Comparing the Three Systems of Geometry (Euclidean and non-Euclidean) Euclidean Geometry Hyperbolic Geometry Elliptic Geometry Euclid (300 B.C.) Lobachevsky, Bloyai (1830) Riemann(1850) Given a point not on a Given a point not on a line, there There are no parallel line, there is one and are an infinite number of lines lines only one line through through the point that do not the point parallel to the intersect the given line. given line. Geometry is on a Geometry is on a pseudosphere: Geometry is on a plane: sphere: Section 10.7 Comparing the Three Systems of Geometry (Euclidean and non-Euclidean) Euclidean Geometry Hyperbolic Geometry Elliptic Geometry Euclid (300 B.C.) Lobachevsky, Bloyai (1830) Riemann(1850) The sum of the The sum of the measures of The sum of the measures of the angles the angles of a triangle is measures of the of a triangle is 180 . less than 180 . angles of a triangle is greater than 180 . Section 10.7 Rotational Symmetry • A symmetry of an object is a motion that moves the object back onto itself. – In symmetry, you cannot tell, at the end of the motion, that the object has been moved. • If it takes m equal turns to restore an object to its original position and each of these turns is a figure that is identical to the original figure, the object has m-fold rotational symmetry. Example: A pinwheel has fourfold rotational symmetry. 4/20/2010 Section 13.2 4 Groups Definition of a Group: 1. The set of elements in the mathematical system is closed under the given operation, represented in this box by . 2. The set of elements is associative under the given binary operation. If a, b, and c are any three elements of the set, a b c a . b c 3. The set of elements contains an identity element. 4. Each element of the set has an inverse that lies within the set. 4/20/2010 Section 13.2 5 Congruence Modulo m a is congruent to b in modulo m, written a b mod m means that if a is divided by m, the remainder is b. Example: Is 22 ? Another4 mod words,6 “if 22 is divided by 6, is the remainder 4?” 22 6 3 remainder 4 Since the remainder is 4, then the statement is true. 4/20/2010 Section 13.2 8 Modular Additions How to add in a modulo m system: 1. Add the numbers using ordinary arithmetic. 2. If the sum is less than m, the answer is the sum obtained. 3. If the sum is greater than or equal to m, the answer is the remainder obtained upon dividing the sum in step 1 by m. Example: Find the sum (2+4)(mod 7). Solution: To find the sum, we first add 2+4 to get 6. Because this sum is less than 7, then 2 4 6 mod 7 4/20/2010 Section 13.2 9 .