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The article about dance formalization of Argentine Tango has been finished. It is actually a small book of 110 pages available online: http://www.virtuar.com/tango/theory/tb0.htm
It is for dancers and about dancing, but the "pattern generator" idea is used there broadly. I have put it in the center of dance figure formation. Folloing a pattern generator one produces many figures. This is especilly important for an improvisational dance. Sorry, it is more about dancing than mathematics, since I am writing it for my dancing friends.

I am working on an article about dance formalization on example of Argentine Tango. Significant work is going with the notion of dance "pattern generators" which are related to mechanizms of  human body and their mapping to geometrical patterns of movements on the floor which they generate. These "pattern generators" are subgroups of a group of all body movements.  I am working on this topic for the last 10 years. The article contains about 80 pages. I hope to finish it soon.

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It seems I have to specify what patterns I'd like here to be considered.
Here is a defintion from Longman Dictionary of Scientific Usage (shortened):
"A repeated or regular arrangement of objects, events, properties, behaviour or characteristics. A pattern can be formed by repeated arrangmements in space ( configurations) e.g. the sodium and chloride ions in sodium chloride crystals form a three-dimensional pattern , or x-rays passed through a crystal form a pattern of spots on a photographic film;  
The recurrent configurations form a static pattern i.e. a pattern which is standing still and can be seen all at once. A pattern of recurrent events or processes is a dynamic pattern, i.e. a pattern which appears at regular intervals of time."
Let us take this as an intuitive description what is pattern.
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Here a definition from Webster which I will NOTconsider as a pattern here:
1. Anything proposed for imitation; an examplar.
2. Anything designed as a guide for making things; a model; plan;
Essentially, not a design. Design is design. A pattern can serve as an example, but it is not an example.

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Experimenting with a new brown scale in this crochet photo pattern. Not sure if I like the shading in her face, but I love the overall pattern. 31 of 80 rows completed.
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Ice pattern on boulder in the small stream
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Near the centre of #Antarctica, measurements from #CryoSat show an unusual pattern in the ice sheet’s elevation. Scientists have now found the reason for this pattern – and the discovery is leading to even more accurate measurements from ESA’s ice mission.

Read more:
http://www.esa.int/Our_Activities/Observing_the_Earth/CryoSat/CryoSat_detects_hidden_Antarctic_pattern
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I do not still know a defintion of what is pattern. I do not think patterns are nesessarily symmetircal or repeatable. Patterns are elements interconnected with some governing regularity, law, rule, formula. A pattern as a collection and its rule form a system. AxxAxxAxx is a pattern, periodical pattern, but AxAxxAxxxAxxxx is a pattern too.
A pattern can manifest, show, represent a rule: when there is a pattern it should be a rule. On another hand, a pattern can set up a rule, a rule defined by a pattern itself.
Intelligent things seach for patterns in sourroundings to discover governing rules of it. Patterns happen where there is repeatability. A pattern is a footprint of a system acting around.
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