MAADHYAM - Gifted Educationgiftededucation.co.in/assets/files/students/4. November...Some people have made certain discoveries about the Great Pyramid, using maths: When using the

MAADHYAM Nurturing Gifted Minds

INSIDE HIGHLIGHTS:

Mathematics Behind Pyramid and Taj Mahal

Exploring mathematically designed buildings around the

world

Why architects use triangular structures?

Maadhyam News Corner

MONTH:NOVEMBER ISSUE NO:2015(4)

Printed under Gifted Education Abhiyaan

An initiative by the office of principal Scientific

Advisor to the Government of India

Have you ever wondered how buildings are constructed?

Are our buildings shaped by

sacred numbers and hidden

codes?

What factors determine the

maximum height of a

monument? ?

Is there any building which is

mathematically designed?

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"In mathematics, if a pattern occurs we can go on and ask 'Why does it occur?' 'What does it signify?' and we can find answers to these questions." W.W.Sawyer Mathematics is full of beautiful patterns. Sacred geometry, or spiritual geometry, is the belief that numbers and patterns such as the divine ratio have sacred significance. Many mystical and spiritual practices, including astrology, numerology, tarot, and feng shui, begin with a fundamental belief in sacred geometry. Architects and designers may draw upon concepts of sacred geometry when they choose particular geometric forms to create pleasing, soul-satisfying spaces. Architecture begins with geometry. Since earliest times, architects have relied on mathematical principles. The ancient Roman architect Marcus Vitruvius Pollio believed that builders should always use precise ratios when constructing temples. "For without symmetry and proportion no temple can have a regular plan,“ From the pyramids in Egypt to the new World Trade Center tower in New York, good architects use the same essential building blocks as your body and all living things. Moreover, the principles of geometry are not confined to great temples and monuments. Mathematicians say that when we recognize geometric principles and build upon them, we create dwellings that comfort and inspire.

Mathematics In Architecture

To read more about M.V.Pollio, visit

http://www.vitruvius-pollio.com/

Strange Before you dismiss the idea of sacred geometry, take a few moments to reflect on the ways some numbers and patterns appear time and again in every part of your life.

MONTH:NOVEMBER ISSUE NO:2015(4)

The Pythagorean theorem and the 3-4-5 right triangle was used by rope stretchers and Egyptian

engineers. Rope was knotted into 12 sections that stretched out to produce a 3-4-5 triangle. For e.g. If

you tie 12 equally spaced knots into a rope, fix the rope at the 5th knot to the ground, go with the longer

side 4 knots in one direction, fix the rope there too, and then try to bring both lose ends together, you

will automatically get a right angle. The Egyptians also used length/height-ratios to construct pyramid

slopes. Mostly used were 1:22 and 1:21-ratios, one cubit height to 21 or 22 fingers length. If you

substitute the cubit with the fingers you get for Kephrens pyramid a 28:21-ratio. Divide this by 7, and

you get a "3 units length by 4 units height"-ratio. These are the short sides of a holy triangle, so the third

side must then be in a "5 units"-ratio in respect to the other sides. Therefore, all pyramids in Egypt

constructed in the 1:21-ratio are carrying the 3:4:5-ratio quite naturally.

The Great Pyramid was laid out with geometric precision - a near-perfect square base, with sides of 230

meters that differ from each other by less than twenty centimeters, and faces that sloped upwards at an

angle of 51º to reach an apex nearly 150 meters above the desert floor.

Some people have made certain discoveries about the Great Pyramid, using maths:

When using the Egyptian cubit the perimeter is 365.24 - the amount of days in the year.

The height x 10 to the power of 9 gives approximately the distance from the earth to the sun.

The perimeter divided by 2 x the height of the pyramid is equal to pi - 3.1416

The weight of the pyramid x 10 to the power of 15 is equal to the approximate weight of the earth.

When the cross diagonals of the base are added together, the answer is equal to the amount of time

(in years) that it takes for the earth's polar axis to go back to its original starting point - 25,286.6

years.

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,

mathematics continues to feature prominently in building design.. We are not just talking about mere measurements, — though elements like that are integral to architecture.

Thanks to modern technology, architects can explore a variety of exciting design options based on complex mathematical languages, allowing them to build groundbreaking forms.

Now take a look at several structures in the past that were modeled along mathematics.

DO YOU KNOW ???

The Great Pyramid of Giza in 4700

B.C. is with proportions according to

a “sacred ratio”. The ancient

Egyptians constructed the Great

Pyramids in such a way that the ratio

(b : h : a) is approximately equal to

(1 : √φ : φ).

To know more on mathematical proof visit the link: https://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html;http://www.halexandria.org/dward106.html 2

MONTH:NOVEMBER ISSUE NO:2015(4)

Mathematics in Taj Mahal Symmetry is found in the world around us: in nature, in artwork and even in buildings we see everyday.

You can see it in structures such as Taj Mahal in India. You might not even realize it, but you probably

like images that are symmetrical. Designers, architects, and artists understand this, which is why they

often use symmetry to create images that are pleasing to us.

The next time you visit the Taj Mahal in India, waiting to get that iconic photo in front of this beautiful

building. Look closer and you’ll 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. The Taj Mahal (completed in 1648) exhibits symmetry, a mathematical property claims that

Divine Proportion was used in the construction of the Taj Mahal.

The Taj Mahal displays golden proportions in the width of its grand central arch to its width, and also

in the height of the windows inside the arch to the height of the main section below the domes.

Divine Proportion...

What is this?

Golden ratio, also known as the golden section, golden mean,

or divine proportion, in mathematics, the irrational number

(1 + √5)/2, often denoted by the Greek letters τ or ϕ, and

approximately equal to 1.618.

The origin of this number and its name may be traced back to

about 500 BC and the investigation in Pythagorean geometry

of the regular pentagon.

We find the golden ratio when we divide a line into two parts

so that: the longer part divided by the smaller part is also

equal to the whole length divided by the longer part.

In terms algebra, letting the length of the shorter segment be

one unit and the length of the larger segment be x units gives

rise to the equation (x + 1)/x = x/1; this may be rearranged to

form the quadratic equation X2 – x – 1 = 0, for which the

positive solution is x = (1 + √5)/2, the golden ratio.

To know more about Golden Ratio visit

the given links: 1. https://www.youtube.com/watch?v=lm9zWqJ6-Rg;

2.https://www.youtube.com/watch?v=1_dDMdPjt60,

3. https://www.mathsisfun.com/numbers/golden-

ratio.html;

4. http://www.basic-mathematics.com/what-is-the-

golden-ratio.html;

5. http://mathart.wikidot.com/golden-ratio2

6. http://www.cut-the-

knot.org/do_you_know/GoldenRatio.shtml

We will now prove that the ratio of the lengths of

two diagonals is indeed the Golden ratio.

Assume that rectangle ABCD is a Golden

Rectangle.

Hence, AD/AB =AE/ED. But, FE = AE, and so

FE/ED=.

Hence, rectangle FCDE is a Golden Rectangle.

We have two similar rectangles and so since =

AD/EF then BD/CE = .

An interesting thing happens when we work with

these rectangle. Suppose we take a rectangle of side 1

unit and a rectangle of side 2 units and we put them

side to side in the following way and draw the

appropriate segments to form a rectangle. If we

continue to create rectangles in this way we will get a

series of rectangles like in figure. The following

picture shows several such rectangles, and the lengths

of their sides.

If we take ratios of the length we will see that the series of

whirling rectangles will begin to estimate the Golden Ratio.

2/1 = 2 3/2= 1.5 5/3 = 1.666... 8/5 = 1.6 13/8 = 1.625 and so on.

Hence as we increase the number of squares we get a figure that

begins to look more and more like the Golden Rectangle. It might

also be noticed that there is something special about the sides of

the squares. If we list them we have, 1, 2, 3, 5, 8, 13, ... This of

course is the famous Fibonacci sequence. Visit the link for

mathematical proof http://www.basic-mathematics.com/what-is-

the-golden-ratio.html

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Image Source: www.phimatrix.com

MONTH:NOVEMBER ISSUE NO:2015(4)

The British physicist and mathematician, Roger Penrose, developed a periodic tiling which incorporates the golden section. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B). When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the “Golden Ratio." In addition to the unusual symmetry, Penrose Tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean. Visit link :

http://www.ams.org/samplings/feature-column/fcarc-penrose

4

Geometrical Construction of Golden Ratio

Now we will construct the Golden Rectangle.

1) First construct a square ABCD. 2) Construct the midpoint E of DC. 3) Extend DC. With center E and radius EB, draw an arc crossing EC extended at C. 4) Construct a perpendicular to DF at F. 5) Extend AB to intersect the perpendicular at G. 6) AGFD is a Golden Rectangle. 7) Now measure the length and width of the rectangle. Then find the ratio of the

length to the width. This should be close to the Golden Ratio (approximately 1.618).

You can also find golden ratio in

Modern abstract art such as Penrose

Tilings.

MONTH:NOVEMBER ISSUE NO:2015(4)

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).

Sydney Harbour Bridge

•The Sydney Harbour bridge is a magnificent structure of mathematical genius, located in what has to be the world’s most beautiful city.

•The mathematics associated with the Sydney Bridge, including deriving the Quadratic Equations for both the lower and upper parabolic arches of the bridge.

•Read more for mathematical proof of bridge: http://passyworldofmathematics.com/sydney-harbour-bridge-mathematics/

5

Taj Mahal

Sydney Harbour Bridge

Akshardham

- a 3D fractal like

temple

The Great

Pyramid of Giza

Morbius Strip Temple

Magic Square

Cathedral Gherkin's curves, London

The Eden Project, Cornwall,

UK

Parthenon

Cube Village

Experimental Math-Music

Pavilion

MONTH:NOVEMBER ISSUE NO:2015(4)

6

Bahá'í-House of worship, New Delhi

The beautiful concept of the lotus, as conceived by the architect, had to be converted into definable geometrical shapes such as spheres, cylinders, toroids and cones. These shapes were translated into equations, which were then used as a basis for structural analysis and engineering drawings.

Unlike conventional structures for which the elements are defined by dimensions and levels, here the shape, size, thickness, and other details were indicated in the drawings only by levels, radii, and equations.

The resultant geometry was so complex that it took the designers over two and a half years to complete the detailed drawings of the temple.

Follow the link to know more: http://www.bahaihouseofworship.in/architectural-blossoming

The Eden Project, Cornwall, UK

•It’s little surprise that the building has taken its inspiration from plants, using Fibonacci numbers(0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... where every number is the sum of the previous two) to reflect the nature featured within the site.

•The outer layer is made of hexagons (the largest is 11 metres across), plus the odd pentagon. The inner layer comprises hexagons and triangles bolted together.

•The geodesic concept provided for least weight and maximum surface area on the curve – with strength.

•Read more: https://www.edenproject.com/eden-story/behind-the-scenes/architecture-at-eden

Mobius Strip Temple

•The Mobius Strip is a surface with only one side and only one boundary.

• An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop.

•The Möbius strip has the mathematical property of being non-orientable, while the Möbius strip has several curious properties.

•A line drawn starting from the seam down the middle will meet back at the seam but at the “other side”. If continued the line will meet the starting point and will be double the length of the original strip.

•This single continuous curve demonstrates that the Möbius strip has only one boundary.

•Read more : http://www.arch2o.com/buddhism-temple-miliy-design/

MONTH:NOVEMBER ISSUE NO:2015(4)

Have you noticed that many

architectures have used triangular

structures!

Yes true! But why are

triangles used in construction of buildings?

As we have read, the most famous triangular structures are, of

course, The Great Pyramids of Egypt and South America. The

strongest part of a pyramid is the wide base. Each successive row has

less weight to support above it.

And also some of these monumental

tetrahedrons have been standing for tens of thousands of years.

A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed.

Properties:

Interior angles of triangles (angles on the inside) sum up to 180°.

Triangle Inequality Theorem

Relationship between measurement of the sides and angles in a Triangle

To know more in detail visit http://www.mathwarehouse.com/geometry/triangles/

Triangle does not easily deform and is able to balance the stretching

and compressive forces inside the structure.

For economic reasons: since the triangle obviously has only 3 sides, it

requires little material to make a support, thus minimizing the costs.

7

Why are triangles so strong?

A square lacks the

rigid strength of a

triangle.

But by adding diagonal

bracing, a common feature in

bridges and buildings, the

structure can again rely on the

strength of a triangle to hold

its shape.

Try Yourself

Triangles are inherently strong because they form a fixed rigid

shape.

This can be demonstrated by building a triangle out of garden canes,

securing the corners with rubber bands. The shape is fixed by the

length of the sides and the triangle withstands quite substantial forces

applied to it.

However if you built a square in the same way, a gentle push at one

corner could easily change the shape into a parallelogram. There are

infinitely many four-sided shapes with equal sides, a square is just one

of them, and so the shape is easily transformed from one to the other

with minimal force.

What could you conclude from this?

To know more

Visit the links:

http://www.pbs.

org/wgbh/build

ingbig/bridge/b

asics.html

http://www.fac

ulty.fairfield.ed

u/jmac/rs/bridg

es.htm

MONTH:NOVEMBER ISSUE NO:2015(4)

M.C.ESCHER Art and Math may at first seem to be very different things, but people who enjoy math tend to look for mathematics in art. They want to see the patterns and angles and lines of perspective. This is why artists like M.C. Escher appeal to mathematicians so much. There is a large amount of math involved in art, not to mention basic things like measuring and lines, but the intricacies of art can often be described using math. Visit the given link to know more about M.C. Escher http://www.mcescher.com/about/ Escher is a famous artist who created mathematically challenging artwork. He used only simple drawing tools and the naked eye, but was able to create stunning mathematical pieces. He produced polytypes, sometimes in drawings, which cannot be constructed in the real world, but can be described using mathematics. His drawings caught the eyes and looked possible by perception, but were mathematically impossible. His particular drawing, Ascending and Descending, was one of the masterpieces. In this drawing, Escher creates a staircase that continues to ascend and descend, which is mathematically impossible, but the drawing makes it seem realistic.

Let’s Know About

8

Relativity

Image Source:

http://www.mcescher.com/gallery/back-in-holland/relativity/

M.C. Escher ~ Cycle Art Print

Image Source: www.leninimports.com

Tessellating Prints - An Introduction - David Bailey s

World of Escher-like Tessellations.

Image Source: http://phoenix-588-3.iaafmt.org/

Mathematical Tessellation- Shells and Starfish

Image Source: http://urioste.weebly.com/tessellations.html

MONTH:NOVEMBER ISSUE NO:2015(4)

Noida boy makes India proud, wins biggest

maths puzzle championship in US Published:

Thursday, June 11, 2015, 17:50 [IST]

Noida's Gaurav Pandey doubled the celebration by

becoming KenKen international champion. KenKen

is a grid-based numerical puzzle that uses basic

math operations while challenging the logic and

problem-solving skills of participants. It is

recognised by NCTM (National Council of

Teachers of Mathematics, USA), an independent

authority, as a powerful tool to build reasoning

skills in kids.

Read more at:

http://www.oneindia.com/india/noida-boy-

makes-india-proud-wins-biggest-maths-puzzle-

championship-in-us-1774508.html

Indian-origin mathematician Manjul

Bhargava wins Fields Medal

Manjul Bhargava was awarded the prestigious

medal for ‘developing powerful new methods in

geometry of numbers’. Manjul Bhargava, a

Canadian mathematician of Indian origin, has

been awarded the prestigious 2014 Fields Medal

at the International Mathematical Union’s (IMU)

International Congress of Mathematicians held

in Seoul.

Read more at:

http://www.livemint.com/Politics/76RVvYH

Nx7neqcW1gEmCNN/Indianorigin-

mathematician-Manjul-Bhargava-awarded-

Fields-M.html

Visit the following link to know about latest

updates on mathematics and science

https://plus.maths.org/content/News

9

Maadhyam News Corner

2014 Nobel Prize in Chemistry: Super-resolved fluorescence microscopy

Life on Earth likely started 4.1 billion years ago,

much earlier than scientists thought

University of California -Los Angeles geochemists

have found evidence that life likely existed on Earth

at least 4.1 billion years ago -- 300 million years

earlier than previous research suggested. The

discovery indicates that life may have begun shortly

after the planet formed 4.54 billion years ago.

To know more:

http://www.sciencedaily.com/releases/2015/10/1

51019154153.htm

http://universityofcalifornia.edu/

2014 Nobel Prize in Chemistry: Super-

resolved fluorescence microscopy

The 2014 Nobel Prize in Chemistry has been

awarded to Eric Betzig of Janelia Farm

Research Campus, Howard Hughes Medical

Institute; Stefan W. Hell of Max Planck Institute

for Biophysical Chemistry and the German

Cancer Research Center; and William E.

Moerner of Stanford University "for the

development of super-resolved fluorescence

microscopy."To know more:

http://www.sciencedaily.com/releases/2014/1

0/141008085419.htm

http://www.kva.se/en/ Deep-sea bacteria could help neutralize

greenhouse gas

A type of bacteria plucked from the bottom of the

ocean could be put to work neutralizing large

amounts of industrial carbon dioxide in the Earth's

atmosphere, a group of University of Florida

researchers has found.

To know more:

http://www.sciencedaily.com/releases/2015/10/1

51022141716.htm

http://www.ufl.edu/

MONTH:NOVEMBER ISSUE NO:2015(4)

Response Sheet

Q1: Consider the following diagram,

Given that the areas of A1, A2 and A3 in the diagram

are equal, show that 𝑅𝑋

𝑋𝑆=

𝑅𝑌

𝑌𝑄=

5 2+1

2

so that the points X and Y divide the sides of the rectangle

in the golden ratio.

____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ Q2: Why are quadrilaterals unstable? Compare their stability with triangular structures.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Q3.In order to test how strong a triangular structure is, Atul has done an experiment.

First of all, He made an accordion fold of an A4 sheet of paper.

Used two books as supporters and put the paper on top of them.

Now, He put a load on it. What can you conclude from this?

Will the A4 paper be able to support two books which are far heavier

than its own weight or not? Give reason.

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q4.

The diagram shows a rhombus PQRS with an internal point O such that OQ=OR=OS=1unit. Penrose used this rhombus, split into two quadrilaterals, a dart and a kite, to make his famous tiling which fills the plane but, unlike a tessellation, does not repeat itself by translation or rotation.

Find all the angles in the diagram, show that POR is a straight line and show that triangles PRS and QRO are similar. Hence prove that the length of the side of the rhombus is equal to the Golden Ratio.

__________________________________________________________________________ ___________________________________________________________________________

__________________________________________________________________________ ___________________________________________________________________________

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MONTH:NOVEMBER ISSUE NO:2015(4)

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RESEARCH TEAM

Geetu Sehgal , Shilpi

Bariar, Jyoti Batra,

Uzma Masood ,Ritu

Verma,Meenakshi

Gifted Education Abhiyaan

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