Symmetry & regular patterns
From GenerativeArt
Contents |
What is Symmetry?
The common notion of symmetry is usually descriptive. It refers to an image or object where one part is the mirror image of an opposing part.
Mathematicians use a more precise definition. For them symmetry refers to invariance under a specified group of transformations or operations. So symmetry is always (1) relative to a set of operations, (2) in the context of a given dimension, and (3) refers to the subset of forms that don't change when those operations are applied.
Symmetry Operations
All regular patterns can be built up from 4 basic symmetry operations. When these operations are applied with reference to a point, line, or plane, one can generate patterns which exhibit radial symmetry, linear symmetry, and planar symmetry.
These operations are:
- Translation - units are shifted by a constant vector.
- Rotation - units are rotated by a constant angle which divides 360 degrees with a 0 remainder.
- Reflection - units are duplicated as mirror images.
- Glide Reflection - units are reflected and then translated.
In illustrating symmetry relationships these operations are often depicted as follows:
Translation - shown simply by showing multiple instances of a given mark. Rotation - shown using a solid dot between the rotated instances of a given mark. Reflection - shown using a solid line to designate the "mirror" between the reflected marks. Glide Reflection - shown using a dashed line designating the "mirror" and in parallel to the path of translation.
Viewed this way symmetry is not a matter of passive description but rather active creation. Symmetry operations are essentially algorithms for creating patterns using a single object or unit as a starting point.
In other words, symmetry operations are generative systems that can be used to create generative art.
Symmetry Groups
Application of the symmetry operations to a point, line, or plane create sets of distinct symmetry groups. The term "group" has a specific mathematical meaning, but for the purposes of generative design something close to the everyday meaning is sufficient. A symmetry group includes all the possible designs that can be executed with the same set of symmetry operations.
The Point Groups
There are an infinite number of ways to apply symmetry relative to a point. 2, 3, 4, ... marks can be rotated around a point, with or without a reflection operation. A circle exhibits infinite symmetry relative to a point.
The codes shown with each illustration are another way of describing the operations that have been applied. Here a number refers to the number of arcs the 360 degrees of a full rotation have been divided into. A single "m" denotes a reflection between the units. A double "mm" denotes a reflection between the units and then a second reflection applied to each resulting unit.
The 7 Line Groups
There are only 7 distinct symmetry groups relative to a line. Such a pattern is sometimes called a frieze (say "freeze") as in the architectural formation. The operation codes shown correspond to translation ("t"), reflection ("m"), a 180 degree rotation ("2"), and glide reflection ("tg").
The 17 Plane Groups
There are only 17 distinct symmetry groups relative to a plane. These are the so called "wallpaper patterns". Here they are by way of examples:
And here they are as generalized abstractions:
