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

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

21. Frieze Patterns also known as Border Patterns

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

23. Frieze Patterns

25. Rotate the Tires Tires One Tires Two

26. Rotate the Tires

27. Rotate the Tires RRH RRV H V

28. Flip the Mattress

29. 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 D C

31. When does 9+4 =1 ?

32. Modular Arithmetic Where numbers "wrap around" upon reaching a certain value—the modulus. Our clock uses modulus 12 mod 12

33. What would time be like if we had a mod 24clock?

34. What would time be like if we had a mod 6 clock?