Transformations & Symmetry Essay, Research Paper

What is transformation? Transformation is a one-to-one function from one plane on to

another plane or to a different area on the same plane. A transformation describes a change in

appearance of points in a plane. It is a transfer from the pre-image to the image. There are

many types of transformations that I will be describing.

The first type of transformation is known as a reflection. A reflection maps each point

from one plane and creates it on another plane in the same manner and order. One of the main

characteristics of reflection is reverse orientation. This means that whatever order the points

were in, they transformed to be the opposite. This concept is the same as when you look into a

mirror, all the points are reversed.

Another type of transformation is known as translation. A translation is a transformation

formed by the composition of two reflections in which the lines of the reflection are parallel.

According to my understanding of this concept, in order to have the lines parallel, the figures

must be placed side by side. In this type of transformation the orientation of the figure is

changed but then changed back. The first reflection reverses the orientation, then the second

reflection reverses it back to the way it first was. When you have more than one transformation

of one figure you are then, performing a composition of transformations.

The third type of translation is called a rotation. A rotation is a transformation formed

by the composition of two reflections in which the lines of reflection intersect. This is

accomplished by using two reflections or a composition. The concept of this transformation is

that it is reflected at an angle, therefore causing the perpendicular lines to intersect at a single

point, sort of like a glass prism.

Another type of transformation is known as a dilation. A dilation is known as a

transformation that expands or contracts the points of the plane in relation to a fixed point.

This expansion or contraction is depicted by a ratio or also known as a scale factor. The

change in size of the figure depends upon the scale factor. All the angles in the figure keep the

same measure, therefore the figures should have the same shape but no longer the same size.

Figures that are the same shape but not the same size are known as similar figures.

One more type of transformation is known as an isometric transformation. An isometric

transformation is one that preserves distance. Saying that it preserves distance means that the

figure is always exactly the same size as the pre-image. Examples of isometry are reflection,

translation and rotation. To keep an image the same throughout some properties must be

preserved such as distance, collinearity of points, betweenness of points, angle measure, and

parallelism. These must all be considered when working with isometry. A dilation is not

isometric for a number of reasons. First of all, dilations do not preserve distance and therefore

cannot be isometries. The only reason that dilations would be considered to be isometric would

be because they preserve shape, but they do not preserve size either. A dilation can only

produce similar figures while a transformation that preserves size and shape can produce an

isometry.

There is a certain form of a reflection that is known as symmetry. A figure has line

symmetry when each half of the figure is the image of the other half under some reflection in a

line. This line is called the axis of symmetry. An example of line symmetry is when you place a

half of a seashell on a mirror, the shell is mirrored so that it coincides with the actual shell. The

mirror, in this example would be the axis of symmetry.

Symmetry can also be achieved by a concept known as rotational symmetry. A figure

has rotational symmetry when the image of the figure coincides with the figure after a rotation.

The amount of rotation must be less than 360 degrees. An example of this is a starfish. You

can turn it and it will still have the same basic starfish shape, therefore depicting rotational

symmetry.

One last type of symmetry is called point symmetry. Point symmetry is actually

rotational symmetry but only of 180 degrees. This means that an object or figure can be rotated

180 degrees and appear the same. An example of this is a football.

This chapter had alot of information in it that was hard to understand, but with

concentration and determination, it became easier. There are many laws that are important to

these concepts and they must all be considered to be sure that you have reached teh correct

answer. This report was a learning experience and helped me to understand the concepts of

transformations and symmetry.