1. Transformations, Symmetries, and Tilings
11.1 Rigid Motions and Similarity
Transformations
11.2 Patterns and Symmetries
11.3 Tilings and Escher-like Designs

2. 11.1
Rigid Motions and Similarity Transformations

3. TRANSFORMATION OF THE PLANE
A one-to-one correspondence of the set
of points in the plane to itself is a transformation of the plane.
If point P corresponds to point
is called the image of P under the
transformation. Point P is called the preimage of

4. RIGID MOTION OF THE PLANE
A transformation of the plane is a rigid motion if, and only if, the distance between any two points equals the distance between their image points.
A rigid motion is also called an isometry.

5. THREE BASIC RIGID MOTIONS
Translation, or slide
Rotation, or turn
Reflection, or flip

6. TRANSLATIONS
A translation, or slide, is the rigid motion in which all points of the plane are moved the same distance in the same direction.
A slide arrow or translation vector defines the translation by giving:
the direction of the slide as the direction of the arrow and
the distance moved as the length of the arrow.

7. A TRANSLATION

8. ROTATIONS
A rotation, or turn, is the rigid motion in which one point in the plane is held fixed and the remaining points are turned about this center of rotation through the same number of degrees, called the angle of rotation.

9. CENTER OF ROTATION IS POINT O ANGLE OF ROTATION IS x

10. REFLECTIONS
A reflection, or flip or mirror reflection, is the rigid motion determined by a line in the plane called the line of reflection.
Each point P of the plane is transformed to the point P on the opposite side of the mirror line m and at the same distance from m.

11. REFLECTIONS

12. A REFLECTION
Note that B is located so that m is the perpendicular bisector of

13. DEFINITION: CONGRUENT FIGURES
Two figures are congruent if, and only if, one figure is the image of the other under a rigid motion.

14. DEFINITION: DILATION, OR SIZE TRANSFORMATION
Let O be a point in the plane and k a positive real number. A dilation, or size transformation, with center O and scale factor k is the transformation that takes each point of the plane to the point
on the ray for which
and takes the point O to itself.

15. A SIZE TRANSFORMATION

16. DEFINITION: SIMILAR FIGURES
A transformation is a similarity transformation if, and only if, it is a sequence of dilations and rigid motions.
Two figures F and G are similar, written
, if, and only if, there is a similarity transformation that takes one figure onto the other figure.

17. SIMILAR FIGURES
A dilation centered at O followed by a reflection define a similarity transformation taking figure F onto figure G.

18. 11.2
Patterns and Symmetries

19. DEFINITION: A SYMMETRY OF A PLANE FIGURE
A symmetry of a plane figure is any rigid motion of the plane that moves all the points of the figure back to points of the figure.

20. REFLECTION SYMMETRY
A figure has reflection symmetry if a reflection across some line is a symmetry of the figure.

21. Example 11.8 Identifying Lines of Symmetry
Identify all lines of symmetry for each letter.
M N O X
M = 1 vertical
N = none
O = 2; one vertical and one horizontal
X = 4: one vertical, one horizontal, and the two lines given in the figure itself.

22. ROTATION SYMMETRY
A figure has rotation symmetry, or turn symmetry, if the figure is superimposed on itself when it is rotated through a certain angle between 0 and 360. The center of the turn is called the center of rotation.

23. POINT SYMMETRY
A figure has point symmetry if it has 180 rotation symmetry about some point O.

24. Example 11.10 Identifying Point Symmetry
What letters, in uppercase block form, can be drawn to have point symmetry?
H, I, N, O, S, X, and Z.
The letters H, I, O, and X also have two perpendicular lines of mirror symmetry. Only N, S, and Z have just point symmetry.
Slide 11-24 24

25. PERIODIC PATTERNS
Border patterns have a repeated motif that has been translated in just one direction to create a strip design.

26. PERIODIC PATTERNS
Wallpaper patterns have a motif that has been translated in two nonparallel directions to create an all-over planar design.

27. 11.3
Tilings and Escher-like Designs

28. DEFINITION: TILES AND TILING
A simple closed curve, together with its interior, is a tile. A set of tiles forms a tiling of a figure if the figure is completely covered by the tiles without overlapping any interior points of the tiles.
In a tiling of a figure, there can be no gaps between tiles. Tilings are also known as tessellations.

29. TILING WITH REGULAR POLYGONS
Any arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure.
Equilateral triangles form a regular tiling because the measures of the interior angles meeting at a vertex figure add to 360.

30. TILING WITH EQUILATERAL TRIANGLES
One interior angle of an equilateral triangle has measure 60.
At a vertex angle:

31. TILING WITH SQUARES
One interior angle of a square has measure 90.
At a vertex angle:

32. TILING WITH REGULAR HEXAGONS
One interior angle of a regular hexagon has measure
At a vertex angle:

33. TILING WITH REGULAR PENTAGONS?
One interior angle of a regular pentagon has measure
At a vertex angle:

34. THE REGULAR TILINGS OF THE PLANE
There are exactly three regular tilings of the plane:
by equilateral triangles,
by squares, and
by regular hexagons.

35. TILING THE PLANE WITH CONGRUENT POLYGONAL TILES
The plane can be tiled by:
any triangular tile;
any quadrilateral tile, convex or not;
certain pentagonal tiles (for example,
those with two parallel sides);
certain hexagonal tiles (for example,
those with two opposite parallel sides of
the same length).

36. SEMIREGULAR TILINGS OF THE PLANE
An edge-to-edge tiling of the plane with more than one type of regular polygon and with identical vertex figures is called a semiregular tiling.

37. Dutch artist Escher created a large number of artistic tilings.
TILINGS OF ESCHER TYPE
Dutch artist Escher created a large number of artistic tilings.
ESCHER’S BIRDS
ITS GRID OF PARALLELOGRAMS

38. TILINGS OF ESCHER TYPE MODIFYING A REGULAR HEXAGON WITH ROTATIONS
CREATES: