Dear @James, this is about  pattern-based software systems.

"Patterns. Patterns are everywhere. There’s thousands of patterns already recognized, documented and made available for reuse. Although patterns provide us with answers – they also lead to many questions. Should we use them? If so, which should we use? How do we find the right ones? How much time should we invest in finding, learning and then applying the pattern? What about new patterns? Are there patterns to be found and used on my project? In my organization? Should we take the time to recognize them, document then and make them available for reuse? What about automation? Is there some way(s) in which I can take these patterns and automate how they are used? How should they be automated?...

PBE is a systematic, disciplined and quantifiable approach to using pattern specifications and implementations in the development and delivery of software solutions..."

http://patternsbasedengineering.net/

http://patternsbasedengineering.net/2010/06/23/what-is-pbe/

Dear James,

That is a good question that has huge implications not only in mathematics but in natural sciences and the arts including music. A pattern is an orderly arrangement of things in space or events and behaviors in space time or things that are made by humans to appeal to our intellect, senses and imagination.

A pattern can be natural, seen from inanimate objects and non human animate animals to the ones constructed by humans. Seasons and natural disasters that are coming with them are some natural patterns that recur in cycles. Galactic objects in the universe follow a pattern of cyclic existence and evolution, symmetries we see in nature are mesmerizing patterns. 

From among patterns of structures humans built is mathematics, as it is a study of patterns, structures of order obeying logic of reasoning that are developed from  fundamental axioms.

I also attach patterns that are made by Ethiopian traditional art works and a pattern on the wall of the 2000 years old tall Ethiopian obelisk at Aksum. 

Things like the number "pi" is its own pattern. Yet indeed a pattern.  And any number not perfectly divisible by 7, when divided by 7 always bears the curious fingerprint ...142857142857... at some point in the stretch of numerals after the decimal point. A double swirl galaxy when stretched north and south, emulates the double helix of DNA.  'Comets and planets' seem to smack of 'sperm and egg'. Pattern could be in the eye of the beholder, so vantage would be important. Any perceivable element, given enough room for analogy, can be likened to a number of other seemingly unrelated contexts. The brain is certainly hard-wired to perceive patterns for survival reasons, and perhaps a major element of imagination, creativity and problem solving relies on our innate gift of pattern-finding.  Kekule looking into the fire to see aromaticity in the form of snakes biting their tails, the dual 'uncovery' of 'calculus' by Liebnitz and Newton, the dual discovery of laser light, the digital computer - by individuals at a distance, certainly indicates the importance of pattern recognition. Perhaps all patterns have, or indicate, 'meaning' - whether tangible or intangible. Pattern recognition is most likely the unavoidable result of human perception. Patterned behavior, social patterns, quantum pattern, universe-size pattern.  Radiolarians seemed to have invented/suggested all possible physical patterns in and of themselves, see: https://www.youtube.com/watch?v=S0hX0Yx7Nk0

Physical patterns ... do they each have a metaphysical component or meaning as well?

Cymatics is another amazing field imbued with pattern:  sound/different pitches causing unique patterns/shapes/forms within physical matrices; configurations as diverse as the radiolarian. Perhaps being kindred to vibration itself, without pattern, nothing would exist.  And it perhaps is either the natural trick and/or the convincing element of pattern itself which acts as the enticing elixir or the actual proof by which humans choose to either believe or to not believe in a supreme being. If "pattern" is expected in all possible universes... is it an indication of a supreme being, or is it something more scientifically usual and expected? Pattern is certainly a notion to be contended with~! 

Pattern - repeated rhythmic structure in the evolving scale-invariant multiverse (open nonlinear dynamic system) that has a certain steadiness to interference (memory of evolution).

This is the most concise definition that can be formulated, perhaps... welcome to the general theory of chaos.

I don't believe that  for the time being I can consistently contribute to the definition of the pattern. But maybe to other aspects patterns have all in common. 

I believe that a pattern is a macroscopic regularity that appears as a consequence of the action of very simple microscopic rules, which are usualy locally acting and more or less hidden to the observer - or in any case not explicitly present there. The observer is  overwhelmed by a macroscopic regularity, but he usually must do some work of research to find out the real reason of the pattern. And very often, the simplicity of the local rules that generated the pattern seem not to be sufficient to explain its complexity, inspite the fact that they are really the only one reason for the pattern.

I believe that we must separate an abstract research domain about pattern generation and different applications in the real world. We need classification results of a generality we can only dream about for the moment... Another problem with patterns is that mathematics did not define so far enough notions in order to be able to study them. If you want to prove something, you must first formulate what you have observed in the form of some rigorous conjecture. Well, in many cases we lack even the best definitions which would make such statements possible.

@Miguel Ángel Montes: The idea of randomess can be considered as the opposite to the idea of pattern...

Looking for the opposite of a pattern is akin to a huge study of separation of sets in a topological space.    The story starts with what is known as a Hausdorff space in which distinct points belong to disjoint neighbourhoods.    What I am suggesting is that there is a parallel between near vs. far (separated) sets and patterns vs. non-patterns.    What you have mentioned would definitely be an example of a non-pattern.   That is, a set of  random events do not describe (fit into) a pattern.    The other good example of a non-pattern is pi, whose digits do not have a pattern.    

Dear @James, this is about  pattern-based software systems.

"Patterns. Patterns are everywhere. There’s thousands of patterns already recognized, documented and made available for reuse. Although patterns provide us with answers – they also lead to many questions. Should we use them? If so, which should we use? How do we find the right ones? How much time should we invest in finding, learning and then applying the pattern? What about new patterns? Are there patterns to be found and used on my project? In my organization? Should we take the time to recognize them, document then and make them available for reuse? What about automation? Is there some way(s) in which I can take these patterns and automate how they are used? How should they be automated?...

PBE is a systematic, disciplined and quantifiable approach to using pattern specifications and implementations in the development and delivery of software solutions..."

http://patternsbasedengineering.net/

http://patternsbasedengineering.net/2010/06/23/what-is-pbe/

Dear James, great justification for your invitation to this post. Congrats!

The study of patterns is a very recent phenomena even though we can trace several historical backgrounds in history - such as mosaics, f.i. I have been studying a bit the subject and I have the feeling that this corresponds to a wonderful shift in science and arts - at large, namely the turn from asymmetry to symmetry. Patterns do have a symmetry, sometimes static, sometimes dynamic. Symmetry is fantastic topic that was "discovered" very recently - that leads us all, indeed, to the study of patterns, and ion the other side, networks.

Thank you for your post, truly.

Dear @James and friends, this is fine resource about patterns and symmetry in nature!

Patterns – The Art, Soul, and Science of Beholding Nature

"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..."

http://www.patternsinnature.org/Book/PatternsContainSymmetry.html

I still full-heartedly contend that things such as the number "pi" are indeed, recognized patterns. The Euler equation is another way of capitalizing on this notion. Everything that exists is imbued with "pattern" by default. Anything, otherwise, is relegated directly to the realm of the imperceptible and indescribable. Perhaps 'tis only a matter of simple semantics which finds need to manifest the sport of mathematic fence-drawing. Pre-defined criteria may often fail in good-will effort to describe hard reality; such brave rules oft' wax arthritic in their attempt to achieve the aim. Once such rules have been invented - so have been any of the multivariate inadvertent boundaries associated with them. In other words, perception, itself, is pattern. The heart beat of a big bang every ~37 billion years can be perceived... as a pattern. An elephant's heart-beat 4 times a minute, is a pattern. The hummingbird's heart-beat 1200 times per minute is another recognizable pattern. Vantage is important where pattern is concerned, therefore a 'wide-eyed lens' is probably a good idea. The wide-eyed mathematical equation ... even better.

Dear @Jack, thanks for fine examples. As I am fan of PI, I do enclose nice discussion on PI as a pattern. "The pattern in pi can be found by a strange yet effective method. It also helps reveal how it is tied in with the golden ratio through the Fibonacci sequence..."

http://mymathforum.com/applied-math/41337-pattern-pi.html

Dear James I think, pattern is synthetic machine-made. See also textile printing!

@Christian E. Jacob: [sometimes]  pattern is synthetic machine-made.

Good point!  I agree.  This is the case with a robotized automobile assembly line.   If the entire assembly line is robotized, then the assembly line exemplifies a synthetic, machine-made pattern.

The hybrid synthetic-machine-made and human-intervention pattern is also of interest.  The penultimate example is a computerized sewing machine operated by a seamstress. 

Perhaps the followers of this thread can point to other  hybrid synthetic-machine-made and human-intervention patterns.

Dear All, a dynamic waveform is a pattern?

Patterns are just something we may classify... By increasing our ability to detect patterns, we detect novel patterns. I'm Machian in my opinion: patterns are just a complex of sensations.  Now I'm Feyerabendian: even if we recognize patterns and they work in our Science, we cannot never be sure they are just artificial creations of our brain, or true.  (The old, unresolved problem of the Thruth of entities and of the concepts...) 

Dear Arturo, if you define a pattern mathematically, it has its own degree of reality (a Plato-like reality in a separate "world of Ideas" or Aristote-like as a potential form in Nature) independently of being sensory or not, true or not. IOW, reality does not depend on sensations or on truth.

dear Alfredo, a lot of philosopher do not believe so... Examples: Avenarius, Mach, Husserl, Carnap, Maturana, gestalt, Hume..

A pattern is a value or qualia that is not so close to heart to have representation in everything and not so distant to have a single representation as the self is seen to have a single unique identity.  

I feel more comfortable with a definition of a pattern as a logical function, with a domain in a set of well-formed representations of a kind of entities. If the value of the logical function is true we say that the pattern matches, or that we have found the pattern in a particular representation and, as a consequence in the object that's being represented. In this view those patterns that match all well-formed representations of a kind of physical entities, for example, are what we call laws of nature. With respect to a particular domain of phenomena, we might be able to give a definition of a non-trivial pattern that matches all possible realizations of the corresponding representation and, therefore, in a way describes whatever we are studying. We might not be able to do this, because of the limitations that are implicit in a formal language.

When we say "There is a pattern.", in my opinion, we are not speaking appropriately. We should say, we have found a pattern that matches whatever we are observing.

@Oscar Chavoya-Aceves: I feel more comfortable with a definition of a pattern as a logical function....

It is also possible to define a pattern as a specified arrangement of objects of interest.  Here are examples:

Adjacent digital image pixels with a particular colour such a patch of blue surrounded by a patch of green.

Birds with large eyes such as owls.

Mosques with twin towers made with white stone such as the blue mosque in Instanbul.

And so on.

@Oscar Chavoya-Aceves   "..for example are what we call laws of nature."

       I think a no so obvious connection (excuse the pun) divides natural law and "pattern". Patterns are  embodied to the eye and mind with a continuous, e.g. connect the dots" nature.  i do not  believe though that the patterns we discern are composed of contiguous connections but appear so only.  Fot instance the constellations in the night sky are actually the composite o light emissions tht differ in the proximities (can be very far divided) observed by the eye though all elements obey correctly established laws of nature (i.e. newtons laws of motion).  i think  nature does not really possess (temporly) emerged "points" that are physically contiguous with regards to structure. If Newtons laws are considered a (basic) pattern, they are within everything and have little to with observation.  Thus are three distinct facets, observation, explanation, and pattern.  The explanation for an observed pattern, however simple or complex, may or may not be traceable to natural laws for explanation to a finite solution,  possibly accounted for only.  Sensory dependence of observation precludes explanation if the senses themselves function beyond awareness in a similar manner. In my own research I envision an egg shape of space from Newtons laws, but on reflection it is not a pattern.   

I compare patterns with glasses. Once you put them on, you may see everything on the same way no matter what you stare at. You should try as many times possible and many glasses too, just to be sure to find the right patterns and how they behave. Then you can infer. For example, in a recent paper of my own authorship I use mainly, nonlinear chaotic patterns in nature to demonstrate that the probability to find intelligent life tends to zero than to one.

Article Evolution through the stochastic dyadic Cantor Set: the uniq...

 In international trade theory, pattern of trade (or pattern of specialization)  means a subgraph T of the complete bipartite graph Km,n, where m stands for the number of countries and n the number of goods. T is defined as a set of couples (i, j) where the production of good j in country i is competitive.

In a Ricardian trade theory, the pattern has m+n-1 elements or less and forms a tree, if the input coefficient matrix is in a general position.

I do not know why pioneers of  international trade theory recognized T as a pattern. They may have called pattern some special subgraph which is rather a small subset of the complete graph  and kept stable.

In a totally different field of mathematics called oriented matroid, sign set of a point is called sign patterns. For example, "Oriented matroid" in Wikipedia contains an expression as follows: 

• The theory of oriented matroids was initiated by R. Tyrrell Rockafellar to describe the sign patterns of the matrices arising through the pivoting operations of Dantzig's simplex algorithm; Rockafellar was inspired by Albert W. Tucker's studies of such sign patterns in "Tucker tableaux". [12] (Italic is mine.)
• [12] Rockafellar, R. T. (1969). "The elementary vectors of a subspace of R^N (1967)"  

It is by a pure coincidence that patterns of trade (in international trade theory) can be interpreted as a special kind of "sign patterns", not of a classical oriented matroid but of a tropical oriented matroid. This sign pattern is usually called "types" in tropical and subtropical oriented matroid.

See  the paper below.

https://www.researchgate.net/publication/280646264_International_trade_theory_and_exotic_algebra

Article International trade theory and exotic algebra

A pattern is other than noise. 

@William R. Buckley: A pattern is other than noise. 

An interesting observation.   Then answer the question What is noise?

The other, related question is this: Is every pattern observable?

Tessellation is an example for the pattern.

@Mehmet Niyazi Cankaya: Tessellation is an example for the pattern.

I agree that one can find many patterns in the tessellation of a surface.   For example, we can look for the following patterns in a Voronoi (Dirichlet) tessellation of a plane surface:

maximal Voronoi region pattern: a maximal Voronoi region (maxVR) is polygon with the highest number of sides (an adjacent) polygons.   It is possible to have more than one.   In the case where there is more than one maxVR, this defines a pattern containing two or more maxVRs.

minimal Voronoi region pattern:  a minimal Voronoi region (minVR) is polygon with the lowest number of sides (an adjacent) polygons. It is possible to have more than one. In the case where there is more than one minVR, this defines a pattern containing two or more minVRs.

@Waldemar Koczkodaj:  ... a pair may be a magic pattern that you are looking for...

Yes, I agree that pairwise similar objects define a pattern.   Perhaps other followers of this thread can suggest some examples of patterns containing pairwise similar objects. 

Ornamental design has been, at least from Neolithic remains, our idea of pattern. Of course, some patterns are perceived rather than made: like the binary recurrence of night and day that led us to eventually understand the seasonal pattern. Conception-to-Childbirth is a pattern that probably led to the perception of the planting-a-seed-and-getting-a-fruit pattern we call "agriculture". Rituals condense many of the life-cycle events as patterns. Ritual is always a pattern.

Here a lovely mayan piece of ornament.

Best regards, Lilliana

It seems that all are discussing the repetition of a physical trait or mathematical sequence. What about pattern recognition, the ability to link seemingly random bits of information or images? It is something that computers still have trouble doing. What about non-random random patterns?

a pattern is a conceptualized ordering of things that can recall things explicitly or implicitly named 'faster' than without this layer of higher - order ... so a pattern has to emerge out of a necessity to recall within a scope or constraint ... if not to provide definite proof of certain pre - conceived notion... 

Pattern in psychology. Behavioral pattern - a recurrent way of acting by an individual or group toward a given object or in a given situation...

http://www.dictionary.com/browse/behavior-pattern

Patterns in engineering!

Detecting and learning from patterns and trends in data, processes, and asset conditions allows organizations to improve their production processes and maintenance practices. The challenge plants face is how to process the massive volume of available data, seeing through the noise and uncovering actionable intelligence. The rapid growth in devices and systems that make up the Industrial Internet of Things (IIoT) is heightening the complexity of the task...

“Discovering behavioral and failure patterns is only one step in preventing process and mechanical equipment degradation,” remarks Michael Brooks, President and COO at Mtell. “The real value comes from detecting a root cause from the earliest symptoms automatically, in real-time, along with prescriptive action to stop the damage from happening in the first place.”...

http://www.plantservices.com/articles/2016/technology-toolbox-trends-are-telling/?show=all

I think satisfying a novelty