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

11.1 Rigid Motions and Similarity Transformations

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

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.

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

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.

A TRANSLATION

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.

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

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.

REFLECTIONS

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

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

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.

A SIZE TRANSFORMATION

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.

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

11.2 Patterns and Symmetries

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.

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

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.

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.

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

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

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

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

11.3 Tilings and Escher-like Designs

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.

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.

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

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

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

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

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

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).

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.

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

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