Abstract Geometry
The ancient Egyptians and Babylonians discovered abstract Geometry. They developed these ideas that were used to build pyramids and help with reestablishing land boundaries. While, the Babylonians used abstract geometry for measuring, construction buildings, and surveying. Abstract geometry uses postulates, rules, definitions and propositions before and up to the time of the Euclid.
Abstract geometry is deductive reasoning and axiomatic organization. Deductive reasoning deals with statements that have already been accepted. An example of deductive reasoning is proving the sum of the measures of the angles of a quadrilateral is 360 degrees. Another example of deductive reasoning is proving the sum of the angles of a trigon is equal to 180 degrees. From this we get, any quadrilateral can be divided into two trigons. Axioms, which are also called postulates, are statements that can be proved true by using deductive reasoning.
Measurement geometry contains theories that exist and can have supporting ideas to back them up, and cannot be disproved. Hyperbolic geometry and elliptic geometry are two examples of measurement geometry. Non-Euclidean geometry can be considered measurement geometry, since it is a branch in which the fifth postulate of Euclidean Geometry is replaced by one of the two alternative postulates. Mathematicians in the nineteenth century showed that it is possible to create consistent geometries with Euclid's postulates.
An example of the difference in the abstract geometry and the measurement geometry is the sum of the measures of the angles of a trigon. The sum of the measures of the angles of a trigon is 180 degrees in Euclidian geometry, less than 180 in hyperbolic, and more than 180 in elliptic geometry. The area of a trigon in hyperbolic geometry is proportional to the excess of its angle sum over 180 degrees. In Euclidean geometry all trigons have an angle sum of 180 without respect to its area. Which means similar trigons with different areas can exist in Euclidean geometry. It is not possible in hyperbolic or elliptic geometry. In two-dimensional geometries, lines that are perpendicular to the same given line are parallel in abstract geometry, are neither parallel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic geometry. The appearance of the lines as straight or curved depends on the postulates for the space.
At the present time mathematics is trying to figure out which of the three is the best representation of the universe.
Deductive reasoning is a logical way to increase the set of facts that are assumed to be true. The purpose of Deductive reasoning is to end up at a logical conclusion based on the subject of discussion. Deductive Reasoning uses statements that are logically true in order to omit other statements that contradict the logically true statement, which is to deduce, subtract or takeaway. What
According to Roland Shearer (1992) the release of non-Euclidean geometries at the end of the 19th Century copied the announcement of art movements occurring at that time, which included Cubism, Constructivism, Orphism, De Stijl, Futurism, Suprematism and Kinetic art. Most of the artists who were involved in these beginnings of Modern art were directly working with the new ideas from non-Euclidean geometry or were at least exposed to it – artists such as Picasso, Braque, Malevich, Mondrian and Duchamp. To explain human-created geometries (Euclidean, non-Euclidean), it is a representation of human-made objects and technology (Shearer
One of the first ever documented estimates for the area of a circle was found in Egypt on a paper known as the Rhind Papyrus around the time of 1650 BCE. The paper itself was a copy of an older “book” written between 2000 and 1800 BCE and some of the information contained in that writing might have been handed down by Imhotep, the man who supervised the building of the pyramids.
The construction phase would not be possible without the knowledge of basic geometry. Points, lines, measurements and angles are often used to lay out the building in accordance to the architect drawings.
Euclidean distance was proposed by Greek mathematician Euclid of Alexandria. In mathematics, the Euclidean distance or Euclidean metric is the distance between two points, which is shown as a length of a line segment and is given by the Pythagorean theorem. The formula of Euclidean distance is a squ...
The Greeks were able to a lot of things with only a compass and a straight edge (although these were not their sole tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found means to construct parallel lines, to bisect angles, to construct various polygons, and to construct squares of equal or twice the area of a given polygon. However, three constructions that they failed to achieve with only those two tools were trisecting the angle, doubling the cube, and squaring the circle.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
Gear ratios is one way geometry is applied in mechanics. Everything from rings and pinion ratios to transmission gear ratios and even tire size come into play for the final drive ratio .These ratios are what determine what the speedometer says at any given speed. Geometry is used to determine the size of the engine itself. Combustion chamber size must be known and calculated as well as oil pan capacity and the cooling system capacity. Every tolerance inside an engine requires geometry in order to work correctly. Cam and crankshaft are dialed in using special gauges,
The most important application of mathematics was in astronomy since it helped guide them in the desert. The translated works of the Greek astronomer Ptolemy greatly helped in inspiring the Arabs to study astronomy. The Arabs scientists made use of a Greek device known as an astrolabe to help them compute “the position of the stars and the movement of the planets” and also helped them to keep track of time.
As stated by John Fuchs, “The triangle is the only two dimensional polygon that if constructed of rigid members with hinged corners is absolutely
Sumer, the southernmost region of Mesopotamia was known as the “cradle of civilization”. It was said to be the birthplace of writing, the wheel, the arch and many other innovations. When civilizations began to settle and develop agriculture, Sumerian mathematics quickly developed as a response to needs for measuring plots of lands, the taxation of individuals, and keeping track of objects. Through time, the Sumerians and Babylonians developed a mathematical system called the Base 60 numerical system in which they made such extraordinary advances in mathematics and astrology. Even today, Sumerians mathematic ways still have a strong influence in our modern mathematics.
Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclicalphenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
The foundations of mathematics are strongly rooted in the history and way of life of the Egyptian people, dating back to the fourth millennium B.C. in Egypt. Egyptian mathematics was elementary. It was generally arrived at by trial and error as a way to obtain desired results. As such, early Egyptian mathematics were primarily arithmetic, with an emphasis on measurement, surveying, and calculation in geometry. The development of arithmetic and geometry grew out of the need to develop land and agriculture and engage in business and trade. Over time, historians have discovered records of such transactions in the form of Egyptian carvings known as hieroglyphs.
In India, there was an era called “the Golden Age of Indian Mathematics. At this period, several refined and advanced mathematics were recorded. The concept of sine, cosine, and tangent in land surveying and navigations were already known to them. In addition, the use of trigonometry to calculate the distance between the earth, the moon, and sun was already part of Hindu’s culture. As the western civilization made some innovations in astronomy, Indian had already grasped the idea that the sun, moon, and the earth form a right-angled triangle when the moon is in half full and situated directly opposite the sun. It is really surprising th...
Trigonometry is one of the branches of mathematical and geometrical reasoning that studies the triangles, particularly right triangles The scientific applications of the concepts are trigonometry in the subject math we study the surface of little daily life application. The trigonometry will relate to daily life activities. Let’s explore areas this science finds use in our daily activities and how we use to resolve the problem.