Patterns In Nature Contain Symmetry
"Tyger! Tyger! burning bright In the forests of the night, What immortal hand or eye, Dare frame thy fearful symmetry?"
-- William Blake. (1794)

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Because we humans are pattern seeking organisms, symmetry has always fascinated us. We view symmetrical objects such as the regular repeating patterns of ancient pottery, weavings, and tilings as pleasing, proportioned, balanced, and harmonious. In addition to offering aesthetic pleasure, symmetry has a role in pattern formation. Patterns in nature possess some form of symmetry in space or in time.

Since the description or classification of many patterns in nature is difficult,  symmetry is a handy "device" for classifying and organizing information about patterns by classifying the motion or deformation of both pattern structure and pattern processes. Symmetry transformations help explain or classify the preservation (the invariance) of a natural pattern.

Symmetry, according to the American Heritage Dictionary is an "Exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis". If it is impossible to to distinguish between the original and final positions of a moved object, we say it has symmetry. Said another way, an object possesses symmetry if it retains its form or shape after some form of transformation or change.

The star fish is an example of radial symmetry. Rotating this animal one-fifth of a turn doesn't change the object, its pattern, or its appearance even though the positions of the arms have changed. In other words, the pattern remains the same even though the animal is rotated. This pattern is said to be "invariant" under rotation around its center.

The Common Buckeye butterfly is an example of mirror (sometimes called "bilateral") symmetry. It is symmetrical along its longitudinal axis (head to tail). Its left side is a mirror image of its right side. This pattern is said to be " invariant" under mirror reflections along this axis. Nature has preserved the symmetrical pattern of this creature. The shape and pattern of the left side is the same as a mirror image of its right side. The human body contains bilateral symmetry.

Basic maneuvers such as rotating the star fish or looking at mirror images of a butterfly are called symmetry operations. Symmetry operations provide a basis for classifying objects or patterns in terms of symmetry. It is for this reason that a study of patterns in nature is also a study of symmetry.

Symmetry is a pattern classification scheme. There are various symmetry classifications including:

  • Bilateral (mirror) symmetry is symmetrical with respect to its reflection. The butterfly and most mammals are symmetrical along the main body axis.
  • Radial (rotational) symmetry, as seen in a starfish, is where similar parts are regularly arranged around a central axis and the pattern looks the same after a certain amount of rotation.
  • Translational symmetry, such as repeating tiles or wallpaper patterns, means that a particular translation of an object to another location does not change its pattern.
  • Scaling symmetry which is the property of a pattern where each part of which  is identical to the whole as seen at different magnifications. This is commonly called self similarity -- a property that characterizes a fractal shape.  Our lungs and tree branches are examples of scaling symmetry.
  • Time symmetry, such as the periodic behavior of ocean waves or music, involves changes in time. Symmetry can also be a description of non-geometric forms such as time and space.

The study of symmetry is the study of invariance in nature. It is the study of nature's regularities during natural events or changes. It is the study of preservation (or conservation) when change does takes place. Symmetry and the study of patterns in nature are closely related because patterns in nature are manifestations of resistance to change. By understanding the symmetry of natural patterns,  we have the opportunity and means to classify types of patterns in nature even though we may not fully understand the underlying pattern formation processes.

So, symmetry is the description and the study of order, of likeness, similarity of structure, regularity in form and arrangement, orderly and similar distribution of parts. Symmetry describes rules for moving things around without changing their pattern.

Symmetry ideas are also used to conceptualize pattern formation. In particular, It is said that patterns in nature are formed through "symmetry breaking" -- making something less symmetrical (having fewer symmetries than its predecessor).

A sphere is an example of "perfect symmetry". Certainly it is bilaterally symmetrical and it also has radial symmetry. It is symmetric in rotation meaning that it looks the same no matter how you rotate it. It is said that a sphere is highly symmetrical because it has all symmetries. It remains invariant in shape under certain classes of transformations such as rotation, reflection, inversion, or more abstract operations. A sphere is said to be "homogeneous", a technical word meaning "having lots of symmetry".

But, the deformation of a sphere results in the breaking of that "perfect symmetry". Shown is a series of computer generated distortions of the sphere that are that are similar to cellular development. Each distortion has its own symmetry but a lesser symmetry than the perfect symmetry of the sphere.

All but the perfect sphere are no longer invariant under certain classes of transformations such as rotation and reflection. This distortion or process of change is called "symmetry breaking". Yet each object is a new or different pattern with its own symmetry. For this reason, scientists regard the process of symmetry breaking (some loss in symmetry) to be the process of new pattern formation. Broken symmetries are important because they help us classify unexpected changes in form. Through the process of symmetry breaking, new patterns in nature are formed. New structure is gained as symmetry is lost.

The above figure is an illustration of the actual patterns of development at various stages of cleavage where the symmetry of the sphere (fertilized egg) is broken and new patterns are formed with less than perfect symmetry.

The subject of symmetry is very well covered in the literature. Check out the links to various references.

Useful References


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