Reflection symmetry is the similarity of forms about an axis or line of reflection. These forms are similar but flipped, as seen in the reflection of a mirror. Mirror symmetry and bilateral symmetry are terms synonymous with reflection symmetry. Some mathematicians describe reflection symmetry as another form of rotation symmetry, because the line of reflection is an axis of three-dimensional rotation to flip the form. An isosceles triangle shows reflection symmetry. In addition, people have reflection symmetry with two legs, two arms, two eyes, and two nostrils on either side of the nose.
Objects with rotation symmetry are similar about a point. An object rotates around the point, which describes an axis normal to the two-dimensional plane of rotation, creating similar forms. Rotation symmetry is also called cyclical symmetry and pinwheel symmetry. Regular polygons have rotation symmetry. Furthermore, bicycle spokes and flower petals are examples of rotation symmetry.
Multiple, similar objects in the same orientation but in different positions in a plane or volume have translation symmetry. Translation symmetry is the movement or copying of a form from one position to another. Some mathematicians describe translation as the rotation of an object about an axis at an infinite distance; so some mathematicians describe all forms of symmetry only through rotation. Circles, with equal diameters, on a plane shows translation symmetry. Also, ants walking in a line are examples of translation symmetry.
The combination of reflection and translation symmetries creates glide reflection symmetry. Alternating bilaterally symmetrical elements arranged along a path shows glide reflection symmetry. A zig-zag line has glide reflection symmetry. Also, footprints in the sand and zippers are examples of glide reflection symmetry.
The combination of rotation and translation symmetries creates helix symmetry. An extruded form that rotates along a path shows helix symmetry. A spiral or spring has helix symmetry. In addition, corkscrews, drill bits, and candy canes are examples of helix symmetry.
Multiple forms that are similar but have different sizes have scalar symmetry. Concentric circles have scalar symmetry. Also, the small, medium, and large sizes of an article of clothing have scalar symmetry.