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
Tiling of Hyperbolic Plane
Symmetry correspondence of distance between various parts of an object
Symmetry •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 Broader definition as of mid-1800’s 1. Transformation Groups - Symmetric Figures 2. Discrete –topology3. 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
Mirror
Rotation
Symmetry of Finite FiguresHave no Translation Symmetry
Do nothing
Rotation by turn
Rotation by turn
13
23
Reflection by mirror m1
Reflection by mirror m2 Reflection by mirror m3
Symmetry of Figures
With a Glide
And a Translation
Vertical Mirror Symmetry
Horizontal Mirror Symmetry
Rotational Symmetry
=
Vertical and Horizontal
Mirrors
Number TheoryWhy numbers?
Number TheoryWhy zero?
Why subtraction?
Why negative numbers?
Why fractions?
Sharing is caring ½ + ½ = 1
Why Irrational Numbers?
Set: items students wear to school
Set: items students wear to school
{socks, shoes, watches, shirts, ...}
{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 SetØ or {}
Set of piano keys on a guitar.
Set A is {1,2,3} Elements of the set 1 A
5 A
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 numbersSet B = {4, 2, 1, 3}
Which one of the following sets is infinite?
A. Set of whole numbers less than 10
B. Set of prime numbers less than 10
C. Set of integers less than 10
D. Set of factors of 10
= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is finite
= {2, 3, 5, 7} is finite
= {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is infinite since the negative integers go on for ever.
= {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. 3B. 4C. 5D. 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 6Z is the set of multiples of 9
Which one of the following is true?( means "subset")⊂
A. X Y⊂B. X Z⊂C. Z Y⊂D. 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 6B is the set of prime factors of 6C is the set of proper factors of 6D is the set of factors of 3Which of the following is true?
A is the set of factors of 6 = {1, 2, 3, 6}
Only 2 and 3 are prime numbersTherefore B = the set of prime factors of 6 = {2, 3}
The proper factors of an integer do not include 1 and the number itselfTherefore 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
A. A = BB. A = CC. B = CD. C = D
Rock Set Imagine numbers as sets of rocks.
Create a set of 6 rocks.
Create Square Patterns
Find the Pattern1. Form two rows2. Sort even and odd
Work with a partnerShare your rocks.Form the odd numbered sets into even numbered sets.What do you observe?
Odd + Odd = Even
Odd numbers can make L-shapesStack successive L-shapesWhat shape is formed?
when you stack successive L-shapes together, you get a square
Sum the numbers from 1-100
Create a Cayley table for the sum of all the numbers from 1 to 10.
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 Patternsalso known as Border Patterns
What are the rigid motions that preserve each pattern?
Frieze Patterns
Flip the Mattress
Flip the MattressMotion 1
A B
C D
Flip the Mattress Motion 2
D C
B A
Flip the Mattress Motion 3
B A
D C
Flip the MattressMotion 4
A B
C D
Operation Identity Rotate Vertical Flip Horizontal FlipIdentity Identity Rotate Vertical HorizontalRotate Rotate Identity Horizontal VerticalVertical Flip Vertical Horizontal Identity RotateHorizontal Flip
Horizontal Vertical Rotate Identity
Flip the BedWords to describe movement/operations.
1. Identity 2. Rotate3. Vertical Flip4. Horizontal Flip
Cayley Table
Rotate the Tires
Tires One Tires Two
Rotate the Tires
Rotate the Tires - options Do nothing 90 Rotations
Operations
1. Identity2. Step 1 903. Step 2 1804. Step 3 270
5 TiresRotation Problem
9+4 =1 ?When does
Modular Arithmetic
Where numbers "wrap around" upon reaching a certain value—the modulus.
Our clock uses modulus 12mod 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
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