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

Human FaceMirror Symmetric?

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