Pranay Dutta's document provides an overview of geometry from ancient India. It discusses how geometry originated in ancient texts like the Sulba Sutras, which contained early examples of Pythagorean triples and statements of the Pythagorean theorem. It then covers the contributions of mathematicians like Brahmagupta, who formulated rules for cyclic quadrilaterals and rational triangles. Finally, it briefly introduces Euclid and his highly influential Elements, which presented geometry in an ideal axiomatic form.

1. Mathematics
Project
(Geometry)
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2. Geometry is a part of mathematics concerned with questions of
size, shape, and relative position of figures and with properties of
space. It is also a branch of mathematics that deals with the
deduction of the properties, measurement, and relationships of
points, lines, angles, and figures in space from their defining
conditions by means of certain assumed properties of space.
BASICS OF GEOMETRY
Point: A point is a location in space. It is represented by a dot. Point is
usually named with a upper letter. For example, we refer to the
following as "point A“
Line: A line is a collection of points that extend forever. The following is
a line. The two arrows are used to show that it extends forever.
We put two points in order to name the line as line AF. However, there
are an infinite amount of points. You can also name it line FA………….
Line segment: A line segment is part of a line. The following is a
segment. A segment has two endpoints. The endpoints in the following
segments are A and F. Notice also that the line above has no endpoints.
Geometry ?
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3. Ray: A ray is a collection of points that begin at one point (an endpoint) and
extend forever on one direction. The following is a ray.
Angle: Two rays with the same endpoint is an angle. The following is an
angle.
Plane: A plane is a flat surface like a piece of paper. It extends in all
directions. We can use arrows to show that it extends in all directions
forever. The following is a plane.………………………………………………………..
Parallel lines When two lines never meet in space or on a plane no matter
how long we extend them, we say that they are parallel lines The following
lines are parallel.……..………………………………………..
knjhhjji
Intersecting lines: When lines meet in space or on a plane, we say that they
are intersecting lines The following are intersecting lines.
Vertex: The point where two rays meet is called a vertex. In the angle
above, point A is a vertex.
Geometry ?
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4. GEOMOTRICAL FIGURES-
2D GEOMETRICAL FIGURES-
A 2D geometric model is a geometric model of an object
as two-dimensional figure, usually on
the Euclidean or Cartesian plane.
Even though all material objects are three-dimensional, a
2D geometric model is often adequate for certain flat
objects, such as paper cut-outs and machine parts made
of sheet metal.
2D geometric models are also convenient for describing
certain types of artificial images, such as
technical diagrams, logos, the glyphs of a font, etc. They
are an essential tool of 2D computer graphics and often
used as components of 3D geometric models, e.g. to
describe the decals to be applied to a car model.
Geometry ?
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5. 3D GEOMETRYCAL FIGURES-
These are three-dimensional shapes. Their sides are made of flat
or curved surfaces
3D shapes
spheres
cubes
cones
pyramids
hemispheres
cuboids
cylinders
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6. INDIAN GEOMETREY
Vedic period
Rigveda manuscript in Devanagari.
The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric
constructions that are similar to the Sulba Sutras.[4]
The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c.
700-400 BCE) list rules for the construction of sacrificial fire altars.[5] Most
mathematical problems considered in the Śulba Sūtras spring from "a single
theological requirement,"[6] that of constructing fire altars which have
different shapes but occupy the same area. The altars were required to be
constructed of five layers of burnt brick, with the further condition that each
layer consist of 200 bricks and that no two adjacent layers have congruent
arrangements of bricks.[6]
According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest
extant verbal expression of the Pythagorean Theorem in the world, although
it had already been known to the Old Babylonians."
The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both
which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes>
produce separately."[7]
Since the statement is a sūtra, it is necessarily compressed and what the
ropes produce is not elaborated on, but the context clearly implies the square
areas constructed on their lengths, and would have been explained so by the
teacher to the student.[7]
They contain lists of Pythagorean triples,[8] which are particular cases
of Diophantine equations.[9] They also contain statements (that with hindsight
we know to be approximate) about squaring the circle and "circling the
square."[10]
Geometry ?
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7. Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba
Sutra, the best-known Sulba Sutra, which contains examples of simple
Pythagorean triples, such as: (3,4,5),(5,12,13),(8,15,17),(7,24,25)
and (12,35,37)[11] as well as a statement of the Pythagorean theorem
for the sides of a square: "The rope which is stretched across the
diagonal of a square produces an area double the size of the original
square."[11] It also contains the general statement of the Pythagorean
theorem (for the sides of a rectangle): "The rope stretched along the
length of the diagonal of a rectangle makes an area which the vertical
and horizontal sides make together."[11]
According to mathematician S. G. Dani, the Babylonian cuneiform
tablet Plimpton 322 written ca. 1850 BCE[12] "contains fifteen
Pythagorean triples with quite large entries, including (13500, 12709,
18541) which is a primitive triple,[13] indicating, in particular, that there
was sophisticated understanding on the topic" in Mesopotamia in
1850 BCE. "Since these tablets predate the Sulbasutras period by
several centuries, taking into account the contextual appearance of
some of the triples, it is reasonable to expect that similar
understanding would have been there in India."[14] Dani goes on to
say:
"As the main objective of the Sulvasutras was to describe the
constructions of altars and the geometric principles involved in them,
the subject of Pythagorean triples, even if it had been well understood
may still not have featured in theSulvasutras. The occurrence of the
triples in the Sulvasutras is comparable to mathematics that one may
encounter in an introductory book on architecture or another similar
applied area, and would not correspond directly to the overall
knowledge on the topic at that time. Since, unfortunately, no other
contemporaneous sources have been found it may never be possible
to settle this issue satisfactorily."[14]
In all, three Sulba Sutras were composed. The remaining two,
the Manava Sulba Sutra composed by Manava (fl. 750-650 BCE) and
theApastamba Sulba Sutra, composed by Apastamba (c. 600 BCE),
contained results similar to the Baudhayana Sulba Sutra.
Geometry ?
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8. In the Bakhshali manuscript, there is a handful of geometric problems
(including problems about volumes of irregular solids). The Bakhshali
manuscript also "employs a decimal place value system with a dot for
zero."[15] Aryabhata's Aryabhatiya (499 CE) includes the computation of
areas and volumes.
Brahmagupta wrote his astronomical work Brāhma Sphuṭa Siddhānta in
628 CE. Chapter 12, containing 66 Sanskrit verses, was divided into two
sections: "basic operations" (including cube roots, fractions, ratio and
proportion, and barter) and "practical mathematics" (including mixture,
mathematical series, plane figures, stacking bricks, sawing of timber, and
piling of grain).[16] In the latter section, he stated his famous theorem on
the diagonals of a cyclic quadrilateral:[16]
Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that
are perpendicular to each other, then the perpendicular line drawn from
the point of intersection of the diagonals to any side of the quadrilateral
always bisects the opposite side.
Chapter 12 also included a formula for the area of a cyclic quadrilateral
(a generalization of Heron's formula), as well as a complete description
of rational triangles (i.e. triangles with rational sides and rational
areas).………………………………………………………………….
Classical Period
Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides
of lengths a, b, c, d, respectively, is given by
where s, the semiperimeter, given by:
Brahmagupta's Theorem on rational triangles: A triangle with rational
sides
for some rational numbers u, v and w [17]
Geometry ?
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9. Father of geometry-Euclid
Statue of Euclid in the Oxford University Museum of Natural History.
Geometry ?
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10. Woman teaching geometry. Illustration at the beginning of a medieval
translation of Euclid's Elements, (c. 1310)
Euclid (c. 325-265 BC), of Alexandria, probably a student of one of
Plato’s students, wrote a treatise in 13 books (chapters), titled The
Elements of Geometry, in which he presented geometry in an
ideal axiomatic form, which came to be known as Euclidean geometry.
The treatise is not a compendium of all that
the Hellenistic mathematicians knew at the time about geometry; Euclid
himself wrote eight more advanced books on geometry. We know from
other references that Euclid’s was not the first elementary geometry
textbook, but it was so much superior that the others fell into disuse and
were lost. He was brought to the university at Alexandria by Ptolemy I,
King of Egypt.
The Elements began with definitions of terms, fundamental geometric
principles (called axioms or postulates), and general quantitative
principles (called common notions) from which all the rest of geometry
could be logically deduced. Following are his five axioms, somewhat
paraphrased to make the English easier to read.
1.Any two points can be joined by a straight line.
2.Any finite straight line can be extended in a straight line.
3.A circle can be drawn with any center and any radius.
4.All right angles are equal to each other
Geometry ?
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11. Geometry is used everywhere. Everywhere in the world there is
geometry, mostly made by man. Most manmade structures today are in
a form of Geometric. How, you ask? Well some examples would the CD,
that is a 3-D circle and the case would be a rectangular prism. Buildings,
cars, rockets, planes, maps are all great examples.
GEOMETRY IN DAILY LIFE
Here's an example on how the world uses Geometry in buildings and structure:-
1.This a pictures with some basic geometric
structures. This is a modern reconstruction
of the English Wigwam. As you can there
the door way is a rectangle, and the
wooden panels on the side of the house are
made up of planes and lines. Except for
really planes can go on forever. The panels
are also shaped in the shape of squares.
The house itself is half a cylinder.
(1)
2.Here is another modern reconstruction if
of a English Wigwam. This house is much
similar to the one before. It used a
rectangle as a doorway, which is marked
with the right angles. The house was made
with sticks which was straight lines at one
point. With the sticks in place they form
squares when they intercepts. This English
Wigwam is also half a cylinder. (2)
3. This is a modern day skyscraper at MIT.
The openings and windows are all made
up of parallelograms. Much of them are
rectangles and squares. This is a
parallelogram kind of building.
(3)
Geometry ?
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12. 4. This is the Hancock Tower, in Chicago. With this
image, we can show you more 3D shapes. As you
can see the tower is formed by a large cube. The
windows are parallelogram. The other structure is
made up of a cone. There is a point at the top
where all the sides meet, and There is a base for
it also which makes it a cone.
5. This is another building at MIT. this building
is made up of cubes, squares and a sphere. The
cube is the main building and the squares are
the windows. The doorways are rectangle, like
always. On this building There is a structure on
the room that is made up of a sphere.
6.This is the Pyramids, in Indianapolis. The
pyramids are made up of pyramids, of
course, and squares. There are also many
3D geometric shapes in these pyramids.
The building itself is made up of a pyramid,
the windows a made up of tinted squares,
and the borders of the outside walls and
windows are made up of 3D geometric
shapes.
7. This is a Chevrolet SSR Roadster
Pickup. This car is built with geometry.
The wheels and lights are circles, the
doors are rectangular prisms, the main
area for a person to drive and sit in it a
half a sphere with the sides chopped off
which makes it 1/4 of a sphere. If a
person would look very closely the
person would see a lot more shapes in
the car. Too many to list.
Geometry ?
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13. Symmetry
Symmetry in common usage generally conveys two primary
meanings. The first is an imprecise sense of harmonious or
aesthetically-pleasing proportionality and balance; such that it
reflects beauty or perfection. The second meaning is a precise and
well-defined concept of balance or "patterned self-similarity" that can
be demonstrated or proved according to the rules of a formal
system: by geometry, through physics or otherwise.
Leonardo da Vinci's Vitruvian Man (1492)
is often used as a representation of
symmetry in the human body and, by
extension, the natural universe.
Reflection symmetry,
Rotational symmetry is symmetry with
respect to some or all rotations in m-
dimensional Euclidean space
Helical symmetry is the kind of symmetry
seen in such everyday objects as springs,
Slinky toys, drill bits, and augers. It can be
thought of as rotational symmetry along
with translation along the axis of rotation,
the screw axis).
Symmetry in religious symbols
Symmetry in architecture (Eg. Qutub Minar, etc)
A drill bit with helical symmetry
Geometry ?
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14. IMPORTANCE OF GEOMETRY
Geometry must be looked at as the consummate, complete and
paradigmatic reality given to us inconsequential from the Divine
Revelation. These are the reasons why geometry is important:
•It hones one's thinking ability by using logical reasoning.
•It helps develop skills in deductive thinking which is applied in all other
fields of learning.
•Artists use their knowledge of geometry in creating their master pieces.
•It is a useful groundwork for learning other branches of Mathematics.
•Students with knowledge of Geometry will have sufficient skills
abstracting from the external world.
•Geometry facilitates the solution of problems from other fields since its
principles are applicable to other disciplines.
•Knowledge of geometry is the best doorway towards other branches of
Mathematics.
•It can be used in a wide array of scientific and technical field.
The importance of Geometry is further substantiated by the requirement
that it is incorporated as a basic subject for all college students. An
educated man has within his grasps mathematical skills together with the
other qualities that make him a gentleman. Finally, what is the importance
of Geometry? From a philosophical point of view, Geometry exposes the
ultimate essence of the physical world.
Geometry ?
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15. GEOMETRY MATHEMATICIANS
EUCLID PYTHAGORAS
RENE DESCARTS
HIBERT
C.F. GAUSS
CLIFFORD
ARYABHATTA
ARYABHATTA
Geometry ?
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16. How Is Geometry Used in Real Life?
Understanding of geometry takes years of study and
involves many related sub-fields.
Geometry is the mathematics of space and shape, which is
the basis of all things that exist. Understanding geometry is
a necessary step in understanding how the world is built.
Most people take geometry in high school and learn about
triangles and vertical angles. The application of geometry in
real life is not always evident to teenagers, but the reality is
geometry infiltrates every facet of our daily living.
Geometry ?
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17. Geometry and Children
Geometry is not generally covered in grades kindergarten through
eight, but children are introduced to shapes and spaces in a variety of
ways. In initial school activities, kindergarten students are asked to
color triangles and circles. By the end of elementary school, most
students are able to make scale drawings. Students are able to
connect locations with coordinates, which is analytical geometry.
Visualization and spatial reasoning skills assist students with problem
solving.
Geometry in the Real World
In the real world, geometry is everywhere. A few examples include
buildings, planes, cars and maps. Homes are made of basic
geometric structures. Some skyscrapers have windows made of
rectangles and squares. The John Hancock Tower in Chicago is made
of a long cube. On a car, the wheels and lights are circles. The great
pyramids of Egypt of made of geometric shapes.
Symmetry in Science
Symmetry is a sense of harmony, proportion and balance. It
reflects beauty and perfection. In the scientific sense, symmetry
is defined as a sense of self-similarity through rules of a formal
system, such as geometry or physics. Symmetry is the basic
concept in the study of biology, chemistry and physics. Systems of
laws in physics and molecules in stereo chemistry reflect the
concepts of geometry. Some have difficulty grasping how
geometry relates to sciences. Since the 1870s, the study of
transformation and related symmetry are parallel to geometric
studies.
Geometry ?
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18. Along with new manufacturing, several changes have been made
including a re-designed upper control arm and other updates to weight,
strength and function.
“Increased demand for pre-engineered IRS suspensions has prompted us
to update and continue our line of Advanced Geometry Systems for the
1999-2004 SVT Cobra", according to Kenny Brown. “The quantum-leap
forward in technology that the Cobra IRS represents provides the perfect
platform for open-track enthusiasts who can now find these cars for a
reasonable price. We raced IRS Cobras starting in 1999 all the way
through 2004, and we still have several customers who compete with it
today. We know exactly how to make that architecture work for the
street and race track, and we are the only company that has ever
supported it whole heartedly".
Kenny’s Advanced Geometry Independent Rear Suspension is designed
to replace the OEM suspension components offering; racing-inspired
suspension geometry, 40 percent reduction in weight, and conversion to
coil-over shock design. A complete independent rear suspension
upgrade consists of; Tubular Rear Lower Control Arms, Tubular Rear
Upper Control Arms, IRS Rear Steer Kit, IRS Forward Torque Brace,
Aluminum IRS Differential Bushings, and Coil-Over Rear Upper Shock
Mounts. There are also a range of coil-over shocks available depending
on the application.
Geometry ?
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19. The extended performance benefits of the Kenny Brown Advanced
Geometry Independent Rear Suspension are; improved handling,
reduced wheel-hop, eliminates rear steer, reduced weight, improved
strength, and clearance for coil-over shocks. Everything you need to
transform the car for aggressive street or full blown open-track is
available in one simple system, or combination of components depending
upon your application.
Tubular Rear Upper and Lower Control Arms are designed to replace the
1999-2004 Cobra OEM control arms and are available for street and
competition applications. The tubular rear control arms feature urethane
bushings for reduced deflection and upgraded rear sway bar links for
improved strength and performance. The new control arms eliminate the
rear spring seats, allowing for conversion to coil-over shocks and much
greater adjustability.
IRS Rear Steer Kit is designed to replace the OEM rear tie rods and inner
tie-rod ends with heavy-duty competition-grade hardware. The rear
steer kit eliminates the rear steer factor allowing the car to exit corners
better with improved grip and helps to eliminate wheel-hop.
IRS Forward Torque Brace is designed to strengthen the area where the
differential assembly mounts to the IRS carrier assembly. The forward
torque brace improves traction and helps eliminate wheel-hop.
Aluminum IRS Differential Bushing Kit is designed to replace the OEM
rubber bushings to eliminate deflection at the rear differential
housing. The aluminum bushings improve traction, help eliminate
wheel-hop and act as a heat soak to pull critical temperature away from
the fragile rear differential.
Geometry ?
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20. Coil-Over Rear Upper Shock Mounts attach at the OEM
rear upper shock mount and allow for fitment of
competition coil-over rear shocks. The rear shock mounts
are bolt-on and will work for most aftermarket coil-over
shocks. When running coil-over shocks in the rear it is
strongly recommended that you also use the Heavy-Duty
Rear Shock Tower Brace.
Advanced Geometry Independent Rear Suspension
Components for 1999-2004 SVT Cobra Mustangs along
with other world-class chassis and suspension
components for the popular SN-95 platform are available
through authorized Kenny Brown Performance Parts
Dealers, online at www.kennybrown.com or by calling
Kenny Brown Performance direct – (855) 847-4477.
Geometry ?
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21. GEOMETRY AND EARTH
An interesting topic in 3-dimensional geometry is Earth geometry.
The Earth is very close to a sphere (ball) shape, with an average
radius of 6371 km. (It's actually a bit flat at the poles, but only by a
small amount).
Earth geometry is a special case of spherical geometry. When we
measure distances that a boat or aircraft travels between any 2
places on the Earth, we do not use straight line distances, since we
need to go around the curve of the Earth from one place to another.
(Think about the direct or straight-line distance between London and
Sydney, through the Earth. That's going to be a lot less than the
distance a plane flies around the surface of the Earth.)
Let's start with an example. What distance does a plane fly between
Beijing, China and Perth, Western Australia?
Geometry ?
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22. Thanking you
Pranay Dutta
X-D
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