Symmetry (geometry)

In geometry, symmetry describes a property where an object remains unchanged after a specific movement. This movement is called a transformation. When an object undergoes a transformation but remains indistinguishable from its original state, it possesses an immunity to change.

Mathematicians study these transformations to understand the underlying structure of shapes. The set of all transformations that leave an object symmetric forms a mathematical structure known as a symmetry group. This group includes the original transformation and its inverse, which undoes the movement.

Most geometric symmetries belong to the Euclidean group of isometries. Isometries are transformations that preserve the distance between points in space. These are commonly applied in two-dimensional plane geometry or three-dimensional solid geometry. The basic types of isometries include reflections, rotations, and translations. A combination of these operations can also result in more complex movements. According to the Cartan–Dieudonné theorem, any orthogonal transformation in n-dimensional space can be represented by combining at most n reflections.

Reflectional symmetry, often called mirror or bilateral symmetry, occurs when an object is flipped across a specific boundary. In one dimension, this is a point of symmetry. In two dimensions, it is an axis of symmetry, which is a line. In three dimensions, it is a plane of symmetry.

For a two-dimensional figure, an axis of symmetry is a line where any two points at equal distances from the axis are identical. A square possesses four such axes. A circle is unique because it has infinitely many axes of symmetry passing through its center.

Rotational symmetry occurs when an object looks the same after being turned around a central point. These rotations are direct isometries because they preserve the orientation of the object.

For example, a point reflection in a two-dimensional plane is equivalent to a 180-degree rotation.

In three dimensions, a point reflection is more complex. It changes the orientation of the space, such as turning a right-handed coordinate system into a left-handed one. Because of this, physicists often refer to this as P-symmetry, or parity symmetry.

Some patterns use more complex combinations of movements, such as glide reflection symmetry. In two dimensions, a glide reflection combines a reflection in a line with a translation along that same line.

A common example of this can be seen in the pattern of human footprints walking in a straight line. Another complex type is rotoreflection symmetry, which is found in three-dimensional objects.

This involves a rotation about an axis combined with a reflection in a plane perpendicular to that axis. A pentagonal antiprism is a geometric example of this property.

Helical symmetry is another sophisticated type of symmetry found in three-dimensional geometry. It involves a screw axis, which is a combination of rotation and translation along that rotation axis.

This creates a coiling effect seen in objects like springs or drill bits. The coiling angle is determined by the relationship between the speed of rotation and the speed of translation.

Some helical structures are regular, while others, like the Boerdijk–Coxeter helix, are nonperiodic.

These structures can be studied by looking at how they repeat or change as they move through space.

Symmetry is deeply connected to the fundamental laws of the universe. In physics, the concept of rotational invariance means that physical laws do not change based on the direction in space. This principle is linked to Noether's theorem. This theorem proves that the rotational symmetry of a physical system is equivalent to the conservation of angular momentum. Whether looking at the simple bilateral symmetry of a butterfly or the complex symmetries of particle physics, these mathematical rules help define the order of our world.