Importance of Mathematics and Geometry in Architecture

F. Acerbi À 04 survey Byzantine recensions of greek mathematical and astronomical texts: e Estudios bizanticos. 2016, 4, p 133 - 213 Lela Alexidze (Tbilisi) AZ co 5 A Lo 6 Phasis 13-14, 2010-2011 p 9-26 ORPHISCHE THEOGONIE UND PLATONISCHE KOSMOLOGIE IN DEN PROKLOSKOMMENTAREN http://www.poetry.net/poem/16414 Hans Christian Andersen “Formens evige Magie 3p Berges, S. Why don't women philosophers count as philosophers? A? 0 Y Posted on June 12, 2013 by Sandrine Berges Feminist history of philosophy. NL GENESIS OF A PYTHAGOREAN UNIVERSE Alexey Burov* and Lev Burov* FERMILAB-PUB-13-533-ad 8p Chapter Trick or Truth? Part of the series The Frontiers Collection pp 157-170 \ as Importance of Mathematics and Geometry in Architecture p 1-11 Ashish Choudhary1 4 Journal of Recent Activities in Architectural Sciences Volume 1 Issue 1. 2016 ds À LI Collier, J. THE Mythological Picture OF CEBES THE THEBAN. The emperor Marcus Antonius his conversation with himself 1708

Importance of Mathematics and Geometry in Architecture Ashish Choudhary1, Naresh Tomar2, Shubhanshu Maheshwari1 1 2 MITS, Gwalior, India Department of Architecture Engineering, Amity University Rajasthan, India E-mail: choudharyashish30@yahoo.com Abstract Intentionally or unintentionally, from ages, architects, builders and construction experts have used mathematics as a very basic yet important tool for the soulful purpose of design, execution and finalization of building projects. In the history, architects were mathematicians and also some mathematicians were architect too. Vitruvius was a very well-known architect as well as famous mathematician. Mathematical readings of Pythagoras were later used in building proportions. Well known worker and user of golden ratio Leonardo Da Vinci along with many achievements was an architect too. The approach of this research paper is to come up with findings on importance of mathematics in architecture, as in geometry, from very important site analysis to final design of elevation or façade. Aim of the whole research is to come up with mathematical functions related to mensuration of building construction and Architectural Engineering. This paper is an initial part of the same research. Keywords: Geometry, form order and function, mensuration, golden section, pythagoras studies, patterns, history of architecture INTRODUCTION dealing with studies of geometric figures, The fundamental study of forms, shapes shapes and forms as elements and at the and spaces and their order along with their same place proportions, differences, angles geometry contributes to the process of positions and transformations as relations composition and designing of any element between of in composition is built by structures. The architecture begins with space developing word is derived from the Latin idea and their relations. Geometry and its study “structura” which means to associate make an important input to this process by together architecture. 1 Composition them. in The order. foundation Mathematics

crucially geometry can be seen as a HISTORY OF USE OF GEOMETRY specific study of structures by considering IN ARCHITECTURE collective sets of architectural elements Men in ancient time built to accommodate and their relations as well as operations. his This concept for an example was the buildings as a royalty. Along with background for an innovative approach to ornamental the possessed a natural geometry-based in part composition of Mr. Richard Buckminster Fuller [1, 2]. spatial needs. Royal highlights their men used dwellings on the structural characteristics of the materials that we available in appropriate Geometry can be stated as the science to diameter. As an example, we take the describe structures and spaces. Max Bill in Sumer Reed house around 4000 B.C. The his artwork plays with geometric structures strong tall reeds of Euphrates delta were as method, for example in his variations of used as standard structural elements. These a single theme of project, the process of were bunched into bundles and bent to transformation from triangle to octagon form either a circular or pointed arch. was the final product and worked well [3, Reed matting was used as filler and the 4]. whole house was skinned with mud [7, 8]. House in its simplicity with geometric Max Billwas one of great artists who gave elements a thought on the relationship between art elements of the Romanesque or Gothic and structure. Through history of geometry cathedral styles. The pyramids of Egypt and that of architecture there were are the best example of Egyptians’ developed some rules and standards based Understanding of Geometry. Pyramids on geometry which formed the very basic reflect their attempt to model their human concepts for architectural composition. In world on “cosmic order” to symbolize the following paper we will analyse the their role sequential strongly related to religion and astronomy. architectural design processes through The whole architecture of that time was several based on “occult geometry system”. This of geometry in examples of the contemporary contains stability. a all the Geometry system of structural there was architecture and along with them examples was measurements, from history of architecture [5, 6]. dimensions and proportion, which were considers as sacred and divine by religious leaders. In the Greek Period the tapering of

columns and manifestation of proportion smart use of geometry and shape to get and visual effects of building according to more and more from any piece of land the dimension of columns of different without wasting any inch square. order was the main concept of building design. The geometrical and technical CONCEPTS OF ARCHITECTURAL achievements of Gothic Period were in GEOMETRY brief, Golden Section differentiating between bearing columns and non-bearing walls, utilizing Such a fundamental principle of harmony the pointed arch, use of vault-supporting derived from nature, applied in art, ribs, development of flying buttresses, architecture and music can be seen in the large use of glass and tracery in various golden section. The idea of the golden th forms and shapes. The 13 century mason section did not solve his structural problems with composition and geometry. This idea steps software and analysis as a modern long time through history of architecture. engineer would, but by very skilful and Hippasos of Metapont (450 B.C.) found it insightful by in his research about the pentagon and the experience and geometric rules of design. relation of its edge length and the The design depended more on getting the diagonal. Euclid (325–270 B.C.) was the shape and then calculate the magnitude of first who described the golden section forces acting on it. In the same way precisely also as a continuous division. In Renaissance Period also seen its own use the following time golden section was seen of geometry in design of buildings. That as the ideal proportion and the epitome of was the period of start of architectural and aesthetics and harmony. Especially in the engineering drawings. The development of renaissance, harmonic proportions were projective geometry (orthographic and based on the geometric relations according perspective) was an event that was of both the golden section in art, architecture as architectural and geometrical importance. well as in music. Filippo Brunelleschi built On the other hand the modern idea of Santa Maria del Fiore in Florence 1296 building the based on the golden section and the functionality of building using materials Fibonacci Numbers. Golden Rectangle, like concrete, steel and glass and is Golden Triangle, Golden Spiral, Penrose reflected Tiling, Pentagon and Pentagram are some trial-and-error, design on focused building of aided on period of shows the modernism and even after, which makes 3 coherence

kind of Golden geometry based on golden Function System (IFS) is the most ratio [9, 10]. common, general and powerful mathematical tool that can be used to generate fractals. The iteration procedure must converge to get the fractal set [11, 12]. Therefore, the iterated functions are limited to strict contractions with the Fig. 1: Golden Ratio or Golden Section Banach fixed-point property. Cantor set, Sierpinski Ratio. Dragon Triangle, curve, Menger Space sponge, filling curve, Fractal Geometry Mandelbrot set are some of best examples The mathematical history of fractals began of fractal geometry. 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, Fig. 2: Examples of Fractal Progression. which is now called the Koch snowflake. In 1915, Waclaw Sielpinski constructed self-similar patterns and the functions that generate them. Georg Cantor also gave an example of a self-similar fractal. In the late 19th and early 20th, fractals were put further by Henri Poincare, Felix Klein, Pierre Fatou and Gaston Julia. In 1975, Mandelbrot brought these work together and named it 'fractal'. Fractals can be constructed through limits of iterative Moreover, IFS provides a connection between fractals and natural images. It is also an important tool for investigating fractal sets. In the 4 an introduction to some basic geometry of fractal sets will be approached from an IFS perspective. In a simple case, IFS acts on a segment to generate contracted copies of the segment which can be arranged in a plane based on certain rules [13, 14]. schemes involving generators of iterative functions on metric spaces following, Iterated

architecture you find this misunderstanding of the role of geometry. In his historical study he refers to the Fig. 3: An Example of Cantor's Set. relations between Gaspard Monge’s Descriptive Geometry and Jean-Nicolas Louis Durand’s theory of architecture. Durand taught architecture at l’École Polytechnique in Paris at the same time as Monge around 1800. Durand developed a universal planning grid for architecture. Evans describes architecture Fig. 4: Fractal Application in Temple Design. that is Durand’s based on grid the misunderstanding of the spatial coordinate system. Instead of understanding the coordinate system in an abstract way, he IMPORTANCE OF GEOMETRY IN ARCHITECTURE transformed the coordinate planes directly in architecture as floor and walls [15, 16]. Geometric and Architectural Space Concepts Geometry for Strength The architectural space is based on a geometric space concept. Especially, in the creation process architecture is thought in relation to a geometric space. Robin Evans analyses the relationship between geometry and architecture: “The first place anyone looks to find the geometry in architecture is in the shape of buildings, then perhaps the shape of the drawings of the buildings. These are the locations where geometry has been, on the whole, stolid and dormant. But geometry has been active in the space between and the space at either end.” According Evans, in history 5 Foundations being most important part of building for strength are constructed simple rectangle based cubes as they are easy to construct and give maximum efficiency and are good with the form working. On the other hand pile foundations are constructed cylindrical as they are drilled in earth minimum friction is required and at the same volume and lesser surface area cylinder is best. Triangle on the other hand is said to be the most stable shape. And for the same reason is used as most reliable members of building when it comes to load bearing and

stable structures may it be geodesic domes Geometry for Aesthetics and Ambience or big trusses. The market success of industrial products strongly depends on their aesthetic character, i.e., the emotional reaction that the product is able to evoke. To achieve their aim designers have to act on specific Fig. 5: Rectangular and Pile Foundation. shape properties, but at present they are not directly supported in this by existing digital tools for model definition and Geometry for Performance Geometric application for performance in architecture can be simply understood as shape selection for maximum output. Like dome, or hemispherical roof have numbers of advantages over flat roof as it gives column less space as the load of roof is directly transferred to the rim of hemisphere and also the hemispheric space being useless can hold the hot air with stack effect and can gradually can be disposed from the roof. And on other side conical roof are generally used to cover manipulation, mainly because of the still missing mathematical formalisation of the properties themselves. The European project FIORES-II (GRD1-1999-10785Character Preservation and Modelling in Aesthetic and Engineering Design), started in April 2000, aims at investigating and identifying the links between emotional shape perception and geometry and to create, through their mathematical formalization, more user friendly tools for aesthetic design tower like structures and pitched of triangular prism like roof is used when run off of water and snow is main concern. Relationships between a Physical Form and its Emotional Message In order to develop modelling tools to allow designers to quickly attain the desired emotional message, it is necessary to understand the procedures they follow to achieve their objectives. Within the FIORES-II project, design activities in different Fig. 6: Pitched Roof and Domed Roof Coverings. 6 industrial fields have been analysed in depth and the language used in different phases of the design cycle has

been studied. It emerged that the terms geometrical strictly related to emotional values (e.g, mechanical laws of the universe that create dynamic, aggressive, etc.) that express the the geometrical patterns in nature. Many objectives to be achieved by the end Gothic product are mainly used when designers proportions derived from the geometry talk with marketing people. On the other inherent in the cube and double-cube; this hand, during the creation and modification tradition continues in modern Christian of churches the digital model, designers laws cathedrals and quantum were to built the using present communicate their aesthetic intent using a time. Churches, temples, mosques, religiou more detailed and restricted set of terms s monuments, altars, tabernacles; as well corresponding to shape properties. In this as phase they provide instructions on which as temenoi, sacred elements and properties have to be greens and holy wells, and the creation changed to realise their objective (e.g, of religious making a curve a bit more accelerated, or symbolic and sacred meanings are ascribed decreasing the tension of its curvature) and to certain geometric shapes and certain to fulfil marketing directives. This second geometric proportions, according to Paul set of terms represents the first link Calter and others. for sacred art. spaces such groves, village In sacred geometry, between low-level geometric properties and the high-level features of a product. Hinduism Therefore, in order to identify links The Agamas are a collection of Sanskrit, between message and geometric shape, we Tamil envisage a two-level mapping: the first constituting level links geometric properties to design construction and creation of idols, worship terms; the second links these latter to the means of deities, philosophical doctrines, emotional message. meditative practices, attainment of six fold and Grantha, scriptures chiefly the temple methods of desires and four kinds of yoga. Elaborate Geometry and Religion rules are laid out in the Agamas for Silpa Sacred geometry is used as a religious, (the art of sculpture) describing the quality philosophical, and spiritual term to explain requirements of the places where temples the fundamental laws of the universe are to be built, the kind of images to be covering Pythagorean geometry and the installed, the materials from which they perceived are 7 relationships between to be made, their dimensions,

proportions, air circulation, lighting in the The square and rectangle play a significant temple complex etc. The Manasara and role in Islamic architecture. Some of the Silpasara are some of the works dealing reason for this is façades built from with these rules. The rituals followed in rectangular worship services each day at the temple brickwork casts shadows in the strong also follow rules laid out in the Agamas. desert sunlight and creates a threebricks. This ornamental dimensional effect. A recurring motif is a small central square turned 45 degrees within a larger square. Another source for the square motif is woven baskets. Fig. 7: Application of Agamas in Temple Planning. Islam Islamic decoration makes great use of Fig. 8: Cosmic Geometric Patterns in geometric patterns which have developed Islam. over the centuries. Many of these derived from various earlier cultures: Greek, The Persianate world is the main area with Roman, Byzantine, Central and buildings with decorative Persian. They are usually distinguished especially during the from for the Great Mosque of Cordoba is another decoration in Islamic art based on curving example further west. The eight-pointed and But star is another common motif in Islamic sometimes foliage and linear geometric architecture, often found in tile-work and patterns are combined in a single design, other media. Star patterns are extremely and some purely abstract linear patterns complex when the outer points are joined adopt designs that seem clearly derived together and other intersections connect in from The a systematic way. The Alhambra palace geometric designs have evolved into in Granada, Spain is a famous example of beautiful and highly complex patterns, still repeating motifs which occur in the tile used in many modern day settings. and stucco decoration. Octagons appear in the arabesque, branching vegetal 8 the vegetal arabesque Asian, term forms. ones. Page 1-11 © MAT Journals 2016. All Rights Reserved brickwork, Seljik period;

Islamic architecture in various shapes. design is not a new concept and has They frequently occur in marble floors. always formed a part of architecture and The design. The consideration of changing Citadel of Aleppo in Syria contains marble opus forces such as climate, setting, culture, and sectile floors, which utilize the square and use has always formed part of the design the eight-pointed star. Pierced screens process. (jali in India) are another common location for geometric decoration. Fig. 10: Example of Parametric Design. Parametric modelling systems can be Fig. 9: Geometry in Jali Designs. divided into two types of systems:  Propagation based systems where you Going Beyond Geometry compute from known to unknowns Parametric design is a process based on with a dataflow model. algorithmic thinking that enables the expression of parameters and rules that,  Constraint systems which solve sets of continuous and discrete constraints. together, define, encode and clarify the relationship between design intent and CONCLUSION design response. Parametric design is a The relationship between architectural paradigm in design where the relationship design and geometry starts with the notion between elements is used to manipulate of harmony as the principle for all sciences and complex and creations. The analysis of the antique geometries and structures. The term comprehension of harmony shows the 'Parametric' originates from mathematics geometrical root and the superior idea of (Parametric equation) and refers to the use this concept for all sciences and designing of certain parameters or variables that can disciplines. Today, the various sciences be edited to manipulate or alter the end and arts are in most cases strongly result of an equation or system. Parametric separated. Therefore, there is the risk that inform 9 the design

powerful relationship between more efforts necessary in the future to geometry and architecture gets lost. Steven work out this relationship in detail, Holl, who refers in his architectural work historical to geometry and other sciences, noticed: architectural and a geometrical point of “For view as well as to experience and apply it example Johannes Kepler’s Mysterious Cosmo-graphical united art, and theoretical, from an in the practice of architectural design. science, and cosmology. Fig. 11: Application of Golden Rati in Roman Temple Design. Today, specialization segregates the fields; yawning gaps prohibit potential crossBy fertilization.” remembering the historical relations between geometry and Fig. 12: Identifying Geometrical Characteristics in a Building. architectural design we help to keep the background of our culture but also to understand between the fruitful geometrical architectural combination thinking designing. By designing curriculum we in the should New York; 1987. 2. this impulses for geometrical based designing in architecture. Only few examples were shown here in an overview. There are Berkel, Ben van, Caroline Bos: Move. UN Studio Amsterdam; 1999. architecture reflect Alberti Leon Battista. The Ten Books of Architecture. Dover Publications, integrating relationship and try to develop new 10 1. and experiments on using geometric structures for REFERENCES 3. Bill, Max: Struktur als Kunst? Kunst als Stuktur? In: Georg Braziller: Struktur in Kunst und Wisssenschaft. Éditions de la Connaissance, Brüssel;

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