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A wide and sometimes chaotic variety of patterns in nature are shown to us by nature and individually described in many books and articles. However, the hidden unity of patterns in nature is not always apparent and is rarely described.

Nonetheless, there appears to be one unifying principle by which all patterns in nature operate. All patterns in nature are connected to other patterns. In order to develop, to operate, and to survive, there is interdependence and feedback between connected patterns.  This interdependence is described by E.O.Wilson and Bert Holldubler in “The Superorganism” as follows:

“Life is a self-replicating hierarchy of levels. Biology is the study of the levels that compose the hierarchy. No phenomenon at any level can be wholly characterized without incorporating other phenomena that arise at all levels. Genes prescribe proteins, proteins self-assemble into cells, cells multiply and aggregate to form organs, organs arise as parts of organisms, and organisms gather sequentially into societies, populations and ecosystems. Natural selection that targets a trait at any of these levels ripples in effect across all the others.”

Connected patterns in nature can be portrayed as nodes in a network where the links, patterns themselves, are conduits for the interdependent dynamic processes and feedback mechanisms that relate one pattern to the next. These networks are special in that they are formed through “preferential attachment” -- meaning that one pattern will connect to another pattern only if it wants to do so. Most certainly, a flying goose will join a flock of other geese of like species rather  than connect to a school of fish or a herd of buffalo. In our world of the Internet, someone may be more likely to connect to Amazon rather than to someone’s personal web site. Preferential attachment (sometimes called “aristocratic” attachment) causes a network to behave in way that is very different than a network where there is an equal likelihood of attachment (sometimes called “egalitarian” attachment).

These dynamic, aristocratic networks associated with patterns in nature are said to be “self-organizing”. Fish schools, bird flocks, and animal herds are  examples of this self-organizing process. Here, there are no leaders, yet the group moves in an orderly fashion. Each creature in the group locally applies a set of simple rules that govern its speed, distance, and direction with respect to its nearest neighbors. Each creature’s sensors interrelate with its nearest neighbors by collecting and responding to proximity and speed information. It is this recursive, aristocratic action of each individual that results in the emergent behavior and organization of the group.

Self-organizing systems of patterns in nature are composed of many levels of interdependent patterns. A plant leaf has an overall pattern that is composed of a pattern of veins. In turn, the veins connect to both leaf cells and to the plant’s transport system. Leaf cells connect to the atmosphere and to the veins. Each level can exhibit its own self-organization (e.g. cell chemistry, cells, and veins). Within each level of organization, no matter how far you magnify or reduce your view,, the organization of parts or the processes looks similar to the whole. This characteristic is called “self-similarity”.

Trees in the winter provide a good  example of self similarity. Each large branch is the same basic pattern as the entire tree. Each smaller branch has the same shape as the larger branch and, in turn, the same shape as the entire tree. Self similarity is this sameness at all levels of organization from the smallest twig to the whole tree. The dynamic recursive processes of self organization  lead to self similarity. The word “fractal” is used to describe patterns in nature that exhibit self-similarity at many levels of magnification.

Different patterns in nature display a characteristic unity when they contain geometric shapes and physical processes that either don’t change over time and space or, at least, change in the same way according to the same "law". Mathematicians and scientists call these changes “scaling”. When looking for unifying factors, we say that we are looking for “scale invariant” or “scale-free” factors where there is no change in form or function as the scale changes. In a network with preferential attachment, scale invariant characteristics are prevalent.

Self-similar objects look approximately the same no matter what their magnification level. It is said that self-similar objects are scale invariant because the same structure or process exists over all levels of magnification.  Scale invariance means that changes in size (scale) do not cause structures or processes to change shape or function. Since self-organized systems are self-similar, it can also be said that self-organizing systems are scale invariant.  Scale invariance is ubiquitous in nature including the constant ratio of mammalian body size to lung volume, blood volume, blood hemoglobin concentration, and red cell size.

The power law is a mathematical model of the scale invariance that is so common in nature’s patterns. It models systems that are aristocratic and contain connecting components that are interdependent and employ feedback – just like patterns in nature. The power law appears to model a unifying principle associated with patterns in nature.

According to Andriani and McKelvey: “Power laws seem ubiquitous – they appear in leaves, coastlines, and music. Cities follow a power law when ranked by population. The structure of the Internet follows a power law, as does the size of firms. Power law signatures range from atoms to galaxies, DNA to species, and networks to wars. “

Current research indicates that a huge number of patterns in nature are “aristocratic” and scale invariant. This suggests that the power law may be the signature of the unifying principle of connectivity in patterns in nature.

Much of the recent work on this subject has been performed by Geoffrey West and others. West points out that power law exponent values that model the relationships between various processes are a unifying characteristic amongst many diverse patterns in nature. Many of these exponents are simple multiples of ¼. This unifying idea is called “Quarter Power Scaling”. He says about power laws that: “… biological systems obey a host of remarkably simple and systematic empirical scaling laws which relate how organismal features change with size over many orders of magnitude. These include fundamental quantities like metabolic rate (the rate at which energy must be supplied to sustain an organism) , time scales (like lifespan and heart rate) and sizes (such as the length of the aorta or the height of a tree trunk). It is remarkable that all of these can be expressed as power law relationships with exponents that are simple multiples of ¼ (e.g. ¼, ¾, 3/8) . They appear to be valid for all forms of life whether it be mammalian, avian, reptilian, unicellular or plant-like. These “laws”  are clearly telling us something important about the way life is organized and the constraints under which life has evolved. ”

George Johnson, in his New York Times article entitled Of Mice And Elephants: A Matter Of Scale , states “scaling emerges from the geometrical properties of the internal networks animals and plants use to distribute nutrients “. West’s research group suggests that, within these internal network connections of self organizing, scale invariant systems, there lies a unity that explains the quarter power scaling phenomena.  West proposes that scaling laws are the unifying feature of patterns in nature. These scaling laws are signatures of the connectivity associated with both interdependency and feedback within patterns.

We’ve characterized these self-organizing, self-similar, scale invariant systems in a very general way as networks composed of nodes and links which are, in themselves, all patterns in nature.  We have suggested that there is a unifying principle in all patterns in nature. The principle is that all patterns are connected and can be modeled by a power law. We’ve specified the dynamics of that connectivity by stating that:

• The nature of the connections is recursive -- meaning that there is interdependence between patterns and a feedback mechanism within each pattern.
• The recursive process produces self organized systems which are self similar and scale invariant.
• The resulting scale invariant network of connected patterns is created by means of preferential attachment by the patterns that join the system.

In summary, the unifying principle behind patterns in nature is connectivity. That unity is carried out by the connective aristocratic networks within nature’s patterns. These networks are manifestations of self-organized, self-similar, scale invariant patterns and systems of patterns.

The reference section on unifying principles contains a list of recent papers describing the latest work on unity within nature’s patterns.