ST.JOSEPH'S COLLEGE OF EDUCATION

(Affiliated to the University of Mysore)

JAYALAKSHMIPURAM, MYSORE-570012

B.Ed choice Based Credit System (CBCS)

MATHEMATICS PROJECT

LIVE WITH POLYGONS Submitted by:

Rashmi R

B.Ed [PM], roll no 72

St. Joseph's college

Education, Mysore

Submitted to:

Priya Mathew ma'am

Lecturer in Physics,

St. Joseph's college of Education, Mysore

CONTENTAcknowledgement

Objectives

Introduction

Part 1

Part 2

Part 3

Further Exploration

Conclusion

Reflection

ACKNOWLEDGEMENTFirst of all, I would like to say thank you, for giving me the

strength to do this project work.

Not forgotten my parents for providing everything, such as money, to buy anything that are related to this project work and their advise, support which are the most needed for this project. Internet, books, computers and all that. They also supported me and encouraged me to complete this task so that I will not procrastinate in doing it.

Then I would like to thank my teacher, Priya ma’am for guiding me and my friends throughout this project. They were helpful that when we combined and discussed together, we had this task done.

OBJECTIVES The aims of carrying out this project work are:

to apply and adapt a variety of problem-solving strategies to solve problems.

to improve thinking skills.

to promote effective mathematical communication. 

to develop mathematical knowledge through problem solving in a way that increases students’ interest and confidence.

to use the language of mathematics to express mathematical ideas precisely.

to provide learning environment that stimulates and enhances effective learning.

to develop positive attitude towards mathematics.

INTRODUCTION In geometry a polygon (/p ɒ l ɪ ɡ ɒ n / ) is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.

The word "polygon" derives from the Greek πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides.

The basic geometrical notion has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the closed polygonal chain and with simple polygons which do not self-intersect, and may define a polygon accordingly. Geometrically two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation). Some other generalizations of polygons are described below.

Historical image of polygons (1699)

History

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.

In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

Polygons in nature

Numerous regular polygons may be seen in nature. In the world of geology, crystals have flat faces, or facets, which are polygons. Quasicrystals can even have regular pentagons as faces. Another fascinating example of regular polygons occurs when the cooling of lava forms areas of tightly packed hexagonal columns of basalt, which may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California.

The most famous hexagons in nature are found in the animal kingdom. The wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a secure place for the larvae to grow. There also exist animals that themselves take the approximate form of regular polygons, or at least have the same symmetry. For example, sea stars display the symmetry of a pentagon or, less frequently, the heptagon or other polygons. Other

echinoderms, such as sea urchins, sometimes display similar symmetries. Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold.

The Giant's Causeway, in Northern Ireland

Starfruit, a popular fruit in Southeast Asia

Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star.

Moving off the earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller body (such as an asteroid or a space station) will remain in a stable orbit. The sun-earth system has five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in its orbit; that is, joining the center of the sun and the earth and one of these stable Lagrangian points forms an equilateral triangle. Astronomers have already found asteroids at these points. It is still debated whether it is practical to keep a space station at the Lagrangian point – although it would never need course corrections, it would have to frequently dodge the asteroids that are already present there. There are already satellites and space observatories at the less stable Lagrangian points.

Polygons are fun in

Math

Many shapes

they have

They are geometrica

l.

PART 1 Examples of polygen

A closed shape consisting of line segments that has at least three sides. Triangles, quadrilaterals, rectangles, and squares are all types of polygons.

The word "polygon" comes from Late Latin polygōnum (a noun), from

Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula. Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine. In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

1.

2.

3.

4.

PART 2

The herb garden is an equilateral triangle of sides 100m with a maximum area of 4330.13m².

CONJECTURERegular polygon will give maximum area.Eg:

Square

Area= base X height

Rectangle

p(m) q(m) θ(degree) Area(m²)

60 140 38.21 2597.89

70 130 49.58 3463.97

80 120 55.77 3968.57

90 110 58.99 4242.53

100 100 60.00 4330.13

110 90 58.99 4242.53

120 80 55.77 3968.57

130 70 49.58 3463.97

140 60 38.21 2597.89

i. 60 < p < 140, 60 < q < 140

ii. When p increases, q decreases,When q increases, p decreases

iii. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem.

PART 3

i. Quadrilateral

2 + 2 = 300 m²

+ = 150 m²

Area =

Area =

10 140 140020 130 260030 120 360040 110 440050 100 500060 90 540070 80 560075 75 5625

The maximum area is 5625 m².

ii. Regular Pentagon

tan

m

m

iii. Regular Hexagon

tan

iv.Regular Octagon

tan 67.5°

tan

v. Regular Decagon

tan

tan

a

FURTHER EXPLORATION

POLYGONAL BUILDINGSUniversity of Economics and Business, Vienna

The new Library & Learning Center rises as a polygonal block from the centre of the new University campus. The LLC’s design takes the form of a cube with both inclined and straight edges. The straight lines of the building’s exterior separate as they move inward,

becoming curvilinear and fluid to generate a free-formed interior canyon that serves as the central public plaza of the centre. All the other facilities of the LLC are housed within a single volume that also divides, becoming two separate ribbons that wind around each other to enclose this glazed gathering space.

The Center comprises a ‘Learning Center’ with workplaces, lounges and cloakrooms, library, a language laboratory, training classrooms, administration offices, study services and central supporting services, copy shop, book shop, data center, cafeteria, event area, clubroom and auditorium.

Vienna, Austria 2008 – 2012

28000 m2

Gross Area: 42000 m²

Length: 136m

Width: 76m

Height: 30m (5 Floors)

Zaha Hadid with Patrik Schumacher

Pittman Dowell Residence, Los Angeles

The project is a residence for two artists. Located 15 miles north of Los Angeles at the edge of Angeles forest, the site encompasses 6 acres of land originally planned as a hillside subdivision of houses designed by Richard Neutra. Three level pads were created but only one house was built, the 1952 Serulnic Residence. The current owners have over the

years developed an extensive desert garden and outdoor pavilion on one of the unbuilt pads. The new residence, to be constructed on the last level area, is circumscribed by the sole winding road which ends at the Serulnic house.

Five decades after the original house was constructed in this remote area, the city has grown around it and with it the visual and physical context has changed. In a similar way, the evolving contemporary needs of the artists required a new relationship between building and landscape that is more urban and contained. Inspired by geometric arrangements of interlocking polygons, the new residence takes the form of a heptagonal figure whose purity is confounded by a series of intersecting diagonal slices though space. Bounded by an introverted exterior, living spaces unfold in a moiré of shifting perspectival frames from within and throughout the house. An irregularly shaped void caught within these intersections creates an outdoor room at the center whose edges blur into the adjoining living spaces. In this way, movement and visual relationships expand and contract to respond to centrifugal nature of the site and context.

Los Angeles, USA Residential - Houses Micheal Maltzan Architecture 3200 SF

Casa da Musica, Porto

The Casa da Musica is situated on a travertine plaza, between the city's historic quarter and a working-class neighborhood, adjacent to the Rotunda da Boavista. The square is no

longer a mere hinge between the old and the new Porto, but becomes a positive encounter of two different models of the city.

The chiseled sculptural form of the white concrete shell houses the main 1,300 seat concert hall, a small 350 seat hall, rehearsal rooms, and recording studios for the Oporto National Orchestra.

A terrace carved out of the sloping roofline and huge cut-out in the concrete skin connects the building to city. Stairs lead from the ground level plaza to the foyer where a second staircase continues to the Main Hall and the different levels above. Heavy concrete beams criss-cross the huge light well above. The main auditorium, shaped like a simple shoebox, is enclosed at both ends by two layers of “corrugated” glass walls. The glass, corrugated for optimal acoustics and sheer beauty, brings diffused daylight into the auditorium. The structural heart of the building is formed by four massive walls that extend from the base to the roof and connect the tilted external walls with the core of the structure. The two one meter thick walls of the main auditorium act as internal diaphragms tying the shell together in the longitudinal direction. The principal materials are white concrete, corrugated glass, travertine, plywood, and aluminum.

Porto, Portugal 22.000 square meters

2005

Concert Hall

Rem Koolhaas and Ellen van Loon

CONCLUSIONPolygons tend to be studied in the first few years of school,

as an introduction to basic geometry and mathematics. But after doing research, answering questions, completing table and some problem solving, I found that the usage of polygons is important in our daily life.

It is not just widely used in fashion design but also in other fields especially in building of modern structures. The triangle, for instance, is often used in construction because its shape makes it comparatively strong. The use of the polygon shape reduces the

quantity of materials needed to make a structure, so essentially reduces costs and maximizes profits in a business environment. Another polygon is the rectangle. The rectangle is used in a number of applications, due to the fact our field of vision broadly consists of a rectangle shape. For instance, most televisions are rectangles to allow for easy and comfortable viewing. The same can be said for photo frames and mobile phones screens.

In conclusion, polygon is a daily life necessity. Without it, man’s creativity will be limited. Therefore, we should be thankful of the people who contribute in the idea of polygon.

REFLECTIONWhile I conducting this project, a lot of information that I

found. I have learnt how polygons appear in our daily life.

Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.

Not only that, I had learned some moral values that I practice. This project had taught me to responsible on the works that are given to me to be completed. This project also had made me felt

more confidence to do works and not to give easily when we could not find the solution for the question. I also learned to be more discipline on time, which I was given about a week to complete this project and pass up to my teacher just in time. I also enjoy doing this project I spend my time with friends to complete this project.

Last but not least, I proposed this project should be continue because it brings a lot of moral value to the student and also test the students understanding in Additional Mathematics.