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== 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. 

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== 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.

<span style="font-size:larger;text-decoration:underline">The Point Groups</span>

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. 

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<span style="font-size:larger;text-decoration:underline">The 7 Line Groups</span>

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"). 

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<span style="font-size:larger;text-decoration:underline">The 17 Plane Groups</span>

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:

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And here they are as generalized abstractions:


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