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"Here's to Looking at Euclid" A Snapshot of Hyperbolic Geometry Nisreen Shiban Lizzy Bradley http://jwilson.coe.uga. Introduction 1. "To draw a straight line from any point to any point." 2. "To produce [extend] a finite straight line continuously in a straight line." 3. "To describe a circle with any centre and distance [radius]." 4. "That all right angles are equal to one another." 5. "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." Introduction 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The Parallel Postulate If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. Proclus' Axiom: "If a line intersects one of two parallels, it must intersect the other also." Playfair's Theorem: "Through a point not on a given line there exists a unique line parallel to the given line." Girolamo Saccheri D C Assume: A AD = BC AD BC B <ADC = <BCD ; not necess. right <s D C Assume: A AD = BC AD BC B Assume <ADC and <BCD > 180 1. AB > CD 2. Sum of a triangle's angles > 180 3. A angle inscribed in a semicircle is always obtuse. Assume <ADC and <BCD < 180 1. AB < CD 2. Sum of a triangle's angles < 180 3. Two parallel lines need not have a common perpendicular. 4. Parallel lines are not equidistant. When they have a common perpendicular they recede from each other on each side of the perpendicular. When they have no common perpendicular, they recede from each other in one direction and are asymptotic in the other. "Out of nothing, I have created a strange new universe." - Johann Bolyai "One geometry cannot be more true than another; it can only be more convenient." - Poincare Beltrami proved that inconsistencies existed simultaneously in Hyperbolic Geometry and in Euclidian -- ending the debate. Theorem 1.1 The upper base angles of the Saccheri quadrilateral are equal. By the SAS (side-angle-side) congruency condition, triangles ABC and BAD are congruent. D C A B Theorem 1.2 The line joining the midpoints of the upper and lower bases is perpendicular to both. Therefore, the upper and lower bases lie on parallel lines sharing a common perpendicular. Theorem 1.3 The angle sum of a triangle does not exceed 180 degrees. Theorem 1.3 continued The angle sum of a triangle does not exceed 180 degrees. Theorem 1.4 Consider a quadrilateral with a lower base that makes right angles with its two arms. i) if the upper base angles are unequal, so are the arms. ii) If the arms are unequal, so are the upper base angles, with the greater upper base angle opposite the greater arm. From neutral to hyperbolic geometry Theorem 1.5 In the Saccheri quadrilateral: i) the altitude is shorter than the arms, and ii) the upper base is longer than the lower base A Theorem to Clarify Parallel Lines Given two lines, if there exists a transversal which cuts the lines so as to form equal alternate interior angles or equal corresponding angles, then the lines are parallel with a common perpendicular. A Yet Another Parallel Theorem The distance between two parallels with common perpendicular is least when measured along that perpendicular. The distance from a point on either parallel to the other increases as the point recedes from the perpendicular in either direction. Dinosaur Math Jokes are the Best Quiz time In hyperbolic geometry, why can there be no squares or rectangles? List three real life applications to hyperbolic geometry If we define a rectangle as a quadrilateral with four right angles, then this is easily shown by cutting the rectangle along the diagonal into two triangles. The sum of the angles of the two triangles must be less than 2*180 = 360, so that we can't possibly have four right angles. Einstein's Theory Of General Relativity is based on a theory that space is curved. The cause is explained by the theory itself. Einstein's General Theory of Relativity can be understood as saying that: Matter and energy distort space The distortions of space affect the motions of matter and energy. If this is true then the correct Geometry of our universe will be hyperbolic geometry which is a 'curved' one. Many present-day cosmologists feel that we live in a three dimensional universe that is curved into the 4th dimension and that Einstein's theories were proof of this. Hyperbolic Geometry plays a very important role in the Theory of General Relativity. Used for navigation around earth by pilots The Sun causes some medium-scale curvature that - thanks to planet Mercury - we are able to measure. Mercury is the closest planet to the Sun. It is in a much higher gravitational field than is the Earth, and therefore, space is significantly more curved in its vicinity. Mercury is close enough to us so that, with telescopes, we can make accurate measurements of its motion. Mercury's orbit about the Sun is slightly more accurately predicted when Hyperbolic Geometry is used in place of Euclidean Geometry. http://www.youtube.com/watch? v=ryTGnBe7CjY