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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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