra and matrix theory, methods of mathematical modeling and optimization. The study program of architects includes mathematical and computer skills.
Sometimes, due to insufficient knowledge of mathematics, an architect has to do a lot of unnecessary work.
Mathematics in Structures Strengthening
In ancient times, people raised their homes and thought primarily about their strength. The strength of a building is associated with longevity. People spent enormous effort on the buildings’ construction and therefore they were interested in their durability – they should stand as long as possible. An example of such a reliable and strong construction that has survived to our days is Greek Parthenon and the Colloseum. Durability of structures is provided not only by the material, from which it is created, but also by the construction that is used as a basis for its design and construction. The strength of the construction is directly connected with the geometric shape, which is the base for it. A mathematician would say that it is very important to select a correct geometrical form (a solid), which fits the structure in the best way.
The most lasting architectural structure has long been considered is the Egyptian pyramids. As we know, they have the form of regular quadrangular pyramids. This geometric shape provides the greatest stability at the expense of a large area of the base. On the other hand, a pyramid shape provides a reduction in weight with increasing height above the ground. These two properties make pyramid stable, and therefore durable in conditions of terrestrial gravity.
The pyramidal shapes were replaced by post-and-lintel systems. In terms of geometry, it is a polyhedron, which is obtained by placing one rectangular parallelepiped onto two vertical rectangular parallelepipeds.
The is one of the first types of structures, which was used in the construction of buildings. It consists of vertical beams and horizontal beams covering them. The first such structure was the dolmen – a religious building. It consisted of two vertical stones with the third vertical stone placed onto them. In addition to the dolmen another structure has survived to our times, which represents a simple post-and-lintel structure - cromlech. It is also a religious building, presumably intended for sacrifice and ritual celebrations. Cromlech consisted of separate stones that were covered by horizontal stones. They form two or more concentric circles. The most famous cromlech is preserved to this day in the town of Stonehenge in England. Some scientists believe that it was an ancient astronomical observatory ("Post-And-Lintel System | Architecture").
It should be noted that the post-and-beam design is so far the most common type of construction in the building industry. Most modern residential houses basically have post-and-beam structure. Stone does not bend, but works well in compression. This has led to the use of architectural arches and vaults. A new type of construction has been developed - the arch-vaulted structure. With the arch-vaulted structure, circles, spheres and circular cylinders have entered the architecture of straight lines and planes. Initially, they were used only in the architecture of semicircular arches and hemispherical dome. This means that the boundary of the arc was a semicircle, the boundary of the dome was a half sphere. For example, a hemispherical dome has the Pantheon - the Temple of all the gods in Rome. The diameter of the dome is 43 meters. The height of the walls of the Pantheon is the radius of the hemispherical dome. In this relation, it turns out that the building of the temple was “put” on a ball with a diameter of 43 m.
This type of design was one of the most popular designs in ancient Roman architecture. The arch-vaulted design allowed Roman architects to build giant structures made of stone. These include the famous Colosseum or Flavian Amphitheatre. It got its name from the Latin word colossus, which translates as a huge or enormous. The same design is used to create the giant Baths of Caracalla and Diocletian, which had accommodated up to three million visitors. This type of design also includes a system and arch culverts and aqueducts with the total length of 60 km.
The next step in the development of architectural designs was a frame (or carcass) system. Flying buttresses were the frame that surrounds the building and take the main load. Arched design was the inspiration of the carcass structure, which is now used as a major in the construction of modern buildings of metal, glass and concrete. Recall the construction of the famous towers: Eiffel Tower in Paris and TV Tower in Shabolovka. TV Tower in Shabolovka consists of several hyperboloids, which stacked one on another. Each piece is made of two families of straight beams. The tower was designed by the engineer Shukhov ("The Shukhov Tower Foundation"). Hyperboloid is a surface formed by the hyperbole rotation, situated symmetrically with respect to a coordinate axis in the rectangular coordinate system around the other axis. Any axial section of hyperboloid is bounded by two hyperbole.
Another interesting geometric surface is hyperbolic paraboloid. This is the surface, which in cross section has a parabola and hyperbola. The emergence of new building materials makes it possible to create a thin reinforced concrete frame and walls of glass. This type of design is used in many US skyscrapers. Glass, concrete and steel frame structures prevailing in the architectural structures of the XX century. They provide a high degree of strength of buildings.
Symmetry as a Measure of Architectural Excellence
This word comes from the Greek word “symmetria”, which means proportionality. Considering the symmetry in architecture, we are interested in the geometric symmetry - the symmetry of shapes such as the proportionality of parts of the whole. It has been observed that under certain transformations of geometric figures, their parts, moved to a new position, will again form the original figure. In axial symmetry, the parts, which substitute each other, formed a straight line. This line is called the axis of symmetry. The analogue of axis of symmetry in the three-dimensional space is a plane of symmetry. Thus, in the space we usually consider symmetry about the plane of symmetry. For example, a cube is symmetrical about a plane passing through its diagonal. Due to the two cases (plane and space), this kind of symmetry is sometimes called the mirror symmetry. The name is justified by the fact that both of the figures, which are on opposite sides of the axis of symmetry or plane of symmetry, are like an object and its reflection in the mirror ("Introduction & Symmetry Operations").
In addition to the mirror symmetry, there is considered central or rotational symmetry. In this case, the transition of the parts to the new position and the formation of the original figure occurs when you turn the figure at a certain angle around the point, which is usually called the center of rotation. Hence, the above mentioned names the type of symmetry. Swivel symmetry can be viewed in space. Cube that turning around the intersection point of the diagonals at an angle of 90 in a plane parallel to any of its faces, switched into itself. Therefore, we can say that the cube is a centrally symmetric figure, it has rotational symmetry.
Architectural structures created by a human for the most part are symmetrical. They are pleasing to the eyes, people find symmetry beautiful. Symmetry is perceived by people as a manifestation of the internal order. Externally, the internal order is perceived as a beauty.
Symmetrical objects have a high degree of expediency, because things are more balanced and have an equal functionality in different directions. All this has led the humankind to believe that that the building is beautiful, if it is symmetrical. Symmetry has been used in the construction of worship places and residential buildings in ancient Egypt. The decoration of these structures are also examples of the use of symmetry. Most clearly symmetry is reflected in the buildings of ancient Greece, in the luxury items and ornaments that adorned them. Since then and up to present times the symmetry in the mind of man has become the objective sign of beauty.
The Golden Ratio in Architecture
It is considered that the concept of the Golden Ratio was introduced into scientific use by Pythagoras, the Greek philosopher and mathematician. There is an assumption that Pythagoras borrowed his knowledge of the Golden Ratio from Egyptians and Babylonians. Indeed, the proportion of the pyramid of Cheops, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun show that Egyptian craftsmen used Golden Ratio in their creations. French architect Le Corbusier found in the relief of the temple of Pharaoh Seti I at Abydos and in relief, depicting the Pharaoh Ramses, the proportions of the figures correspond to the values of the Golden Ratio ("Giza Pyramid Temples & The Golden Section"). Architect Hesira, depicted in relief with wooden boards from the tomb of his name, holding a measuring instruments, which are fixed in the proportion of the Golden Ratio.
Greeks were good geometricians: they even taught their children by using geometric shapes. Pythagoras Square and the diagonal of the square were the basis for building dynamic rectangles. Plato also knew about the gold division. His dialogue "Timaeus" is devoted to mathematical and aesthetic views of the school of Pythagoras and, in particular, on the Golden Ratio. In the facade of an ancient Greek temple of the Parthenon there is a golden proportion. In its excavations, there were found compasses used by architects and sculptors of the ancient world. The Pompeian Museum in Naples is also incorporated a proportion of the Golden Ratio. In literature, Golden Ratio for the first time mentioned in the "Elements" of Euclid. In the 2nd book of "Elements", there is given the construction of the Golden Ratio
In the Renaissance, the interest to the Golden Ratio among scientists and artists is increased in connection with its application both in geometry and in the arts. Especially it is seen in the architecture of Leonardo da Vinci. According to contemporaries and historians of science, Luca Pacioli was a real star, the greatest mathematician of Italy in the period between the Galileo and Fibonacci. Luca Pacioli was a pupil of the artist Piero della Francesca, who wrote two books, one of which was about the "perspective in painting." He is considered the creator of descriptive geometry. Luca Pacioli knew the importance of science to art. In 1509, in Venice he has published a book named "Divine Proportion" with brilliantly executed illustrations. The book was an enthusiastic hymn of the Golden Ratio. Among the many advantages of the Golden Ratio, monk Luca Pacioli called it as "divine essence" as an expression of the divine trinity of God the Son, God the Father and God the Holy Spirit (implying that small segment is the personification of God's son, a greater segment is God the Father, and the entire segment is God the Holy Spirit) (Gleeson-White).
Leonardo da Vinci also paid much attention to the study of the Golden Ratio. He produced sections of a stereometric solid, formed by regular pentagons and each time he received rectangles with the relationship of the parties in the Golden Ratio. The great astronomer XVI centuries Johannes Kepler considered the Golden Ratio as one of the greatest treasures of geometry. He first drew attention to the significance of the Golden Ratio to botany (plant growth and structure).
Golden Ratio is a harmonic proportion, it is a proportional division of the segment into unequal parts, in which the ratio of the whole segment to the bigger part is equal to the ration of the bigger part to the smaller part. It can be written mathematically as follows:
a:b = b:c or c:b = b:a.
The sections of the Golden Ratio are expressed in the irrational infinite fraction 0.618 and 0.382 For practical purposes, architects often use approximate values of 0.62 and 0.38. If the segment is taken as 100 parts, the bigger part is equal to 62 and the smaller one is 38.
In the books about the Golden Ratio, we can find a remark that in architecture and in painting, it all depends on the observer's position. If some proportion of the building on the one hand seems to form the Golden Ratio, then with the other points of view, they will look different. "Golden Section" provides the most tranquil ratio of various lengths. One of the most beautiful works of ancient Greek architecture is the Parthenon (V century BC). The Parthenon has 8 columns on the short sides and 17 on the long sides. The building is made entirely of marble squares. The nobility of the material used in temple building helped the architects to shorten the use of coloring. It only emphasizes the details and forms a colored background (for example, blue or red) for the sculptures. The ratio of building height to its length is equal to 0.618. If we make a division of the Parthenon by the Golden Ratio, we will get the projections of the facade.
Conclusion
Since the times of ancient Greeks, the question of the mathematical assumptions of beauty and the role of mathematics in the art and architecture was of a great importance. In our time, the geometry is a necessary element of general education and culture. Geometry is also of great historical interest, it has fundamental practical application and inner beauty. Mathematical calculations, measurements and constructions are the most important and essential techniques for an architect. There are many similarities between mathematics and architecture. Among them, the units of calculations used both in mathematics and architecture, the tools and concepts, methods and properties. It proves the close relationship between mathematics and architecture again and again. Divine mathematical proportions of the Golden Ratio give the architectural beauty of the building, it pleases the human eyes. So, mathematics has an aesthetic impact on the architecture. Summarizing the above, in terms of architecture, it can be said that mathematics is a big mental structure that simulates the world around us and the phenomenon occurring in it.
Summing up, mathematics offers an architect many tools and general rules of the organization of parts into a whole. It helps to:
locate these parts in space, so that they manifest the order;
set a definite relationship between the size of parts and set (reduce or enlarge) a certain single pattern for the changes of the size that provides the perception of integrity and understanding of the order;
Works Cited
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Gleeson-White, Jane. "Mathematics, Art And God: The Divine Proportion Of Luca Pacioli With Drawings By Leonardo Da Vinci". bookish girl. N.p., 2012. Web. 15 Apr. 2016.
"Introduction & Symmetry Operations". Tulane.edu. N.p., 2016. Web. 15 Apr. 2016.
"The Shukhov Tower Foundation". Shukhov.org. N.p., 2016. Web. 15 Apr. 2016.
"Post-And-Lintel System | Architecture". Encyclopedia Britannica. N.p., 2016. Web. 15 Apr. 2016.
