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 Reflection by mirror m 1 Reflection by mirror m 2 Reflection by mirror m 3 Rotation Do nothing 1 Rotation by turn 3 2 Rotation by turn 3

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

1



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 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. 3 B. 4 C. 5 D. 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. X

⊂

B. X

⊂

C. Z

⊂

D. Z

⊂ Y Z Y 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. A = B B. A = C C. B = C D. C = D A is the set of factors of 6 = {1, 2, 3, 6} 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 A B Flip the Mattress Motion 2 D C Flip the Mattress Motion 3 B A Flip the Mattress Motion 4 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 Identity Identity Rotate Rotate Vertical Flip Vertical Horizontal Flip Rotate Rotate Identity Vertical Flip Horizontal Flip Vertical Horizontal Horizontal Horizontal Identity Horizontal Vertical Rotate Vertical Rotate Identity

Rotate the Tires

Tires One Tires Two

Rotate the Tires

Rotate the Tires - options

Do nothing 90 Rotations

Operations

1. Identity 2. Step 1 90 3. Step 2 180   4. 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