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Excursions in Modern Mathematics Sixth Edition. Peter Tannenbaum. Chapter 11 Symmetry. Mirror, Mirror, Off the Wall. Symmetry Outline/learning Objectives. To describe the basic rigid motions of the plane and state their properties.

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chapter 11 symmetry

Chapter 11Symmetry

Mirror, Mirror, Off the Wall

symmetry outline learning objectives
SymmetryOutline/learning Objectives
  • To describe the basic rigid motions of the plane and state their properties.
  • To classify the possible symmetries of any finite two-dimensional shape or object.
  • To classify the possible symmetries of a border pattern.
symmetry

Symmetry

11.1 Rigid Motions

symmetry symmetries of a triangle
Symmetry- Symmetries of a Triangle

In terms of symmetry, how do these triangles differ? Which one is the most symmetric? Least symmetric?

symmetry6
Symmetry

Let’s say, for starters, that symmetry is a property of an object that looks the same to an observer standing at different vantage points. Thus, we can think of symmetry as a property related to an object that can be moved in such a way that when all the moving is done, the object looks exactly as it did before.

symmetry rigid motion
Symmetry- Rigid Motion

The act of taking an object and moving it from some starting position to some ending position without altering its shape or size is called a rigid motion such as illustrated in (a).

symmetry rigid motion8
Symmetry- Rigid Motion

If the shape is altered, the motion is not rigid such as illustrated in (b).

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Symmetry
  • Equivalent rigid motions – two rigid motions that move an object from a starting position A to an ending position B.
  • Basic rigid motions of the plane – every rigid motion is equivalent to a reflection, a rotation, a translation, or a glide reflection.
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Symmetry
  • Image – denoted by Pand informally means Mmoves P to P.
  • Fixed point – It may happen that a point P is moved back to itself under M , in which case we call P a fixed point of the rigid motion M .
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Symmetry

11.2 Reflections

symmetry reflection
Symmetry- Reflection

A reflection in the plane is a rigid motion that moves an object into a new position that is a mirror image of the starting position. In two dimensions, the “mirror” is a line called the axis of reflection.

symmetry reflections of a triangle
Symmetry- Reflections of a Triangle

The above figure shows three cases of reflection of a triangle ABC. In all cases the reflected triangle A´B´Cis shown in red. In (a) the axis of reflection l does not intersect the triangle ABC.

symmetry reflections of a triangle14
Symmetry- Reflections of a Triangle

In (b) the axis of reflection l cuts through the triangle ABC – here the points where l intersects the triangle are fixed points of the triangle. In (c) the reflected triangle A´B´C falls

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Symmetry- Reflections of a Triangle

on top of the original triangle ABC. The vertex B is a fixed point of the triangle, but the vertices A and C swap positions under the reflection.

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Symmetry

Useful facts about reflection

  • A reflection is completely determined by its axis l.
  • A reflection is completely determined by a single point-image pair P and P(as long as P is not a fixed point).
  • A reflection is an improper rigid motion.
  • If the same reflection is applied twice, every point ends up exactly where it started.
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Symmetry

11.3 Rotations

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Symmetry

A rotation is defined by giving the rotocenter and the angle of rotation The figure on the right illustrates how a clockwise rotation with rotocenter (the point O that acts as the center of the rotation), and the angle of rotation  (actually the measure of an angle indicating the amount of rotation) moves a point P to the point P.

symmetry rotations of a triangle
Symmetry- Rotations of a Triangle

The above illustrates three cases of rotation of a triangle ABC. In all cases the reflected triangle A´B´C is shown in red. In (a) the rotocenter O lies outside the triangle ABC.

symmetry rotations of a triangle20
Symmetry- Rotations of a Triangle

In (b) the rotocenter O is at the center of the triangle ABC. In (c) the 360°rotation moves every point back to its original position – from the rigid motion point of view it’s as if the triangle had not moved.

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Symmetry

Useful facts about rotation

  • A 360° rotation is equivalent to the identity motion.
  • A rotation is a proper rigid motion.
  • A rotation is completely determined by two point-image pairs, P, P and Q, Q .
symmetry22
Symmetry

Useful facts about rotation (continued)

  • A rotation that is not the identity motion has only one fixed point – the rotocenter O.
  • Combining a clockwise rotation with rotocenter O and angle  with a counterclockwise rotation with the same rotocenter and angle gives the identity rigid motion.
symmetry23

Symmetry

11.4 Translations

symmetry translations of a triangle
Symmetry- Translations of a Triangle

This figure illustrates the translation of a triangle ABC. There are three “different” arrows shown in the figure but they all have the same length and direction, so they describe the same vector of translation v.

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Symmetry

Useful facts about translation

  • A translation is completely determined by a single point-image pair P and P .
  • A translation has no fixed points.
  • A translation is a proper rigid motion.
  • Combining a translation with vector v and a translation with vector -v gives the identity rigid motion.
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Symmetry

11.5 Glide Reflections

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Symmetry

A glide reflection is a compound rigid motion obtained by combining a translation (the glide) with a reflection with axis parallel to the direction of translation. Thus, a glide reflection is described by two things: the vector of the translation v and the axis of the reflection l, and these two must be parallel.

symmetry glide reflection of a triangle
Symmetry- Glide Reflection of a Triangle

This figure illustrates the result of applying the glide reflection with vector v and axis l to the triangle ABC. In (a) the translation is applied first, moving triangle ABC to the intermediate position A*B*C*.

symmetry glide reflection of a triangle29
Symmetry- Glide Reflection of a Triangle

The reflection is then applied to A*B*C*, giving the final position A´B´C. If we apply the reflection first, then the triangle ABC gets moved to the intermediate position A*B*C* (b) and then translated to the final position A´B´C .

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Symmetry

Useful facts about glide reflection

  • A glide reflection is completely determined by two point-image pairs, P, P and Q, Q .
  • A glide reflection has no fixed points.
  • A glide reflection is an improper rigid motion.
  • Combining a glide reflection with vector v and axis l with a glide reflection vector -v and axis l gives the identity rigid motion.
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Symmetry

11.6 Symmetry as a Rigid Motion

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Symmetry

A symmetry of an object (or shape) is a rigid motion that moves the object back onto itself.

symmetry33
Symmetry

For two-dimensional objects in the plane, there are only four types of rigid motions and symmetry:

  • Reflection symmetry
  • Rotation symmetry
  • Translation symmetry
  • Glide reflection symmetry
symmetry the symmetries of a square
Symmetry- The Symmetries of a Square

What are the possible rigid motions that move the square in (a) back onto itself?

First, there are reflection symmetries.

symmetry the symmetries of a square35
Symmetry- The Symmetries of a Square

For example, if we use the line l1 in (b) as the axis of reflection, the square falls back into itself with points A and B interchanging places and C and D interchanging places.

symmetry the symmetries of a square36
Symmetry- The Symmetries of a Square

Are there any other symmetries? Yes– the square has rotation symmetries as well as cited in (c).

symmetry the symmetries of a square37
Symmetry- The Symmetries of a Square

All in all, we have found eight symmetries for the square in (a). Four of them are reflections, the other four are rotations.

symmetry the symmetries type z 4
Symmetry- The Symmetries Type Z4

A propeller with symmetry type Z4 (four rotation symmetries, no reflection symmetries).

symmetry the symmetries type z 2
Symmetry- The Symmetries Type Z2

A propeller with symmetry type Z2 (two rotation symmetries, no reflection symmetries).

symmetry the symmetries type d 1
Symmetry- The Symmetries Type D1

Objects with symmetry type D1 (one rotation symmetry plus the identity symmetries).

symmetry the symmetries type z 1
Symmetry- The Symmetries Type Z1

Objects with symmetry type Z1 (only symmetry is the identity symmetry).

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Symmetry

11.7 Patterns

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Symmetry

We define a pattern as an infinite “shape” consisting of an infinitely repeating basic design called the motif of the pattern.

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Symmetry

Border patterns are linear patterns where a basic motif repeats itself indefinitely in a linear direction, as in a frieze, a ribbon, or the border design of a pot or basket.

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Symmetry

Kinds of symmetries in border patterns:

  • Translations
  • Reflections
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Symmetry

Kinds of symmetries in border patterns (continued):

  • Rotations
  • Glide reflections.
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Symmetry

Wallpaper patterns are patterns that fill the plane by repeating a motif indefinitely along several (two or more) nonparallel directions.

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Symmetry

Kinds of symmetries in wallpaper patterns:

  • Translations
  • Reflections
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Symmetry

Kinds of symmetries in wallpaper patterns:

  • Rotations
  • Glide reflections.
symmetry conclusion
Symmetry Conclusion
  • Basic Rigid Motions
  • Symmetry
  • Patterns