Geometry

1. Geometry concerned with questions of shape, size, relative position of figures, and the properties of space.

2. 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

3. Symmetry correspondence of distance between various parts of an object Tiling of Hyperbolic Plane

4. 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

5. Broader definition as of mid-1800’s Transformation Groups - Symmetric Figures Discrete –topology Continuous – Lie Theory and Riemannian Geometry Projective Geometry - duality Symmetry

6. Projective Geometry

7. 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. Symmetric Figures Groups

8. Motion of Figures: Translation Rotation Mirror – vertical and horizontal Glide Symmetry Group

9. Mirror Symmetry

10. Rotation Symmetry

11. Symmetry of Finite Figures Have no Translation Symmetry Mirror Rotation Reflection by mirror m1 Reflection by mirror m2 Reflection by mirror m3 Do nothing Rotation by turn Rotation by turn

12. Symmetry of Figures With a Glide And a Translation

13. Vertical Mirror Symmetry

14. Horizontal Mirror Symmetry

15. Rotational Symmetry = Vertical and Horizontal Mirrors

17. Human Face Mirror Symmetric?

19. Number Theory Why numbers?

20. Number Theory Why zero?

21. Why subtraction?

22. Why negative numbers?

23. Why fractions? Sharing is caring ½ + ½ = 1

24. Why Irrational Numbers?

26. Set: items students wear to school {socks, shoes, watches, shirts, ...} Set: items students wear to school {index, middle, ring, pinky}

27. 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}

28. Null Set or Empty Set Ø or {} Set of piano keys on a guitar.

29. Set A is {1,2,3} Elements of the set • 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 numbers Set B = {4, 2, 1, 3}

30. Which one of the following sets is infinite? Set of whole numbers less than 10 Set of prime numbers less than 10 Set of integers less than 10 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

31. A is the set of factors of 12.Which one of the following is not a member of A? 3 4 5 6 Answer: 12 = 1×12 12 = 2×6 12 = 3×4A is the set of factors of 12 = {1, 2, 3, 4, 6, 12}So 5 is not a member of A

32. X is the set of multiples of 3 Y is the set of multiples of 6Z is the set of multiples of 9Which one of the following is true? (⊂ means "subset") 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⊂XEvery member of Z is also a member of X, so Z⊂XTherefore Only answer D is correct

33. 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 = B A = C B = C C = D

35. Rock Set Imagine numbers as sets of rocks. Create a set of 6 rocks. Create Square Patterns

36. Find the Pattern Form two rows Sort even and odd

37. Work with a partner Share your rocks. Form the odd numbered sets into even numbered sets. What do you observe? Odd + Odd = Even

38. 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

39. Create a Cayley table for the sum of all the numbers from 1 to 10. Sum the numbers from 1-100

41. Geoboard – construct square, rhombus, rectangle, parallelogram, kite, trapezoid or isosceles trapezoid. Complete table below.

43. Frieze Patterns • frieze • from architecture • refers to a decorative carving or pattern that runs horizontally just below a roofline or ceiling

44. Frieze Patterns also known as Border Patterns

45. What are the rigid motions that preserve each pattern?

46. Frieze Patterns

48. Flip the Mattress

49. Flip the Mattress Motion 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 Mattress Motion 4 A B C D

50. Flip the Bed Words to describe movement/operations. 1. Identity 2. Rotate 3. Vertical Flip • Horizontal Flip Cayley Table