A. Understanding Geometry
Geometry (Greek; geo = earth, metria = measure) is a part of mathematics that takes issues regarding the size, shape, and position as well as the properties of space. Geometry is one of the oldest sciences. Beginning a body of practical knowledge that take the weight at a distance, area and volume, but in the 3rd century progressed geometry that is about aksiometik form by Euclid, whose results are influential for the next few centuries.
Geometry is one branch in the mathematical sciences. Geometry of Science literally means measurement of the earth, namely the study of the relationship in space. Indeed, the science of geometry has been studied Ancient Egyptian civilization, the people of the Indus River Valley and Babylonians.
These ancient civilizations are known to have expertise in the swamp drainage, irrigation, flood control and construction of buildings-large buildings. Most of the ancient Egyptians and Babylonians geometry is limited to the calculation of line segments length, area, and volume.
B. A Brief History of Geometry
There are at least six areas that can be regarded as a 'source' contributor knowledge of geometry, namely: the Babylonians (4000 BC - 500 BC), Greece (600 BC - 400 BC), Egypt (5000 BC - 500 BC), Jasirah Arab (600 - 1500 AD), India (1500 BC - 200 BC), and China (100 BC - 1400). Of course there are countries other contributors to the knowledge of geometry, however, is less significant or not recorded in a written tradition.
The Babylonians occupies a fertile area that stretches between the rivers Euphrates and Tigris in the Middle East region. At first, the area occupied by the Sumerians. At that time, 3500 BC, or about 5,000 years ago has been living very advanced. Many buildings are built like a city now. Agricultural irrigation systems and rice fields has also been growing. Geometry considered by engineers for development purposes.
Geometry which was born and developed in Babylon is a result of the desires and expectations of government and religious leaders at the time. It is intended to be able to establish a wide range of solid construction and great. Also hope for the king in order to control the land for the benefit of the tax revenue. The techniques geometry that develops when the generally rough and intuitive. However, quite accurate and can meet the needs of the calculation of various facts about the techniques of geometry is currently loaded in Ahmes more kiurang papyrus written in 1650 BC and was discovered in the 9th century. Relics in the form of this paper is a part of the goods stored by museums in London and New York. In this papyrus contained the formula of calculating the area of a rectangle, right-angled triangle, trapezoid that has legs perpendicular to the base, as well as the formula of calculation approaches the area of a circle. The Egyptians seem to have developed a formula-sumus this in their lives to calculate the area of cultivated fields.
In addition to continuing to develop geometry, they also developed a number system that we now know as 'sexagesimal based' 60. We are still enjoying (and use) the system when talking about time.
They divided the day into 24 hours. One hour divided into 60 minutes. One minute is divided into 60 seconds. We say, for example, is currently at 9, 25 minutes, 30 seconds. When written will be shaped at 9 25 '30 ", and in sexagesimal can be written as 9 5 25/60 30/3600. This system has been using the place value as we use it today (in base 10 rather than in base 60).
The Babylonians developed a way of calculating area and volume. Among them calculate the circumference of a circle is equal to three times the length of its diameter. We know the price is close to the price of three π. Pythagoras formula is also already known at that time.
The Egyptians inhabit very fertile region along the Nile. Agriculture is growing rapidly. The government needs a way to divide the rice plots fairly. Thus, advanced geometry here because it presents various polygon shapes that are customized to the circumstances Walayah along the Nile.
In Greece, geometry endured a period emas'nya. About 2000 years ago, discovered the theory we know today by the name of axiomatic theory. The theory think that bases itself on the most basic things that we take for granted the truth. Truth is what we call the truth of the axiom. Axiom derived from a variety of arguments of both the fundamental assertion or argument of the derivative. From this era, we also inherit geometry book, which until now has not been uncontested, namely Euclidean geometry. Geometry we teach formally in school is a 'coffee-ness' of this Euclidean geometry.
In the early development of Islam, the Islamic leaders suggest that studying as much as possible. We know belajaralah up to China. In that era, Islam spread in the Middle East, North Africa, Spain, Portugal, and Persia. The Islamic mathematicians contributed to the development of algebra, asronomi, and trigonometry. Trigonometry is one approach to menyelesaian algebraic geometry problems. We know into analytic geometry. They also developed a polynomial.
In the eastern region, India and China are known contributors reliable mathematical knowledge. In India, the mathematicians have a duty to make a variety of burning buildings for victims at the altar. One of the conditions is allowed to form (and even should) be different but must be the same breadth. For example, make pangunan pembekaran consisting of five levels and each level consists of 200 bricks. In between the two levels of the order should not be exactly the same brick structure. At that moment appeared geometers in India. Of course, the building was also equipped with a roof. The roof is also part Indian mathematician task. This is where the developing theories of geometry.
Like branches of science others, mathematics (including geometry) also developed by Chinese scientists since 2000 BC (or about 4,000 years ago). While in Europe there is a book 'Elements', Euclidean geometry that can penetrate time 2000 years unchallenged, in eastern China are the Nine chapters on mathematics' created around the year 179 by Liu Hui. This book contains a lot of geometry problems. Among them calculate area and volume. In the book explores the law of Pythagoras. Also talked about the polygon.
In this age of Pertengan, Muslim mathematics Expert contributed much to the development of geometry, especially algebraic geometry and algebraic geometry. Al-Mahani (1853) got the idea to describe the problem geometry such as copying a cube to problems in algebraic form. Thabit ibn Qurra (known as a Thebit in Latin) (836-901) mengendali with arimetikal control given to the quantity ratio geometry, and contribute on the development of analytic geomeri. Omar Khayyam (1048 -1131) for resolution of the geometry to find the cubic equation, and the subsequent investigation of the biggest is the development of not Euclidean geometry.
At the beginning of the 17th century, there were two important developments in geometry. The first, and most important, is the creation of analik geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). It was early that the need for the development of calculus. The second geometric development is the systematic investigation of projective geometry by Girard Desargues (1591-1661). Projective geometry is the investigation of geometry without measurement, Only by probing how the relationship between each other.
Two developments in geometry in the 19th century, changed the way she had learned before. It is a discovery not Euclidean geometry by Lobachevsky, Bolyai and Gaussdan of symmetry formulations as a primary consideration in the Erlangen Program of Felix Klein (who concluded the geometry of Euclid and not Euclid). Two of the experts at the time ialahBernhard geometry Riemann, working in mathematical analysis, and Henri Poincaré, as pengagastopologi algebraik and geometric theory of systems dinamikal.
As a result of this major change in the conception of geometry, the concept of "space" into something rich and different, and the background was originally just a different theory sepertianalisis complex and classical mechanics. A traditional type of geometry was certainly well known as dariruang homogeneous, namely space it has sufficient bekalan symmetry, so that from point to point they look the same.
C. Geometric Figures
1. Thales (640-546 BC)
At first geometry is based solely born by the experience. But the mathematician who first felt dissatisfied with the method that is based solely on experience is Thales (640-546 BC). Thales Mathematical Society now appreciated as a person who always berkarta "Prove it" and even he's always doing that. Of the many theorems are:
- Angles of a triangle samakaki bases are congruent,
- The corners of the bracket is congruent,
- An angle that is expressed in a semicircle is a right angle.
The work and principles Theles obviously been manandai the beginning of an era that develops mathematical advances as alasa logical deductive proof acceptable. Deductive substantiation necessary to lower the theorem of postulates. Furthermore, to formulate a new logical statement.
2. Pythagoras (582-507 BC)
After the death of Thales comes Pythagoras (582-507 BC) follows his followers, known as Pythagoreans move on Thales. The Pythagoreans used method of proving not only to develop the Pythagorean Theorem, but also against the theorems of the angles in a polygon, the properties of the lines are parallel, teorama of amounts that can not be compared, as well as the theorem about five wake solid irregular.
3. Euclid (300 BC)
Not many people are fortunate to have enduring fame as Euclid, Greek geometer great. Although in life figures such as Napoleon, Martin Luther, Alexander the Great, is much more famous than Euclid but in the long term may surpass all the fame they called it.
Besides fame, almost no detailed information about the life of Euclid knowable. For example, we know he has been active as a teacher in Alexandria, Egypt, in about 300 BC, but when he was born and when he died really dark. In fact, we do not know on which continent and in what city he was born. Although he wrote several books, among them there remained, its place in history mainly lies in the great book on geometry called The Elements.
In The Elements, Euclid combine school work she had to know with all the mathematical knowledge he has learned in a systematic comparisons to be an amazing result. Most of the work that are original, as the deductive method he demonstrated most of the required knowledge through reasoning. Euclid in the Elements also explain algebra and number theory as good as he describes the geometry.
The importance of the book The Elements does not lie in the statement of personal formulas are flung. Almost all the theories contained in the book had been written about before, and also has to be substantiated. Euclid contribution lies in the way the setting up of ingredients and formulation problems and thorough in planning the preparation of the book. Here caught, most important, the election of the arguments and the calculations, for example, about the possibility of drawing a straight line between two points.
After that carefully and cautiously he set up the proposition so easily understood by the people afterward. When necessary, he provides instructions for solving matters unresolved and develop experiments to the problems that are missed. It should be noted that the book The Elements other than primarily a development of the field of strict geometry, also in addition containing parts algebra following broad theory of summation.
The book The Elements is already a standard handbook of more than 2000 years and is the most successful books ever devised humans. So great Euclid compiled his book so that its shape alone was able to get rid of the books I've ever made before.
As a coach tool logic of the human mind, the book The Elements is much more influential than all the treatises of Aristotle's logic. The book is a complete example about the structure of deductive and is a stunning thought the fruit of all the creation of the human brain.
Fair if we say that the books of Euclid is an important factor for the growth of modern science. Science is not just a collection of observations carefully and not simply generalize too sharp and wise. Great results were snatched modern science comes from a combination of empirical inquiry work and experiments on the one hand, with careful analysis and conclusions that have a strong base on the other.
Euclid influence on Sir Isaac Newton are felt at all, since Newton wrote a famous book called The Principia in the form kegeometrian, similar to The Elements. Various scientists are trying to identify with Euclid with street shows how all of their conclusions logically traced back to the original assumption. Not unless what is done by mathematicians such as Russel, Whitehead and philosopher Spinoza.
Now, mathematicians already understand that the geometry of Euclid. is not the only system of geometry that is so fundamental guide and steadfast, and which can be planned well, they were advised that during the last 150 years many people who formulated the geometry is not a la Euclid. Actually, since Einstein's relativity theory accepted by, the scientists realized that Euclidean geometry is not always true in real application problems horizon.
At about the proximity of "Black Hole" and --misalnya-- neutron star where gravity is located in a high degree, Euclidean geometry does not give a thorough overview of the world, or do not show the exact translation of the space as a whole. However, these examples are rare, because in many ways the work of Euclid presents the possibility of estimates closer to reality. Advancement of human knowledge these days does not reduce either the result of intellectual effort Euclid and of the importance of his position in history.
4. Scientists-Muslim Scientists
In the era of Islamic Caliphate, the Muslim scientists also helped develop the geometry. In fact, in the medieval era, geometry controlled by Muslim mathematicians. No wonder if the Islamic civilization contributed a significant contribution to the development of the modern branches of mathematics.
Achievement of Islamic civilization in the golden era in the field of geometry really very amazing. Imagine. Researchers in the United States (US) found that in the 15th century AD, Muslim scholars have used crystal-like geometric patterns. In fact, the only modern mathematician discovering new pla geometric design it in the 20th century AD
According to a study published in the journal Science, the Muslim mathematician in the golden era has demonstrated an important breakthrough in the field of mathematics and art design in the 12th century AD "It's very impressive," said Peter Lu, a researcher from Harvard, US, told BBC.
Peter Lu said, the matemetikus and designers Muslim Caliphate era has mamapu make the design of the walls, floors and ceilings using tiles that reflect the use of a mathematical formula that is so sophisticated. '' The theory was only discovered 20 or 30 years ago, "he said.
Design in Islamic art using rules similar geometry to form crystals using symmetric polygon shapes to create a pattern. Until now, the general view is outstanding intricate pattern and star-shaped polygons in Islamic art design is achieved by using a zigzag lines drawn with a ruler and compass.
"You can see the development of this sophisticated geometric design. So they start with a simple design pattern, and eventually become more complex," added Peter Lu. Peter Lu's discovery proves that the Islamic civilization has been able to achieve remarkable progress in the field of geometry.
So how Islamic mathematicians develop geometry? In the 9th century AD, Muslim mathematician named Khwarizmi has developed geometry. Initially, studied the geometry of the leading mathematicians of the book entitled The Elements of Euclid's work. He was later to develop the geometry and encounter many new things in the study of the relationship in space. Al-Khwarizmi invented the term secans and tangens in the investigation trigonometry and astronomy. He also found a number system that is essential for modern number system. In the Number System, al-Khwarizmi contains terms Cosine, sine and tangent to solve trigonometric equations, theorems isosceles triangle, the calculation of the area of triangle, rectangle or circle area calculation in geometry.
Research al-Khwarizmi is regarded as a major revolution in the world of mathematics. He connects the concepts of geometry of ancient Greek mathematics into a new concept. Studies al-Khwarizmi produce a combined theory which enables the rational number / irrational, geometrical magnitudes are treated as objects of algebra.
Research al-Khwarizmi allow systematic application of algebra. For example, application of arithmetic to algebra, and vice versa, and vice versa algebra to trigonometry, algebra to the theory of numbers, algebra to geometry and vice versa. These studies underlie the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, number theory and geometry construction of the equation. The concept of geometry in mathematics introduced by al-Khwarizmi also very important in the field of astronomy. Because of Astronomy is a science that examines the stars including the position, movement, and interpretation related to the star. In order to calculate the position of stars on earth requires the calculation of geometry.
Other Muslim scientists who contributed to develop the geometry is Thabit Ibn Qurra. Muslim mathematicians who went by the nickname Thebit it also was one of the leading Muslim scientists in the field of geometry. He did important discoveries in the field of mathematics as integral calculus, trigonometry, analytic geometry, and geometry non-Eucledian.
One of the works of Thabit phenomenal in the field of geometry is his book The composition of Ratios (composition ratio). In the book, Thabit apply between arithmetic, geometry quantity ratio. This thinking, far beyond the discovery of the ancient Greek scientist in the field of geometry.
Thabit donations to other geometry that is, development of the theory of Pythagorean geometry in which he developed it from the right triangle special to all right-angled triangles. Thabit also studied geometry to support the discovery of the curve required to form an image of the sun.
In addition, other Muslim scientist who contributed to develop the geometry is Ibn al-Haitham. In the field of geometry, Ibn al-Haitham develop analytical geometry connecting geometry to algebra. In addition, he also introduced the concept of movement and transformations in geometry. Ibn al-Haitham theory in the field of the square is a theory that was first in elliptical geometry and hyperbolic geometry. This theory is considered as a sign of the emergence of non-Euclidean geometry. The works of Ibn al-Haitham it affects the work of the experts Persian geometry such as Nasir al-Din al-Tusi and Omar Khayyam. But the influence of Ibn al-Haytham do not stop in the Asian region alone. A number of European experts such Gersonides geometry, Witelo, Giovanni Girolamo Saccheri, and John Wallis was influenced thought al-Haitham. One of his leader in the science of geometry is Kitab wa al-Tahlil al'Tarkib.
Other Muslim scholars who contributed to develop the geometry is Abu Nasr Mansur ibn Ali ibn Iraq or commonly called Abu Nasr Mansur. He merupakana one of the experts who studied geometry spherical geometry (geometry associated with astronomy). Spherical geometry is very important to solve the difficult problems within Islam astonomi. Muslims need to determine the right time for prayers, Ramadan and the feast of both Eid and Eid al-Adha. With the help of spherical geometry, now people Muslimbisa estimate these times with ease. That is one of the scientific heritage of Abu Nasr Mansur for us today.
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