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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Chapter 4 Exploring Transformations

Are you aware that animated movies and computer graphics use a lot of mathematics and geometry? The images are moved by moving key points to new locations on the screen. The process of moving each point in the Euclidean plane to a new, possibly the same, point is called a transformation. Every point is assigned a single landing position; no point is sent to more than one position by a single transformation.
Since it is impossible to show the action of a transformation on every point in the plane, we instead view how a transformation affects a given figure. The images below show two examples of the effect of transformations on the pentomino \(ABCDEFGH\text{.}\)
Pentomino ABCDEFGH is in black. A copy labeled A’B’C’D’E’F’G’H’ rotated 90 degrees counterclockwise about a point outside the pentomino is shown in blue.
Figure 4.0.1. Rotation of pentomino \(ABCDEFGH\)
Pentomino ABCDEFGH is shown in black. Pentomino A"B"C"D"E"F"G"H" is a larger copy of the original shown in blue. It has been reflected across a line and enlarged by a factor of 1.5.
Figure 4.0.2. Reflected dilation of pentomino \(ABCDEFGH\)
The first image shows a transformation which moves objects by a rotation but does not change any lengths or angle measures. Transformations like this one that preserve distance and shape are called isometries or rigid transformations. Even though shape is preserved, the transformation in the second image is not an isometry because it does not preserve distance; the lengths of the sides change.
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