Section 10.7 Comparing the Three Systems of Geometry (Euclidean and non-Euclidean) Euclidean Geometry Euclid (300 B.C.) Hyperbolic Geometry Lobachevsky, Bloyai (1830) Elliptic Geometry Riemann(1850) Given a point not on a line, there is one and only one line through the point parallel to the given line. Given a point not on a line, there are an infinite number of lines through the point that do not intersect the given line. There are no parallel lines Geometry is on a plane: Geometry is on a pseudosphere: Geometry is on a sphere:
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Section 10.7
Comparing the Three Systems of Geometry(Euclidean and non-Euclidean)
Euclidean Geometry
Euclid (300 B.C.)
Hyperbolic Geometry
Lobachevsky, Bloyai (1830)
Elliptic Geometry
Riemann(1850)
Given a point not on a
line, there is one and
only one line through
the point parallel to the
given line.
Given a point not on a line, there
are an infinite number of lines
through the point that do not
intersect the given line.
There are no parallel
lines
Geometry is on a
plane:
Geometry is on a pseudosphere: Geometry is on a
sphere:
Section 10.7
Comparing the Three Systems of Geometry(Euclidean and non-Euclidean)
Euclidean Geometry
Euclid (300 B.C.)
Hyperbolic Geometry
Lobachevsky, Bloyai (1830)
Elliptic Geometry
Riemann(1850)
The sum of the
measures of the angles
of a triangle is 180 .
The sum of the measures of
the angles of a triangle is
less than 180 .
The sum of the
measures of the
angles of a triangle is
greater than 180 .
4/20/2010 Section 13.2 4
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 5
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,
.
3. The set of elements contains an identity element.
4. Each element of the set has an inverse that lies within the set.
cbacba
4/20/2010 Section 13.2 8
Congruence Modulo m
a is congruent to b in modulo m, written
mba mod
means that if a is divided by m, the remainder is b.
Example: Is ? Another words, “if 22 is divided by 6, is the remainder 4?”
6 mod 422
4remainder 3622
Since the remainder is 4, then the statement is true.
4/20/2010 Section 13.2 9
Modular AdditionsHow 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
7 mod 642
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