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Classical Geometry: Euclidean, Transformational, Inversive, and Projective by G. W. Tokarsky, A. C. F. Liu, J. E. Lewis, I. E. Leonard

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CHAPTER 8

THE ALGEBRA OF ISOMETRIES

8.1 Basic Algebraic Properties

Consider a large iron grate (G) in the shape of a right triangle, as shown in the figure below, The grate has to move from its current position to cover the hole (H). It must fit exactly within the dotted lines.

The grate is far too heavy to be lifted by hand, so a machine has been rigged that can lift the grate and flip it around any desired axis, thereby placing it on the other side of the turning axis. The action of the machine is shown in the figure below. If necessary, the axis can pass through the grate.

Find a sequence of flips that will move the grate to cover the hole. What is the minimum number of flips it will take to cover the hole?

The machine is a “reflecting” device—after the machine does its work, the new position of the grate is the reflection through the axis of the original position of the grate. The solution to the problem is that the grate can be moved in a step-by-step manner to cover the hole in three flips, and the minimum number of flips necessary is three. A sequence of flips is depicted in the figure below.

The first step uses a reflection through the line l, which is the right bisector ...

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