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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Mirror, Mirror, Off the Wall
11.1 Rigid Motions
In terms of symmetry, how do these triangles differ? Which one is the most symmetric? Least symmetric?
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.
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).
If the shape is altered, the motion is not rigid such as illustrated in (b).
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.
The above figure shows three cases of reflection of a triangle ABC. In all cases the reflected triangle A´B´Cis shown in red. In (a) the axis of reflection l does not intersect the triangle ABC.
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
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.
Useful facts about reflection
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.
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.
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.
Useful facts about rotation
Useful facts about rotation (continued)
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.
Useful facts about translation
11.5 Glide Reflections
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.
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*.
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 .
Useful facts about glide reflection
11.6 Symmetry as a Rigid Motion
A symmetry of an object (or shape) is a rigid motion that moves the object back onto itself.
For two-dimensional objects in the plane, there are only four types of rigid motions and symmetry:
What are the possible rigid motions that move the square in (a) back onto itself?
First, there are reflection symmetries.
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.
Are there any other symmetries? Yes– the square has rotation symmetries as well as cited in (c).
All in all, we have found eight symmetries for the square in (a). Four of them are reflections, the other four are rotations.
A propeller with symmetry type Z4 (four rotation symmetries, no reflection symmetries).
A propeller with symmetry type Z2 (two rotation symmetries, no reflection symmetries).
Objects with symmetry type D1 (one rotation symmetry plus the identity symmetries).
Objects with symmetry type Z1 (only symmetry is the identity symmetry).
We define a pattern as an infinite “shape” consisting of an infinitely repeating basic design called the motif of the pattern.
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.
Kinds of symmetries in border patterns:
Kinds of symmetries in border patterns (continued):
Wallpaper patterns are patterns that fill the plane by repeating a motif indefinitely along several (two or more) nonparallel directions.
Kinds of symmetries in wallpaper patterns:
Kinds of symmetries in wallpaper patterns: