Reflection (mathematics)

In mathematics, a reflection is a specific type of mapping within Euclidean space. It is an isometry, which means it is a transformation that preserves distances. A reflection uses a specific set of fixed points that do not move during the process. In two-dimensional space, this set is a line called an axis. In three-dimensional space, this set is a flat surface called a plane.

This axis or plane acts as the mirror for the transformation. The resulting image is a mirror image of the original figure.

To perform a reflection, you follow a precise geometric mechanism. If you are reflecting a single point, you first drop a perpendicular line from that point to the axis or plane. This line must hit the mirror at a right angle. You then extend that line an equal distance on the opposite side of the mirror. The new location is the reflected point.

To reflect an entire geometric figure, you simply repeat this process for every individual point within that shape. This ensures the entire object is flipped accurately.

Mathematicians can also construct reflections using a compass and a straightedge. First, you create a circle centered at your original point. This circle should intersect the axis at two points. Next, you draw two new circles centered at those intersection points. These circles will have a radius equal to the distance between the intersection points and the original point. The two points where these new circles meet will be the reflected points.

This method provides a way to find reflections without needing to measure exact distances manually.

Reflections belong to a special category of mappings called involutions. An involution is a transformation that, when applied twice in succession, returns every point to its original location. This means that applying a reflection twice restores any geometric object to its starting state. While most people use "reflection" to mean a flip across a hyperplane, the term can also describe other involutive isometries. For example, a reflection through a single point is an isometry with only one fixed point. This specific operation is known as a central inversion.

There are several distinct ways to view these transformations through different mathematical lenses. In a vector space, a reflection through the origin is identical to vector negation. In higher dimensions, a reflection can occur across a hyperplane. The mathematical properties of these transformations are quite specific. The matrix used to represent a reflection is orthogonal and has a determinant of -1. Furthermore, the eigenvalues for these matrices include -1 and several instances of 1. These values describe how the space is stretched or flipped during the mapping.

Reflections are the fundamental building blocks for other complex movements in space. According to the Cartan–Dieudonné theorem, reflections generate the orthogonal group. This theorem explains that every rotation is the result of an even number of reflections. Conversely, every improper rotation is the result of an odd number of reflections.

This connection shows that complex turns and flips are actually just combinations of simple mirror images. Because of this, groups generated by reflections in affine hyperplanes are known as reflection groups. Some finite versions of these are called Coxeter groups.

We can see the practical effects of reflections by looking at how they change familiar symbols. For instance, a vertical reflection of the lowercase Latin letter "p" results in the letter "q". If you apply a horizontal reflection to that same letter "p", it becomes a "b". A central inversion through a point would turn the letter "p" into a "d".

These changes demonstrate how symmetry and orientation work in a coordinate system. By studying these flips, mathematicians can understand the structure of the Euclidean group, which consists of all isometries in Euclidean space.