Transcript Modern Algebra - Denise Kapler

Geometry
concerned with questions of shape,
size, relative position of figures, and
the properties of space.
Geometry originated as a practical science concerned
with surveying, measurements, areas, and volumes.
Under Euclid worked from point, line, plane and space.
In Euclid's time…
… there was only one form of space.
Today we distinguish between:
•Physical space
•Geometrical spaces
•Abstract spaces
Symmetry
correspondence of
distance between various
parts of an object
Tiling of Hyperbolic Plane
•Area of Geometry since before Euclid
•Ancient philosophers studied symmetric
shapes such as circle, regular polygons,
and Platonic solids
•Occurs in nature
•Incorporated into art
Example M.C. Escher
Symmetry
Symmetry
Broader definition as of mid-1800’s
1. Transformation Groups - Symmetric Figures
2. Discrete –topology
3. Continuous – Lie Theory and Riemannian Geometry
4. Projective Geometry - duality
Projective Geometry
Symmetric Figures Groups
Symmetry Operation - a mathematical operation or transformation that results in
the same figure as the original figure (or its mirror image)
Operations include reflection, rotation, and translation.
Symmetry Operation on a figure is defined with respect to a given point (center of
symmetry), line (axis of symmetry), or plane (plane of symmetry).
Symmetry Group - set of all operations on a given figure that leave the figure
unchanged
Symmetry Groups of three-dimensional figures are of special interest because of
their application in fields such as crystallography.
Symmetry Group
Motion of Figures:
1. Translation
2. Rotation
3. Mirror – vertical and horizontal
4. Glide
Mirror Symmetry
Rotation Symmetry
Symmetry of Finite Figures
Have no Translation Symmetry
Mirror
Rotation
Reflection by mirror m1
Reflection by mirror m2
Reflection by mirror m3
Do nothing
1
3
Rotation by turn
Rotation by
2
3
turn
Symmetry of Figures
With a Glide
And a Translation
Vertical Mirror Symmetry
Horizontal Mirror Symmetry
Rotational
Symmetry
=
Vertical and
Horizontal
Mirrors
Human Face
Mirror Symmetric?
Number Theory
Why numbers?
Number Theory
Why zero?
Why subtraction?
Why negative numbers?
Why fractions?
Sharing is caring
½+½=1
Why Irrational Numbers?
Set: items students
wear to school
{socks, shoes, watches, shirts, ...}
Set: items students
wear to school
{index, middle, ring, pinky}
Create a set begin by defining a set specify the common
characteristic.
Examples:
•Set of even numbers {..., -4, -2, 0, 2, 4, ...}
•Set of odd numbers {..., -3, -1, 1, 3, ...}
•Set of prime numbers {2, 3, 5, 7, 11, 13, 17, ...}
•Positive multiples of 3 that are less than 10 {3, 6, 9}
Null Set or Empty Set
Ø
or {}
Set of piano keys on a guitar.
Set A is {1,2,3}
Elements of the set
A
5 A
1
Two sets are equal if they have precisely the same
elements.
Example of equal sets A = B
Set A: members are the first four positive whole numbers
Set B = {4, 2, 1, 3}
Which one of the following sets is infinite?
A. Set of whole numbers less than 10
= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is finite
B. Set of prime numbers less than 10
= {2, 3, 5, 7} is finite
C. Set of integers less than 10
= {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is infinite since the negative integers go
on for ever.
D. Set of factors of 10
= {1, 2, 5, 10} is finite
A is the set of factors of 12.
Which one of the following is not a
member of A?
A.
B.
C.
D.
3
4
5
6
Answer:
12 = 1×12
12 = 2×6
12 = 3×4
A is the set of factors of 12 = {1, 2, 3, 4, 6, 12}
So 5 is not a member of A
X is the set of multiples of 3
Y is the set of multiples of 6
Z is the set of multiples of 9
Which one of the following is true?
(⊂ means "subset")
A.
B.
C.
D.
X⊂Y
X⊂Z
Z⊂Y
Z⊂X
X = {...,-9, -6, -3, 0, 3, 6, 9,...}
Y = {...,-6, 0, 6,...]
Z = {...,-9, 0, 9,...}
Every member of Y is also a member of X, so Y⊂X
Every member of Z is also a member of X, so Z⊂X
Therefore Only answer D is correct
A is the set of factors of 6
B is the set of prime factors of 6
C is the set of proper factors of 6
D is the set of factors of 3
Which of the following is true?
A is the set of factors of 6 = {1, 2, 3, 6}
A.
B.
C.
D.
A=B
A=C
B=C
C=D
Only 2 and 3 are prime numbers
Therefore B = the set of prime factors of 6 = {2, 3}
The proper factors of an integer do not include 1
and the number itself
Therefore C = the set of proper factors of 6 = {2, 3}
D is the set of factors of 3 = {1, 3}
Therefore sets B and C are equal.
Answer C
Rock Set Imagine numbers as sets of rocks.
Create a set of 6 rocks.
Create Square Patterns
Find the Pattern
1. Form two rows
2. Sort even and odd
Work with a partner
Share your rocks.
Form the odd numbered sets into even numbered sets.
What do you observe?
Odd + Odd = Even
Odd numbers can make L-shapes
Stack successive L-shapes
What shape is formed?
when you stack successive L-shapes together, you get a square
Create a Cayley table for the sum
of all the numbers from 1 to 10.
Sum the numbers from 1-100
Geoboard – construct square, rhombus, rectangle, parallelogram,
kite, trapezoid or isosceles trapezoid. Complete table below.
Frieze Patterns
frieze
•from architecture
•refers to a decorative carving or pattern that runs horizontally
just below a roofline or ceiling
Frieze Patterns
also known as
Border Patterns
What are the rigid motions that preserve each pattern?
Frieze Patterns
Flip the Mattress
Flip the Mattress
Motion 1
Flip the Mattress
Motion 2
Flip the Mattress
Motion 3
Flip the Mattress
Motion 4
A
B
D
C
B
A
A
B
C
D
B
A
D
C
C
D
Flip the Bed
Words to describe movement/operations.
1. Identity
2. Rotate
3. Vertical Flip
4. Horizontal Flip
Cayley Table
Operation
Identity
Rotate
Vertical Flip
Horizontal
Flip
Identity
Identity
Rotate
Vertical
Horizontal
Rotate
Rotate
Identity
Horizontal
Vertical
Vertical Flip
Vertical
Horizontal
Identity
Rotate
Horizontal Flip
Horizontal
Vertical
Rotate
Identity
Rotate the Tires
Tires One
Tires Two
Rotate the Tires
Rotate the Tires - options
Do nothing
90 Rotations
Operations
1.
2.
3.
4.
Identity
Step 1 90 
Step 2 180 
Step 3 270 
5 Tires
Rotation
Problem
When does
9+4 =1 ?
Modular Arithmetic
Where numbers "wrap around" upon
reaching a certain value—the modulus.
Our clock uses modulus 12
mod 12
What would
time be like
if we had a
mod 24
clock?
What would
time be like if
we had a
mod 7 clock?
NASA GPS Satellite
Constellation of GPS System