Application of mathematics in Architecture

1. TOPIC- APPLICATION OF
MATHEMATICS INARCHITECTURE

2. ABSTRACT
Mathematics has various roles in architecture. In mathematics geometry has a crucial part
for designing . Architects uses geometry to define the spacial form of the buildings.
Mathematics is the science that deals with the measurement properties and relationships of
quantities as expressed in either numbets,quatity, geometry and forms.
INTRODUCTION
Mathematics is the science that deals with the measurement, properties, and relationships
of quantities, as expressed in either numbers or symbols. The definition of mathematics is
the study of the sciences of numbers, quantities, geometry and forms. It is an abstract
representational system used in the study of numbers, shapes, structure, change and the
relationships between these concepts.
Galileo Galilei said, "The universe cannot be read until we have learned the language
and become familiar with the characters in which it is written. It is written in mathematical
language, and the letters are triangles, circles and other geometrical figures, without which
means it is humanly impossible to comprehend a single word. Without these, one is
wandering about in a dark labyrinth." French mathematician Claire Voisin states "There is
creative drive in mathematics, it's all about movement trying to express itself."
Some people describe mathematics more of a language in which every symbol and every
combination has precise meaning which can be determined by application of logical rules.
This language can be used to describe and analyze anything in the universes. Mathematics
helps counting. It helps measuring. It helps comparing things.Addition, subtraction ,
multiplication and divisions are the basic operations of the mathematics, through which we
can define and develop many more operations situating our practical situation.
Mathematics is essential in many fields, including natural science, engineering, medicine,
business, banking, architecture, art, social science, cooking, sports, agriculture, insurance,
space research, technologies etc. Mathematics arises from many different kinds of
problems. At first these were found in commerce, land measurement, architecture and
later astronomy; today, all sciences suggest problems studied by mathematicians, and
many problems arise within mathematics itself. Mathematics expresses itself everywhere,
in almost every facet of life - in nature all around us, and in the technologies in our hands.
Mathematics is the language of science and engineering - describing our understanding of
all that we observe.

3. Architecture is both the process and the product of planning, designing, and constructing
buildings and other physical structures. Architectural works, in the material form
of buildings, are often perceived as cultural symbols and as works of art. Historical
civilizations are often identified with their surviving architectural achievements.
Architecture has to do with planning, designing and constructing form, space and ambience
to reflect functional, technical, social, environmental and aesthetic considerations. It
requires the creative manipulation and coordination of materials and technology, and of
light and shadow. Often, conflicting requirements must be resolved. The practice of
Architecture also encompasses the pragmatic aspects of realizing buildings and structures,
including scheduling, cost estimation and construction administration. Documentation
produced by architects, typically drawings, plans and technical specifications, defines
the structure and/or behavior of a building or other kind of system that is to be or has been
constructed.
The link between mathematics and architecture goes back to ancient times, when the two
disciplines were virtually indistinguishable. Pyramids and temples were some of the earliest
examples of mathematical principles at work. Today, math continues to feature prominently
in building design. architects can explore a variety of exciting design options based on
complex mathematical languages, allowing them to build groundbreaking forms.
Architecture begins with geometry. Since earliest times, architects have relied on
mathematical principles. The ancient Roman architect Marcus Vitruvius believed that
builders should always use precise ratios when constructing temples. "For without
symmetry and proportion no temple can have a regular plan," Vitruvius wrote in his famous
treatise De Architectura, or Ten Books on Architecture.
OBJECTIVES OF STUDY
To know about the application of mathematics in architecture.
To know how mathematics and architecture are connected.
To find some examples that shows the application of mathematics in architecture.
To understand which are the mathematical terms or concepts used in architecture.
To understand why mathematics is used in architecture.
DATA COLLECTION
The Great Pyramid of Giza, Cairo, Egypt

4. The superlatives that describe the Great Pyramid of Giza speaks for itself: its
the largest and oldest of the three pyramids and was the tallest man-made structure in the
world for 3,800 years, but there’s also plenty of math behind one of the Seven Wonders of
the Ancient World. Built around 2560 BC, its once flat, smooth outer shell is gone and all
that remains is the roughly-shaped inner core, so it is difficult to know with absolute
certainty. The outer shell remains though at the cone, so this does help to establish the
original dimensions.
There is evidence, however, that the design of the pyramid may embody these foundations
of mathematics and geometry:
Phi, the Golden Ratio that appears throughout nature.
Pi, the circumference of a circle in relation to its diameter.
The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570
– 495 BC), which can be expressed as a² + b² = c².
Taj Mahal, Agra, India
Sitting firmly at the top of many traveler’s wish lists, the Taj Mahal in India is a delight for
tourists, with many waiting to get that iconic photo in front of this beautiful building. But
look closer and we can find a great example of line symmetry – with two lines, one vertical
down the middle of the Taj, and one along the waterline, showing the reflection of the
prayer towers in the water.

5. Pentagonal, Phyllotactic Greenhouse and Education Center
The Eden Project, in South West England, opened in 2001 and now ranks as one of the
UK’s most popular tourist attractions. Although visitors come to check out what’s inside,
the greenhouses –geodesic domes made up of hexagonal and pentagonal cells – are pretty
neat too.
‘The Core’ was added to the site in 2005, an education center that shows the relationship
between plants and people. It’s little surprise that the building has taken its inspiration from
plants, using Fibonacci numbers to reflect the nature featured within the site.
There’s even more math to be found in the building structure, which is derived
from phyllotaxis, the mathematical basis for most plant growth (opposing spirals are found
in many plants, from pine cones to sunflower heads).
Parthenon, Athens, Greece
Constructed in 430 or 440 BC the Parthenon was built on the Ancient Greek ideals of
harmony, demonstrated by the building’s perfect proportions. The width to height ratio of
9:4 governs the vertical and horizontal proportions of the temple as well as other
relationships of the building, for example the spacing between the columns.
It’s also been suggested that the Parthenon’s proportions are based on the Golden
Ratio (found in a rectangle whose sides are 1: 1.618).
The Ancient Greeks were resourceful in their quest for beauty – they knew that if they made
their columns completely straight, an optical illusion would make them seem thinner in the

6. middle, so they compensated for this by making their columns slightly thicker in the
middle.
The Gherkin, London, UK
A Mathematically-Inclined Cucumber in the Sky Standing 591-feet tall, with 41 floors
is London’s skyscraper known as The Gherkin ( like the cucumber). The modern tower
was carefully constructed with the help of parametric modeling amongst other math-savvy
formulas so the architects could predict how to minimize whirlwinds around its base. The
design’s tapered top and bulging center maximize ventilation. The building uses half the
energy of other towers the same size. Any mathematician would be pleased to claim credit
for the building, but architectural firm Foster and Partners might have something to say
about that.
The Gherkin’s unusual design features – the round building, bulge in the middle, the narrow
taper at the top and spiraling design – create an impact in more ways than you might think.
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
doesn’t block out sunlight, helping to maximise natural ventilation and saving 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 a distinctive feature in
London’s city skyline.

7. Chichen Itza, Mexico
Chichen Itza was built by the Maya Civilization, who were known as fantastic
mathematicians, credited with the inventing ‘zero’ within their counting system. At 78 feet
tall, the structure of El Castillo (or ‘castle’) within Chichen Itza is based on the astrological
system.
Some fast facts: the fifty two panels on each side of the pyramid represent the number of
years in the Mayan cycle, the stairways dividing the eighteen tiers correspond to the Mayan
calendar of eighteen months and the steps within El Castillo mirror the solar year, with a
total of 365 steps, one step for each day of the year.
Sagrada Familia,Barcelona,Spain
Designed by Antoni Gaudi, the Sagrada Familia is one of Spain’s top tourist destinations.
There’s plenty of math to get your teeth into too. Gaudi 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.
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.

8. Guggenheim
Museum, Bilbao,Spain
Bilbao may not be the first place you’d think to travel to in Spain, but the Guggenheim
Museum certainly gives you a good excuse to pay this northern port city a visit. Since
opening to the public in 1997, the Guggenheim Museum Bilbao has been celebrated as one
of the most important buildings of the 20th century and it’s not hard to see why.
Intended to mimic a ship, the titanium panels, which look like fish scales, were designed to
appear random but actually relied on Computer Aided Three Dimensional Interactive
Application (CATIA). In fact, computer simulation made it possible to build the sorts of
shapes that architects from earlier years could have only imagined.
The building is a perfect example of the more avant-garde architecture of the twentieth
century and represents a landmark for its innovative architectural design both abroad and
domestically, forming a seductive backdrop for the exhibition of contemporary art. Visits
can be made to both the interior and the exterior of the building.
The design of the building follows the style of Frank Gehry. Inspired by the shapes and
textures of a fish, it can be considered a sculpture, a work of art in itself. The forms do not
have any reason nor are governed by any geometric law. The museum is essentially a shell
that evokes the past industrial life and port of Bilbao. It consists of a series of interconnected
volumes, some formed of orthogonal coated stone and others from a titanium dkeleton
covered by an organic skin. The connection between volumes is created by the glass skin.
The museum is integrated into the city both by it height and the materials used. Being below
the benchmark of the city, it does not surpass the rest of the buildings. The limestone, of a
sandy tone, was selected specially for this aim. Seen from the river, the form resembles a
boat, but seen from above it resembles a flower.

9. Philips Pavilion, Brussels, Belgium
( Experimental Math-Music Pavilion)
This is a construction of asymmetric hyperbolic paraboloids and steel tension cables. This
amazing building appeared at the first Expo after World War II, so it was an important
moment that allowed its creators to show off the technological progress the world had made
since the devastating battle. Philips Electronics Company wanted to create a unique
experience for visitors, so they collaborated with an international group of renown
architects, artists, and composers to create the experimental space. ArchDaily wrote about
the groundbreaking, temporary building, calling it the “first electronic-spatial environment
to combine architecture, film, light and music to a total experience made to functions in
time and space. It was through these visually inspired concepts that elevated the Philips
Pavilion into a complete experience where one could visualize their special movements
through a space of sound, light, and time.” Poeme Electronique was one of the works
prominently displayed at the time.
Tetrahedral-Shaped Church
The tetrahedron is a convex polyhedron with four triangular faces. Basically, it’s a complex
pyramid. You’ve seen the same geometric principle used in RPGs, because the dice is
shaped the same. Famed architect Walter Netsch applied the concept to the United States
Air Force Academy’s Cadet Chapel in Colorado Springs, Colorado. It’s a striking and
classic example of modernist architecture, with its row of 17 spires and massive tetrahedron

10. frame that stretches more than 150 feet into the sky. The early 1960’s church cost a
whopping $3.5 million to construct.
Modern Music-Math Home
A classical violinist commissioned an eccentric, $24 million dollar home located on the
edge of a Toronto ravine. The curved, elegant structure — which also serves as an
incredible concert space for 200 people — was named the Integral House. (Calculus geeks,
represent!) The home’s owner Jim Stewart was a calculus professor who wrote textbooks
and wanted to incorporate the mathematical sign into the home’s name and design.
Undulating glass and wood walls also echo the shape of a violin.
Solar Algorithm Wizardry
Barcelona’s Endesa Pavillion used mathematical algorithms to alter the cubic building’s
geometry, based on solar inclination and the structure’s proposed orientation. Algorithms
can be used to create the perfect building for any location with the right computer program.
For Endesa, the movement of the sun was tracked on site before an architect from the
Institute for Advance Architecture of Catalonia stepped in to complete the picture. The
algorithm essentially did all the planning for him, calculating the building’s optimal form
for that particular location.

11. Cube Village
Cube Village, built by Dutch architect Piet Blom. His tilted, geometric houses — built on
top of a pedestrian bridge to mimic an abstract forest — are split into three levels. The top
has windows on every facade and feels like a separate structure entirely.
Fractal Gas Station Makeover
A fractal is a fragmented geometric shape that is split into several parts, but each of those
components is just a smaller-sized copy of the overall form. Many architects apply this
mathematical principle to their building designs, like this Los Angeles gas station that
recently had a “green” makeover. Everything has been stripped down — including the
filling station’s signs, which are subtle symbols — and the mirrored facade beautifies
ninety solar panels that power the station. Recycled materials and a plant-covered roof
complete the enviro-friendly revamp.

12. The London City Hall
The London City Hall houses the Mayor of London, the London Assembly and the Greater
London Authority. The use of glass and a giant helical staircase in the interior are supposed
to symbolise the transparency and the accessibility of the democratic process. What is most
striking when looking from the outside, though, is the building's odd shape.
Perched on the banks of the river Thames, the building is reminiscent of a river pebble,
with its roundness again hinting at the democratic ideal. But as with the Gherkin, the shape
was not only chosen for its looks, but also to maximise energy efficiency. One way of doing
this is to minimise the surface area of the building, so that unwanted heat loss or gain can
be prevented. As the mathematicians amongst you will know, of all solid shapes, the sphere
has the least surface area compared to volume. This is why the London City Hall has a
near-spherical shape.
The building's lopsidedness is also conducive to energy efficiency: the overhang on the
South side ensures that windows here are shaded by the floor above, thus reducing the need
for cooling in the summer. As with the Gherkin, computer modelling showed how air
currents move through the building and the geometry within the building was chosen to
maximise natural ventilation. In fact, the building does not require any cooling at all and
reportedly uses only a quarter of the energy of comparable office spaces.
Even the helical staircase was not chosen for entirely aesthetic reasons. As part of their
analysis, the SMG modelled the lobby's acoustics, quite appropriately for a building
representing the voice of the people. Initially the acoustics were terrible with echoes
bouncing around the large hall. Something was needed to break up the space. One of Foster
+ Partners' past projects provided a clue: the Reichstag in Berlin also contains a large hall,
but in this case it is broken up by a large spiral ramp. The SMG created a model of a similar
spiral staircase for the London City Hall and the company Arup Acoustics analysed the
acoustics for this new model. As you can see in the animation below, sound is trapped
behind the staircase and echoes are reduced, so the idea was adopted in the final design.
The London City Hall on the river Thames.
ANALYSIS

13. Man has always needed shelter. In the earliest days men were nomads whose main
occupations were hunting and fishing. In order to survive they moved from place to place
very frequently. They were content to live in caves and other temporary shelters. With the
advent of agriculture, men were able to settle in more permanent locations, and they built
lasting structures to use as homes. It was then that architecture came into being.
As years passed, mans knowledge grew and principles of construction improved. No longer
were men satisfied to build houses alone. Now they designed tombs in which to be buried,
monuments to serve as memorials, palaces to house the rulers, and churches where they
could worship their gods. To produce structures that were functional as well as models of
architectural beauty, designers had to apply principles of mathematics in their work. Proper
ratios and proportions related each feature of a building with every other one and with the
whole structure. Various geometric shapes provided maximum use as well as a pleasing
appearance in all types of architecture.
Architects use a variety of shapes in designing buildings, combining various shapes,
modifying a single shape, repeating a single shape, and repeating combinations of shapes.
Analyze the buildings to determine what geometric shapes have been used in designing
each building and how those shapes are used and relate to one another.
Rectangles are found in most buildings. Their shapes can vary greatly.Curves and arcs are
often used in architectural details, columns, windows, and as structural support elements.
Triangles add variety and drama, lengthening lines, and are often found in roofs and
window elements. Circles add interest and contrast from rigid rectangular shapes. They
are often found in windows and decorative elements.
No: Name of the building Mathematical terms/concepts
used
1 Pyramid Phi, the Golden Ratio Pi, the
circumference of a circle in
relation to its diameter.The
Pythagorean Theorem
2 Taj Mahal line symmetry
3 Greenhouse and Education
Center
hexagon and pentagon

14. 4 Parthenon
Golden Ratio
5 Gherkin cylindrical shape, parametric
modeling
6 Chichen Itza astrological system
7 Sagrada Familia hyperbolic paraboloid , Magic
Square
8 Guggenheim Museum Gemetric shapes
9 Philips Pavilion asymmetric hyperbolic
paraboloids
10 Tetrahedral-Shaped Church tetrahedron
11 Modern Music-Math Home curved, elegant structure
12 Solar Algorithm Wizardry mathematical algorithms

15. 13 Cube Village geometrical shapes (cube)
14 Fractal Gas Station fragmented geometric shape
15 London City Hall Helical, spiral shapes
FINDINGS AND CONCLUSION
Mathematics and architecture are related, since, as with other
arts, architects use mathematics for several reasons. Architects use geometry: to define
spatial forms, to create forms considered harmonious, and thus to lay out buildings and
their surroundings according to mathematical, aesthetic and sometimes religious
principles, to decorate buildings with mathematical objects such astessellations, and to
meet environmental goals, such as to minimise wind speeds around the bases of tall
buildings.Towards the end of the 20th century, fractal geometry was quickly seized upon
by architects, as was aperiodic tiling, to provide interesting and attractive coverings for
buildings.
Mathematics and architecture are two sides of the same golden coin.On the one side is
mathematics, and its capacity to enhance the understanding of architecture, both aesthetic
aspects such as symmetry and proportion, and structural aspects such as loads, thrusts, and

16. reactions. On the other side is architecture, as an attractive setting that allows basic abstract
and abstruse mathematics to become visible and more transparent.
Architects use mathematics for several reasons, leaving aside the necessary use of
mathematics in the engineering of buildings.
Firstly, they use geometry because it defines spatial forms.
Secondly, they use mathematics to design forms that are considered beautiful or
harmonious. From the time of the Pythagoreans with their religious philosophy of
number, architects in Ancient Greece, Ancient Rome, the Islamic world and the Italian
Renaissance have chosen the proportions of the built environment – buildings and their
designed surroundings – according to mathematical as well as aesthetic and sometimes
religious principles.
Thirdly, they may use mathematical objects such astessellations to decorate buildings.
Fourthly, they may use mathematics in the form of computer modelling to meet
environmental goals, such as to minimise whirling air currents at the base of tall buildings.
REFERENCE
www.en.m.wikipedia.org
www.quoro.com