Non-Euclidean geometry

Math 451: Axiomatic Geometry Research Paper 12/13/16 History of Hyperbolic Geometry The discovery of non-Euclidean geometry is credited to nineteenth-century mathematicians Carl Friedrich Gauss, Nikolai Ivanovich Lobachevsky, and János Bolyai because they are first to recognize that the negation of Euclid’s Fifth Postulate as an axiom produced another geometry that was as rich and solid as that of Euclidean geometry (Venema, 2012). However, several concepts of Hyperbolic geometry were already known

which architects attempt to build unconventional forms and structures. In the ancient world, geometry was used as a practical tool to solve problems in fields such as architecture and navigation. As fragmented knowledge grew, mathematicians felt the need to approach geometry in a more systematic fashion. (Thorgeirsson 3) The key for understanding the structure of organic forms is abandoning Euclidean geometry, which does not describe the elements of the natural world. Instead it merely simplifies it

The Royal academy 1811-1815 In 1811 Samuel Morse’s father allowed him to attend the Royal Academy in England soon after his education at Yale University. At Yale, Samuel Morse saw demonstrations of electricity, which played an important role in his later life. The Royal Academy was a school where Samuel Morse would be able to study and practice painting. He practiced there from 1811 to 1815 alongside some of the most famous artists of his time. Later, he furthered his artistic career. He painted

Euclid. The geometry we studied in high school was based on the writings of Euclid and rightly called Euclidean geometry. Euclidean geometry is based on basic truths, axioms or postulates that are “obvious”. Born in about 300 BC Euclid of Alexandria a Greek mathematician and teacher wrote Elements. The book is one of the most influential and most published books of all time. In his book the Elements Euclid included five axioms that he deduced and which became the basis for the geometry we now call

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

Ontology deals with questions concerning what entities exist or can be said to exist. What do we know? What are we certain of? What can we prove? What is the nature of existence? Epistemology is the study of knowledge. How do we know what we know? How can we establish truth and certainty? Are their limits to what we can know based on how we come to know it? These epistemological questions when combined with ontological questions have philosophers pondering what exists and how we know it exists.

We find it in the American West through Manifest Destiny, freedom from caste, and in the chance that homesteaders had to acquire virtually free land. We find independence in math through in the building of stronger theoretical foundations, non-Euclidean geometries, and Cantor's infinities. Independence involves breaking from the commonly accepted, traditional views in order to explore the new. It is not necessarily individual people working alone. We can see independence in a community of thought

EUCLID: The Man Who Created a Math Class      Euclid of Alexandria was born in about 325 BC. He is the most prominent mathematician of antiquity best known for his dissertation on mathematics. He was able to create “The Elements” which included the composition of many other famous mathematicians together. He began exploring math because he felt that he needed to compile certain things and fix certain postulates and theorems. His book included, many of Eudoxus’ theorems,

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

postulates prove or define the existence of points, lines and circles and from there on go to define other aspects of geometry based upon the simpler concepts. The Elements consists of thirteen books. Some assumptions are not totally provable, such that there is only one straight line between two points. Euclid makes some assumptions that make his form of mathematics, Euclidean Geometry, sometimes at odds with other forms. His fifth postulate, states that only one line can be made through a point parallel