REFLECTIONS & SYMMETRY

Many of us have seen either in real life or a picture of mountains, trees, buildings, etc. reflected in a body of water.  You have also seen your own reflection in a mirror.  Plants and animals are illustrations of reflections.  In fact, the world around us is permeatied with reflections of some sort.  Floor tiles, art work, chemical compounds are a few examples which possess some sort of reflection.

The following figures illustrate some sort of reflection.

WHY ARE REFLECTIONS IMPORTANT

Many things can be done to figures to transform them into a new figure.  In each of the above figures there is an image and a preimage.  In each case, the figure has been transformed.  The operation with was applied to the figure is called a transformation.  A transformation is sometimes called a mapping.  A transformation maps a preimage onto an image.  Transformations preserve various properties of the figure (between the image and the preimage).

The importance of the image and preimage properties is one reason for studying reflections.

A second reason is that reflections are used to produce other transformations.

The third reason for the importance of reflections is that  there is a close relation to a property we call symmetry. 

TYPES OF REFLECTIONS

We will explore three types of reflections.

            1.         Line reflections.  Reflections about one or more lines.

                        a.         Reflections about parallel lines.  TRANSLATION

                        b.        Reflections about intersecting lines.  ROTATION

            2.         Point reflection.  Reflection about a point.

            3.         Glide Reflection.   Reflection about a line and then a translation.

PROPERTIES OF REFLECTIONS

            Reflections preserve

                        1.         Collinearity

                        2.         Betweenness

                        3.         Distance --length or linear measure

                        4.         Angle measure

            Reflections do not preserve orientation.  Reflections reverse orientation.  This reverse orientation is of importance in the study of chemical compounds.

Notice the opposite orientation in the figures below.

                                                                       

The turtles in first illustration near the beginning of this module represents reverse orientation.  One turtle faces left and the reflection turtle faces to the right.

SYMMETRY

The word symmetry means  same measure.  Reflections about a line preserve length or same measure.  Thus the line of reflection is called a symmetry line or line of symmetry.  There can be many lines of symmetry.  In figure 1, the reflection line could be placed anywhere.  Thus, there could be infinitely many reflection images.

                                                                       

                                                                           

If the symmetry line is placed as shown in below, the image and the preimage are exactly the same set of points.  The reflection image coincides with the preimage.

                                                                       

The pentagon above is said to be reflection-symmetric.  The figure and its image coincide.

Notice that, in the first case, the orientation of the pentagon is reversed.  The vertices labeled A, B, and C are in a counter-clockwise orientation.  The image vertices A', B' and C' are in a clockwise orientation.  This condition could be called right-handed or left-handed.  The reflection of a hand would produce an image of another hand but of opposite handedness.  A term for this is CHIRALITY.  Your left and right hand have opposite chirality.

The butterfly is reflection symmetric. The reflection operation leaves the appearance of the figure unchanged.

                                         

 

In some cases, as below, the appearance of the figure is visually changed.

                        

                                             

TRANSLATION

The reflection is a simple transformation.  More complex transformations can be produced from successive application of simpler ones.  The figure below illustrates the result of a reflection about two parallel lines. 

 

The reflection of a reflection about a second line parallel to the first line of reflection produces a figure that is oriented the same as the preimage.  This new figure is in all ways the same as the first except  it is translated to a new position.  This is called a translation. 

The mathematical description would be stated as:  The composite of two reflections over parallel lines is a translation.

ROTATION

The reflections of a figure over two intersecting lines produces a transformation called a rotation.  The rotation is about the point of intersection of the lines. 

  

The angle of rotation is the angle formed by the intersection of the lines and a point of the figure and its image.  The direction of the rotation in this case is clockwise  (a negative angle rotation).

A rotation of 180 degrees produces a special transformation.  The figure is essentually reflected about or through a point. 

       

Every line segment joining a point of the figure and its image passes through the point and is bisected by the point.  This point is called the point of symmetry.  The transformation is a half-turn rotation and is generally referred to as POINT SYMMETRY.

GLIDE REFLECTION

If a figure is reflected about a line and the image is translated in a direction parallel to the line, the result is a glide reflection.

 

MULTIPLE LINES OF REFLECTION OR MULTIPLE LINES OF SYMMETRY

 

 

The concepts of symmetry and right- or left-handedness are of utmost importance to understanding the basic functioning of life itself.