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Title: Reflections
Book
Description: This
workbook illustrates some facts about motions of the plane that preserve
the distances between any pair of points. Such motions are called "isometries"
or "rigid motions." Examples of these motions are "translations" that
simply slide each point by a certain definite amount in a fixed direction.
Other examples are "rotations" about a fixed point.
These examples have the property that they may be implemented by a continuous
sequence of small motions so that from each step to the next, the variation
between the object and its next location is very small. Such motions preserve
"orientation" in the sense that, say applied to a letter "F" they would
map it into something that was recognizably an F, even if perhaps upside-down.
Another example of an isometry, quite different from these, is a "reflection"
across a fixed line. These are interesting. First, you can easily see
that they are not "orientable." An "F" would flip into something that
is not an F at all, and for which we have no typographical character.
Reflections cannot be obtained by any sequence of "small" motions. Now,
it may be surprising to learn that any rigid motion of the plane can be
obtained by a sequence of reflections. That fact is the topic of this
workbook. In the book, readers may construct "objects" (polygons) and
mirrors, may place them as they like, and observe the effect of reflecting
objects through arbitrary sets of mirrors. They also learn inductively
the rotation that results from a pair of reflections.
Author: James White
Suggested Use: Learn experimentally how to realize a translation as the composition of a pair of reflections.
Topics: geometry, reflections, plane rotations, plane translations
Number of Pages: 5
Animation: Yes
Grade
Level: 