Prelims of Kant get Marx 2.0: a general politics quiz
93639430 additional-mathematics-project-work-2013
1. 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
3. ACKNOWLEDGEMENT
First 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.
4. 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.
5. INTRODUCTION
In geometry a polygon (/pɒlɪɡɒn/) 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)
6. 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
7. 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
geometric
al.
8. 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
9. 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
11. The herb garden is an equilateral triangle of sides 100m with a maximum
area of 4330.13m².
CONJECTURE
Regular polygon will give maximum area.
Eg:
Square
Area= base X height
Rectangle
i. 60 < p < 140,
60 < q < 140
ii. When p increases, q decreases,
When q increases, p decreases
12. 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 =
The maximum area is 5625 m².
Area =
10 140 1400
20 130 2600
30 120 3600
40 110 4400
50 100 5000
60 90 5400
70 80 5600
75 75 5625
m
m
16. POLYGONAL BUILDINGS
University 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
17. 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
18. 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
19. CONCLUSION
Polygons 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.
20. REFLECTION
While 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.