4. architecture, nature & conclusion

Mathematics in

Architecture

Mathematics in

Architecture

Mathematics in Nature


Nurul Khairunnisa Morni08B0534

Nurul Khairunnisa Morni08B0534

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Mathematics in Architecture

Architecture has in the past done great things for geometry.

Together with the need to measure the land they lived on, it was people's need to build their buildings that caused them to first investigate the theory of form and shape. But today, 4500 years after the great pyramids were built in Egypt, what can mathematics do for architecture?


For thousand of years mathematics has been an invaluable tool for design and construction.

Here is a partial list of some mathematical concepts which have

been used in architecture over the centuries:


Pyramids Prisms Golden rectangle Optical illusions Cubes Polyhedra Geodestic dome Triangles

Phytogorean theorem

squares, rectanglesParallelogramsCircles, semi-circlesSpheres, hemispherePolygonsAnglesSymmetryParbolic curvesCatenary curves


The design of a structure is influenced by its surroundings, by the availability

and type of materials and by the imagination and resource upon which

architect can draw.


Historical architecture

Pyramids of Egypt, Mexico and the Yucatan


Machu Picchu


Pathenon


Ancient theater at Epidaurus


Roman architects


Architects of the Byzantine


Architects of Gothic cathedrals


Renaissance


Temple of Athena Parthenos


In the construction of the Temple of Athena Parthenos, Pythagorean ideas of ratios of small numbers were used.


The length of the Temple is 69.5 m, its width is 30.88m and the height at the cornice is 13.72 m. To a fairly high degree of accuracy this means that the ratio width : length = 4 : 9 while also the ratio height : width = 4 : 9. Berger took the greatest common denominator of these measurements to arrive at the ratios

height : width : length = 16 : 36 : 81 which gives a basic module of length 0.858 m


Then the length of the Temple is 92 modules, its width is 62 modules and its height is 42 modules. The module length is used throughout, for example the overall height of the Temple is 21 modules, and the columns are 12 modules high. The naos, which in Greek temples is the inner area containing the statue of the god, is 21.44 m wide and 48.3 m long which again is in the ratio 4 : 9. Berger notes the amazing fact that the columns are 1.905 m in diameter and the distance between their axes is 4.293 m, again the ratio of 4 : 9 is being used.


Modern architecture

With the discovery of new building materials, new mathematical ideas were adopted and used to maximize the potential of these materials.

Using wide range of available building materials architects have been able to design virtually any shape.


Modern architecture

The geodestic structures of

Buckminster Fuller


Formation of the hyperbolic paraboloid


The Gherkin

There are three main features that make it stand out from most other sky-scrapers: it's round rather than square, it bulges in the middle and tapers to a thin end towards the top, and it's based on a spiralling design. All these could easily be taken as purely aesthetic features, yet they all cater to specific constraints.


The London City Hall

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.


Architecture is an evolving field. Architects study, refine, enhance, reuse ideas form the past as well as create new

ones. In the final analysis, an architect is free to imagine any design as long as the mathematics and materials exist to

support the structure.

Mathematics in NatureThere is no branch of mathematics, however abstract, which may not someday be applied

to phenomena of the real world – Lobachevsky

Ever look at… a leaf and wonder why it could be

divided exactly in half?

the spiral growth pattern of a certain shells?

of the growth pattern if hair on a human head?


Nature abounds with examples of mathematical concepts.

Orb spider’s web

Mathematical ideas that appear in the web are – radii, chords, parallel segments, triangles, congruent corresponding angles, the logarithmic spiral, the catenary curve and the transcendental number .


Tortoise shell

The mathematics of triple junction, hexagonal tiling and calculus.


The comb’s walls are made up of cells which are about 1/80 of an inch thick, yet can support 30 times their own weight.

A honeycomb of about 14.5’’ x 8.8’’ can hold more than 5 pounds of honey, while it only requires about 1.5 ounces of wax to construct.

The bees form the hexagonal prisms in three rhombic sections and the walls of the cell meet at exactly 1200 angles.


The comb is built vertically downward and the bees use parts of their bodies as measuring instrument. Their heads act as plummets.

Honeybee has a “compass” Bees’ orientation is influenced by the Earth’s magnetic field. Bees can detect small fluctuations in the Earth’s magnetic field.


This is why the bees occupying a new location simultaneously begin to build the hive in different parts of the new area without any bee directing them.

All the bees orient their new comb in the same direction as their old hive.


Communication

Bees communicate the direction of the food and the distance by transmitting codes in a form of a “dance”.

The orientation of the dance in relation to the sun gives the direction of the food, while the duration of the dance indicates the distance.


Communication

Honeybees “know” that the shortest distance between two point is a straight line. – BEELINE

The honeybee gets its mathematical training via its genetic codes.


Flock of birds

Ever wonder why a flock of birds in flight as they swooped through the air don’t collide?


Heppner F.H. established 4 simple rules based on avian behaviour and used triangles for birds on why flock of birds don’t collide:

1.Birds are attached to a focal point or roost.2.Birds are attached to each other.3.Birds want to maintain a fixed velocity.4.Flight paths are altered by random occurences such as a gust of wind.

As Galileo mentioned. “… the universe stands continually open to our gaze, but it cannot be

understood unless one first learns to comprehend the language and interpret the

characters in which it is written. It is written in the language of mathematics, and its characters

are triangles, circles and other geometric figures, without which it is humanly impossible

to understand a single word of it… “

Mathematics is not just about learning the basic four operations or counting numbers but it

is more than that.

It evolves in the world around us in almost every fields / subjects / professions in this world.

Mathematics is so powerful that from the day it was first used until today it is still considered as one of the fundamental subjects and fields.

MATHEMATICS IN… Everyday life Art & design Architecture

Nature Cooking

Psychology Sports

Medicine

4. architecture, nature & conclusion

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