Transformations, Symmetries, and Tilings

11.1 Rigid Motions and Similarity Transformations11.2 Patterns and Symmetries11.3 Tilings and Escher-like Designs

11.1

Rigid Motions and Similarity Transformations

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TRANSFORMATION OF THE PLANE

'.P

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 thetransformation. Point P is called the preimage of

', then 'P P

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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.

That is, ' ' for all points and .PQ P Q P Q

and P Q

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THREE BASIC RIGID MOTIONS

• Translation, or slide

• Rotation, or turn

• Reflection, or flip

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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: 1. the direction of the slide as the direction of the arrow and2. the distance moved as the length of the arrow.

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A TRANSLATION

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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.

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CENTER OF ROTATION IS POINT OANGLE OF ROTATION IS x

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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.

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REFLECTIONS

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A REFLECTION

Note that B is located so that m is the perpendicular bisector of

.BB

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DEFINITION:CONGRUENT FIGURES

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

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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.

P OP OP

OP k OP

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A SIZE TRANSFORMATION

OP k OP

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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.

~F G

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SIMILAR FIGURES

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

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11.2

Patterns and Symmetries

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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.

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REFLECTION SYMMETRY

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

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Example 11.8 Identifying Lines of SymmetryIdentify all lines of symmetry for each letter.

M N O XM = 1 verticalN = noneO = 2; one vertical and one horizontalX = 4: one vertical, one horizontal, and the two lines given in the figure itself.

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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.

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POINT SYMMETRY

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

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Example 11.10 Identifying Point SymmetryWhat 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.

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PERIODIC PATTERNS

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

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PERIODIC PATTERNS

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

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11.3

Tilings and Escher-like Designs

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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.

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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.

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TILING WITH EQUILATERAL TRIANGLES

6 60 360

One interior angle of an equilateral triangle has measure 60.

At a vertex angle:

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TILING WITH SQUARES

4 90 360

One interior angle of a square has measure 90.

At a vertex angle:

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TILING WITH REGULAR HEXAGONS

3 120 360

One interior angle of a regular hexagon has measure

At a vertex angle:

(6 2) 180 720 120 .6 6

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TILING WITH REGULAR PENTAGONS?

(5 2) 180108

5

One interior angle of a regular pentagon has measure

At a vertex angle:

3 108 324

leaves a gap.

4 108 432

overlaps.

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THE REGULAR TILINGS OF THE PLANE

There are exactly three regular tilings of the plane:

• by equilateral triangles,

• by squares, and

• by regular hexagons.

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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).

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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.

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TILINGS OF ESCHER TYPE

Dutch artist Escher created a large number of artistic tilings.

ESCHER’S BIRDS ITS GRID OF PARALLELOGRAMS

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TILINGS OF ESCHER TYPEMODIFYING A REGULAR HEXAGON WITH ROTATIONS

CREATES:

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