When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate
While the study of curvature is an ancient one, the geometry of curved surfaces is a topic that has been slowly developed over centuries. The Ancient Greeks certainly considered the curvature of a circle and a line distinct, noting that lines do not bend, while circles do. Aristotle expanded on this concept explaining that there were three kinds of loci: straight, circular, and mixed (Coolidge)Then in the third century B.C. Apollonius of Perga found that at each point of a conic section there is
well as quickly learning new languages. Farkas claimed that Janos had learned everything that Farkas could teach him by the time he was fifteen. Janos could speak many languages, and was very knowledgeable in calculus, trigonometry, algebra, and geometry. He was also a student at the Academy of Military Engineering in Vienna at the young age of 16. He studied for 4 years completing his degree in a little over half the time it took most students. Janos became interested in the problem of the axiom
Euclidean Geometry is a type of geometry created about 2400 years ago by the Greek mathematician, Euclid. Euclid studied points, lines and planes. The discoveries he made were organized into different theorems, postulates, definitions, and axioms. The ideas came up with were all written down in a set of books called Elements. Not only did Euclid state his ideas in Elements, but he proved them as well. Once he had one idea proven, Euclid would prove another idea that would have to be true based on
Nikolai Lobachevsky was born on December 1, 1792 near Nizhny Novgorod in Russia. He was born to Polish parents named Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya. He was one of three sons and his family was very poor. When Lobachevsky was only seven years of age, his father, a land surveyor, died. Soon after that his family uprooted and moved to Kazan, Russia, located somewhere near Siberia to try and start a new life and escape poverty. This is where Lobachevsky would
Differences in Geometry Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between
The Ellipse, Parabola and Hyperbola 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
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
Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry. M.C. Escher used his knowledge of geometry, and mathematics in general, to create his tessellations, some of his most well admired works. It is well known that in the past, Renaissance artists received their training in an atmosphere of artists and mathematicians
art in a more definite way – by actually becoming art. The introduction of fractal geometry and tessellations as creative works spawned the creation of new and innovative genres of art, which can be exemplified through the works of M.C Escher. Escher’s pieces are among the most recognized works of art today. While visually stimulating and deeply meaningful, his art reflects many ideas of mathematics through geometry, symmetry, and patterns. Maurits Cornelius Escher was born on June 17th, 1898
Gauss Carl Friedrich Gauss was a German mathematician and scientist who dominated the mathematical community during and after his lifetime. His outstanding work includes the discovery of the method of least squares, the discovery of non-Euclidean geometry, and important contributions to the theory of numbers. Born in Brunswick, Germany, on April 30, 1777, Johann Friedrich Carl Gauss showed early and unmistakable signs of being an extraordinary youth. As a child prodigy, he was self taught in the fields
inspired by the math he read about and his work related to those mathematical principles. This is interesting because he only had formal mathematical training through secondary school. He worked with non-Euclidean geometry and “impossible” figures. His work covered two main areas: geometry of space and logic of space. They included tessellations, polyhedras, and images relating to the shape of space, the logic of space, science, and artificial intelligence (Smith, B. Sidney). Although Escher worked
Combinations in Pascal’s Triangle Pascal’s Triangle is a relatively simple picture to create, but the patterns that can be found within it are seemingly endless. Pascal’s Triangle is formed by adding the closest two numbers from the previous row to form the next number in the row directly below, starting with the number 1 at the very tip. This 1 is said to be in the zeroth row. After this you can imagine that the entire triangle is surrounded by 0s. This allows us to say that the next row (row
well-known division of math, known as Geometry. Thus, he was named ‘The Father of Geometry’. Euclid taught at Ptolemy’s University, Egypt. At the Alexandria Library, It was said that he set up a private school to teach Mathematical enthusiasts like himself. It’s been also said that Euclid was kind and patient, and has a sense of humor. King Ptolemyance once asked Euclid if there was an easier way to study math and he replied “There is no royal road to Geometry”. Euclid wrote the most permanent mathematical
deductive geometry. He also discovered theorems of elementary geometry and is said to have correctly predicted an eclipse of the sun. Many of his studies were in astronomy but he also observed static electricity. Phythogoras was a Greek philosopher. He discovered simple numerical ratios relating the musical tones of major consonances, to the length of the strings used in sounding them. The Pythagorean theorem was named after him, although this fundamental statements of deductive geometry was most
determining absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and navigation, including global positioning systems (GPS). In contrast to triangulation, it does not involve the measurement of angles. In two-dimensional geometry, it is known that if a point lies on two circles, then the circle centers and the two radii
counting and record keeping, and they both developed systems of arithmetic (Allen, 2001, p.1). They used computation to find area, volume, circumference, and both used fractions. For both, the arithmetic was used for distribution of goods and the geometry for building. Their mathematics was very practical. What survives from both civilizations is records of problems solved by example. There is no record of generalizing principles or teaching principles supported by examples. This lack of mathematical
of his system.” Postulate 5, the parallel postulate, is today very controversial. Next, Euclid created a list of five common notions, of which only the fourth sparked a little debate. These common notions were more general and were not specific to geometry. After completing all these “preliminaries,” Euclid proved 48 propositions in Book 1. His first proposition was the equilateral triangle construction. However, this proof sparked a lot of controversy because EUclid didn’t prove that the two circles
Leonhard Euler was a Swiss mathematician born on April 15, 1707 in Basel, Switzerland. His parents were Paul Euler and Marguerite Brucker. Euler had two sisters,named Anna Maria and Maria Magdalena, and he was raised in a religious family and would be a faithful calvinist for the rest of his life because of his father being a priest of the Reformed Church and his mother being raised by a dad who was a pastor. Soon after Leonhard Euler was born, his parents moved
my time learning something that I possibly may never use outside of school?” Well, you’d be surprised if you knew all the different careers and jobs that use advanced math every day. For example, carpenters, contractors, and even optometrists use geometry and algebra quite often. Whether you want to believe it or not, math is around you everyday. The buildings you live in, the glasses you wear, and even furniture you sit on all starts with math. A carpenter is a type a craftsman, usually dealing with