A transformation that keeps a figure’s same shape and size, but moves it to a new location, is called isometry. There are several common types of these transformations. Dilatation is the only transformation that does not create equal figures. It means to take a shape and make it larger or smaller, but keep the same proportions. In terms of a circle, a dilatation creates another circle with the same center, also known as a concentric circle. Reflection is similar to what we called a “flip” in elementary school mathematics—like the flip side of an object. One easy way to see this is by noting one’s reflection in a mirror—the “figure” is on one side of a line and the mirror image on the other. Reflection twice about two parallel lines is synonymous with translation; reflection twice about two intersecting lines is called rotation.

Another type of transformation is rotation. This is simple to understand: in elementary school mathematics it is sometimes called a “turn” or “spin.” In this case, one point on a plane remains unchanged while keeping all the distances between the other points the same. Finally, a translation (or glide transformation) is similar to what is called a “slide” in elementary school mathematics. All the points in the plane move in the same direction and the same distance; or, the figure slides in a single direction. Translation is also considered to be reflection twice across two parallel lines.

René Descartes, who is more often remembered for developing the concept of Cartesian coordinates, also originated the idea of using letters when writing equations that include unknown values. Library of Congress.