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

2. Under Euclid worked from point, line, plane and 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
Wikipedia

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

4. 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

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

6. Projective Geometry

7. 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.

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

9. Mirror Symmetry
Atu.edu 6-13

10. Rotation Symmetry
Atu.edu

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. Vertical and Horizontal Mirrors
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
Conway, Princeton mathematician devised the following names for the different frieze groups:
1 Hop; 2 SpinHop; 3 Jump; 4 Sidle;5 Step; 6 SpinJump; 7 SpinSidle.

25. Rotate the Tires Tires One Tires Two And God said unto baby Noah
"There will be a mild increase in water level, and you must find a tire, 2 cubits in diameter, and take with you two of each stuffed animal."

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
In Spring - spin; in Fall - flip.

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 24 clock?

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