Geometry is simply the study of space. There are Euclidean and Non-Euclidean Geometries. Euclidean geometry is the most common and is the basis for other Non-Euclidean types of geometry. Euclidean geometry is based on five main rules, or postulates. Differences in these rules are what make new kinds of geometries. There is Euclidean, Elliptic, and Hyperbolic Geometry. Euclidean geometry is the study of flat space and was invented by Euclid, a mathematician from Alexandria, in 330 B.C. Euclid described his new ideas and general rules in the book Elements (Bradley 30). The foundation of Euclidean geometry is the concept of a few undefined terms: points, lines, and planes. In essence, a point is an exact position or location on a surface. A point has no actual length or width. A line shows infinite distance and direction but absolutely no width. A line has at least two points lying on it. Euclid’s first postulate is that only one unique straight line can be drawn between any two points. Line segments are lines that have a set length and do not go on forever. Euclid’s second postulate is that a finite straight line, or line segment, can be extended continuously into a straight line. The last of Euclid’s undefined terms is a plane, a flat surface similar to a table top or floor. However, a plane’s area is infinite. It has never ending length and width but has no depth. Lines can intersect each other or they can be parallel. Intersecting lines can be perpendicular, meaning they cross at a right angle. Lines in a plane that do not intersect or touch at a point and have a constant, unchanging distance between each other are called parallel lines. Line segments can be used to create different polygons. As in Euclid’s third postulate, with any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All the angles in a triangle add up to 180 degrees. An acute angle is less than 90 degrees. A right angle is 90 degrees; all...

...HYPERBOLIC GEOMETRY AND OMEGA TRIANGLES
Hyperbolic geometry was first discovered and explored by Omar Khayyam in the 9th century and Giovanni Gerolamo Saccheri in the 15th century. Both were attempting to prove Euclid’s parallel postulate by proving the concept of hyperbolic geometry to be inconsistent, and ironically they discovered it to be a new type of geometry. It wasn’t until the 19th century that it became fully developed...

...is not a problem!
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metri "measurement") "Earth-measuring" is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose...

...Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclideangeometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p.
Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there...

...When it comes to EuclideanGeometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for EuclideanGeometry 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...

... We use Euclidean and Non-Euclideangeometry in our everyday use. In many ways they are similar and different. There are similarities and differences in Euclideangeometry and spherical geometry, Euclid’s fifth postulate applies to both forms, and it is used every day in astronomy. Euclideangeometry is the study of flat space, and can be easily drawn on a piece of...

...Non-Euclideangeometry is any form of geometry that is based on axioms, or postulates, different from those of Euclideangeometry. These geometries were developed by mathematicians to find a way to prove Euclid’s fifth postulate as a theorem using his other four postulates. They were not accepted until around the nineteenth century. These geometries are based on a curved plane,...

...Maddux Willingham
Mr. Warfle
Geometry Honors
26 September 2011
Euclidean & Non-EuclideanGeometry Paper
Isn’t it amazing that we still study the same geometry as people did back nearly twenty-three centuries ago? Euclidean and Non-Euclideangeometry communicates to us through mathematical equations immense amounts of significant information....

...segment PQ:
In Euclideangeometry the perpendicular distance between the rays
remains equal to the distance from P to Q as we move to the right.
However, in the early nineteenth century two alternative geometries
were proposed. In hyperbolic geometry (from the Greek hyperballein,
"to exceed") the distance between the rays increases. In elliptic
geometry (from the Greek elleipein, "to fall short") the distance decreases and...

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