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Why is Euclidean geometry important
The Power of Triangulation
Euclidean geometry terms
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Euclidean Geometry is the study of plane and solid figures based on the axioms and theorems outlined by the Greek mathematician Euclid (c. 300 B.C.E.). It is this type of geometry that is widely taught in secondary schools. For much of modern history the word geometry was in fact synonymous with Euclidean geometry, as it was not until the late 19th century when mathematicians were attracted to the idea of non-Euclidean geometries. Euclid’s geometry embodies the most typical expression of general mathematical thinking. Rather than simply memorizing basic algorithms to solve equations by rote, it demands true insight into the subject, cleaver ideas for applying theorems in special situations, an ability to generalize from known facts, and an …show more content…
Not far removed from the information given above, a geodesic is simply the shortest distance between two points. This is a concept we should all be familiar with. In R2 a geodesic is just a straight line. In spherical geometry a geodesic is a great circle. Unlike hyperbolic geometry, which may appear to have a more abstract beginning, the idea of a geodesic has a more concrete relation to the real world. The word geodesic itself comes from geodesy, which is the science of measuring the shape of the earth. Shifting focus to perhaps a more interesting topic, we can begin the discussion on the mathematics of geodesics. The geodesic dome presented by this group is what is known as a 2V Icosa Alternate or more broadly a Class I Dome. It is created by fitting 4 triangles inside each triangular surface of an icosahedron; which is one of the five solids created by the ancient Greeks. When considering a icosahedron, or any regular polyhedral for that matter, we have the following formulas to consider: 1. V = 10υ2 + 2 2. F = 20υ2 3. E = …show more content…
They also gain strength from the triangular rigidity of the dome which greatly helps to prevent crushing and falling. The shape of a triangle makes it very strong and able to withstand heavy loads. The triangle is in actuality the strongest shape known to man. Box structures, which are the typical conventional homes, on the other hand are easily distorted by heavy loads. When many triangles are connected to form the shell of a dome, they translate to tremendous strength which makes them self-supporting. This in turn eliminates the need for supporting structures. The tremendous strength is a result of triangles distributing strength evenly across the entire structure. Rectangular or box structures for that matter distribute loads at right angles making them considerably
Euclid propositions can be called theorems in common language. In the Book I Euclid main considerations was on geometry. He began with a long list of definitions which followed by the small number of basic statements to take the essential properties of points, lines, angles etc. then he obtained the remaining geometry from these basic statements with proofs. (Berlinghoff, 2015, p.158).
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
What is trigonometry? Well trigonometry, according to the Oxford Dictionary ‘the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.’ Here is a simplified definition of my own: Trigonometry is a division of mathematics involving the study of the relativity of angles and sides of triangles. The word trigonometry originated from the Latin word: trigonometria.
Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it's simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid's elements, which states that, "If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles."
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.
Non-euclidean geometry not only **, but also suggests that the characteristics of “necessity” and “university” could be questioned, as “universality” and “necessity” seems to suggests, once the proposition of
"The Foundations of Geometry: From Thales to Euclid." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 1. Detroit: Gale, 2001. Gale Power Search. Web. 20 Dec. 2013.
Euclid is one of the most influential and best read mathematician of all time. His prize work, Elements, was the textbook of elementary geometry and logic up to the early twentieth century. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians. Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes. Personally, all accounts of Euclid describe him as a kind, fair, patient man who quickly helped and praised the works of others. However, this did not stop him from engaging in sarcasm. One story relates that one of his students complained that he had no use for any of the mathematics he was learning. Euclid quickly called to his slave to give the boy a coin because "he must make gain out of what he learns." Another story relates that Ptolemy asked the mathematician if there was some easier way to learn geometry than by learning all the theorems. Euclid replied, "There is no royal road to geometry" and sent the king to study. Euclid's fame comes from his writings, especially his masterpiece Elements. This 13 volume work is a compilation of Greek mathematics and geometry. It is unknown how much if any of the work included in Elements is Euclid's original work; many of the theorems found can be traced to previous thinkers including Euxodus, Thales, Hippocrates and Pythagoras. However, the format of Elements belongs to him alone. Each volume lists a number of definitions and postulates followed by theorems, which are followed by proofs using those definitions and postulates. Every statement was proven, no matter how obvious. Euclid chose his postulates carefully, picking only the most basic and self-evident propositions as the basis of his work. Before, rival schools each had a different set of postulates, some of which were very questionable. This format helped standardize Greek mathematics. As for the subject matter, it ran the gamut of ancient thought. The
By its definition, a dome is a hemispherical form-resistant structure of small thickness (Salvadori, 2002). Their impressive stability is due to their continuous, curved shape that allows them to withstand bending, tensile, and compressive loads (Levy & Salvadori, 1992). Without the use of reinforcing steel, as is necessitated in modern construction, domes built in antiquity had to be self-supporting. They succeeded in this through two key features: the dome’s meridians and parallels. These double features, both still used today, were developed during antiquity and proved their worth throughout the passage of time (Salvadori, 2002).
Mathematicians, engineers and scientists encounter numerous functions in their work: polynomials, trigonometric and hyperbolic functions amongst them. However, throughout the history of science one group of functions, the conics, arise time and time again not only in the development of mathematical theory but also in practical applications. The conics were first studied by the Greek mathematician Apollonius more than 200 years BC. Essentially, the conics form that class of curves which are obtained when a double cone is intersected by a plane. There are three main types: the ellipse , the parabola and the hyperbola . From the ellipse we obtain the circle as a special case, and from the hyperbola we obtain the rectangular hyperbola as a special case. These curves are illustrated in the following figures. cone-axis
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
As a secondary subject, society often views mathematics a critical subject for students to learn in order to be successful. Often times, mathematics serves as a gatekeeper for higher learning and certain specific careers. Since the times of Plato, “mathematics was virtually the first thing everyone has to learn…common to all arts, science, and forms of thought” (Stinson, 2004). Plato argued that all students should learn arithmetic; the advanced mathematics was reserved for those that would serve as the “philosopher guardians” of the city (Stinson, 2004). By the 1900s in the United States, mathematics found itself as a cornerstone of curriculum for students. National reports throughout the 20th Century solidified the importance of mathematics in the success of our nation and its students (Stinson, 2004). As a mathematics teacher, my role to educate all students in mathematics is an important one. My personal philosophy of mathematics education – including the optimal learning environment and best practices teaching strategies – motivates my teaching strategies in my personal classroom.
“Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid” (Artmann, 2016, para.1). Euclidean geometry was developed by Euclid, who ran his own school in Alexandria, Egypt
Conic sections are the various gemetric figures created by the interection of a plane. They are among the oldest curves in history and is one of the oldest area of study for mathmaticians. conics were discovered by Menaechmus (c. 375 - 325 BC), a Greek pupil of Plato and Exodus. He was trying to solve the famous problem duplicating a cube. Euclid studied them and Appollonius reinforced and expanded previous results of conics into a book he named Conic Sections. It is a series of eight books with 487 propositions. He applied his findings to the study of planetary motion and it was used to advance the development of Greek astronomy. It is because of Appollonius that the name ellipse, parabola, and hyperbole were given to conics. Conics evolved even further during the Renaissance with Kepler’s law of planetary motion, Descarte on his work Geometry and Fermat’s coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, and Pascal. We can see conics in satellite dishes, sharpening pencils, automobile headlights, when a baseball is hit, telescopes, and much more. Physicians apply conics in treating kidney stones. Even, John Quincy Adams used conics to eaves drop on members of the house of representatives from his desk in the U.S. Capitol building.
Euclid, also known as Euclid of Alexandria, lived from 323-283 BC. He was a famous Greek mathematician, often referred to as the ‘Father of Geometry”. The dates of his existence were so long ago that the date and place of Euclid’s birth and the date and circumstances of his death are unknown, and only is roughly estimated in proximity to figures mentioned in references around the world. Alexandria was a broad teacher that taught lessons across the world. He taught at Alexandria in Egypt. Euclid’s most well-known work is his treatise on geometry: The Elements. His Elements is one of the most influential works in the history of mathematics, serving as the source textbook for teaching mathematics on different grade levels. His geometry work was used especially from the time of publication until the late 19th and early 20th century Euclid reasoned the principles of what is now called Euclidean geometry, which came from a small set of axioms on the Elements. Euclid was also famous for writing books using the topic on perspective, conic sections, spherical geometry, number theory, and rigor.