The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

Many important ideas in geometry can be understood through the use of transformations. As a simple example, two figures are said to be congruent if one can be carried rigidly into an exact duplicate of the other. Conceptually, we imagine sliding the first figure around until it lands directly on top of the second. This is shown in the animated graphic below, where a red triangle is manipulated to coincide with a blue triangle.

More formally, the manipulation of a figure in this way is described in terms of a function whose domain and range are subsets of the plane. If `f` is such a function, and if (`x`, `y`) is a point in the domain, then `f` (`x`, `y`) is another point in the plane, say (`u`, `v`). By applying `f` to each point of a figure, we obtain a new figure, referred to as the image under `f` of the original figure, or more briefly, just the image. In the graphic above, for each point (`x`, `y`) in the original red triangle, `f` (`x`, `y`) is the corresponding position in the blue triangle. In this way, the function `f` transforms the red triangle into the blue triangle. Note that the animation is for visualization purposes only, it is not part of the function `f`. Rather, `f` tells us for each point of the starting figure, how to compute the final location in the ending figure. Although it is often helpful to imagine the starting figure evolving into the final figure as in the animation, `f` only tells us how to compute each ending point. It says nothing about the path a point follows in going from its starting position to its final position.

Of particular interest here are geometric transformations which consist of rotations about the origin, translations, and imposition of scale factors. For example, let us consider a rotation about the origin through an angle of π/2, in the counterclockwise direction. Any point (`x`, `y`) in the plane is at a distance from the origin, and makes an angle `θ` with the positive `x` axis, as illustrated in Figure 1 below.

To rotate this point about the origin, we hold the length `r` fixed, but add π/2 to the angle `θ`. A typical point (`x`, `y`) and its rotated image (`u`, `v`) are shown in Figure 2. This figure shows how the rotation effects a single point. But we are usually interested in applying a transformation to every point in some geometrical object. This is illustrated in Figure 3. Each point of the red triangle has been rotated about the origin by π/2 to create the blue triangle. The yellow and blue grids are highlighted to make it easier to visualize that this is indeed a rotation through π/2.

Figure 4 shows an example of a *translation*. Every point of the red triangle is shifted left 5 units and down 1 unit to create the blue triangle. In general, a translation moves every point (`x`, `y`) in the domain a fixed distance in a fixed direction to produce the image point (`u`, `v`).

The third operation corresponds to multiplication by a fixed positive constant. This is easily expressed algebraically: for some positive constant `c`, `f `(`x`, `y`) = (`c``x`, `c``y`). Pictorially, this amounts to either expanding a figure radially outward (if `c` > 1) or contracting it radially inward (if `c` is less than 1). For example, with `c` = 2, a figure's linear dimensions are doubled in size and each point is moved twice as far from the origin. With `c` = 1/2, a figure's dimensions are reduced by half and each point is moved half the distance to the origin. There are illustrated in Figure 5 and Figure 6, in which the original figure is a red polygon, and the transformed image is blue.

Notice that rotations and translations always transform a figure to a congruent image, but a scale factor transforms a figure to a *similar* figure. Here, the word *similar* has a specific mathematical meaning that is more particular than the usual English meaning. Similar figures have exactly the same shape and proportions, differing only in size.

In all of the examples presented above, there are simple mathematical formulas for the transformations involved. In each case, we represent the starting point by (`x`, `y`), and the image is (`u`, `v`) = `f`(`x`, `y`). For the last examples, with a scale factor of 2 we have (`u`, `v`) = (2`x`, 2`y`), while for a scale factor of 1/2 we have (`u`, `v`) = (`x`/2, `y`/2). More generally, for a scale factor of `r` the transformation is given by (`u`, `v`) =` `(`r``x`, `r``y`). For a translation, the transformation has the form (`u`, `v`) = (`x` + `a`, `y` + `b`), where `a` and `b` are fixed constants. In the example of Figure 4, `a` = −5 and `b` = −1, so the transformation is given by (`u`, `v`) = (`x` − 5, `y` − 1). For the rotation example, the transformation is defined by (`u`, `v`) = (−`y`, `x`).