Perhaps this comment could help

In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. For this reason, definitions of complexity often depend on the concept of a "system"—a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time.

Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity. Phenomena of 'disorganized complexity' are treated using probability theory and statistical mechanics, while 'organized complexity' deals with phenomena that escape such approaches and confront "dealing simultaneously with a sizable number of factors which are interrelated into an organic whole". Weaver's 1948 paper has influenced subsequent thinking about complexity.

In science, there are at this time a number of approaches to characterizing complexity. It is important to highlight that even among scientists, there is no unique definition of complexity - and the scientific notion has traditionally been conveyed using particular examples.

Finally, it is important to understand that the concept of complexity should not be confused with the concept of complicatedness, which denotes a situation or event that is not easy to understand, regardless of its degree of complexity.

Perhaps this comment could help

In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. For this reason, definitions of complexity often depend on the concept of a "system"—a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time.

Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity. Phenomena of 'disorganized complexity' are treated using probability theory and statistical mechanics, while 'organized complexity' deals with phenomena that escape such approaches and confront "dealing simultaneously with a sizable number of factors which are interrelated into an organic whole". Weaver's 1948 paper has influenced subsequent thinking about complexity.

In science, there are at this time a number of approaches to characterizing complexity. It is important to highlight that even among scientists, there is no unique definition of complexity - and the scientific notion has traditionally been conveyed using particular examples.

Finally, it is important to understand that the concept of complexity should not be confused with the concept of complicatedness, which denotes a situation or event that is not easy to understand, regardless of its degree of complexity.

Thank you Jorge, I must say that I concur with all you said about complexity, and although this is well elaborated and clear (especially the reference), it somewhat relegates complicatedness to a lesser priority. The remainder of the question is yet to be answered (complicatedness) and I remain hopeful that it will attract more participants

Let me illustrate the difference between complexity and complicatedness with an example. Relative to a manual transmission, a car’s automatic transmission has more parts and more intricate linkages. It is more complex. However, to drivers, it is unquestionably less complicated, but to mechanics who have to fix it, it is more complicated.

Complexity is an inherent property of systems. Complicatedness is a derived property that characterizes an execution unit’s ability to manage a complex system.

Thanks again Jorge. Your illustrative example with the car transmission suggests that complexity and complicatedness are relative to perceptions. I tend to think that complex systems are characterised by interactive causality and emergence which inevitably reflects open systems. A car transmission, whether manual or automatic is a closed system which make the notions of causality and emergence absent. Then on what basis are we able to infer that it is complex?

That complexity is a derived property of systems, I am still struggling with this. What characterises complicatedness?

A car's transmission is also an open system because is part of a more complex system which is the car as such. Other systems of a car interact with the transmission system and vice versa to allow the car to move. In my opinion, both concepts are relative to perceptions, because one system could be complex to you but not to others and vice versa.

I can´t recall where I heard or read this, but as an example it is the one I always use. When you solve a complex problem for the first time, it is often also complicated to solve, but if you have to solve it a second time, it remains equally complex but should no longer be complicated.

I like the discussion thus far. Taking a slightly different tack, I have found that complicated often refers to large numbers of individual components or steps in a process that can be reduced to "stuff" that is relatively simple or straightforward to handle individually (that would explain both Rafael and Jorge's perceptual aspect to 'complicated.)' Complex often means that certain characteristics or parts or steps in the system's behavior are not easily reducible or understandable even when separated out - and that such separation can be difficult.

That is one of the most important problems in complexity, namely:

a) can a linear system (= complicated) be transformed into a nonlinear?, and

b) how does a linear system become complex, i.e. nonlinear

A good example is a catastrophe - natural or cultural, say social

Understand the systemic and use it as key to understanding our world which is neither a simple, nor a complicated system, but rather a complex system , with its laws of operation. The whole is more than the sum of its parts ... it changes our traditional Cartesian understanding of things!

Complexity and complication are two concepts should not be confused ! In a complicated system , there is typically a large number of poorly integrated with one another components , on the one hand and , on the other hand , a very large number of connections , all of rudimentary level ( binary interactions , sequential , unambiguous , etc ... therefore linear ) . Complexity? The key notion here is that the level of non -linear interaction .

The complexity is the opposite of complication. Mayonnaise is complex irreversible made ​​of global live interactions , strong, vital , but it is so simple: egg , mustard and oil . Airbus , it is complicated but it is not complex: it is mechanical, disassembled and reassembled without organic interactions between its components.

Complicate systems are understood and explained via tools such as statistics, vectorials, matrixes, Gaussian bells, Bell curves, averages, standards, and the like.

Complex systems, on the contrary display power laws, indeterminacy, the unavoidable presence of randomness, among other features.

Complex systems might not be as many as complicated systems, but they are by and large the most interesting ones.

I would say that complex systems have permeable boundaries and energy running through it (away from equilibrium).  But if you want an example of transition from complicatedness to complexity under certain conditions?...fugeddaboutit.  May as well explain what life is.  :)

I think complexity can be broken down to more simpler elements using tools/operators while complicatedness like a knotted rope needs a lot of effort to simplify and many a times there could be residuals or remainders - much like having 5 unknowns with only 4 equations...

Complexity refers to dynamic interactive systems. Cause effect relationships can only be fully understood retrospectively.  Complex systems are self-organising. 

Complicatedness refers to (self) imposed orderly systems with known or knowable cause effect relationships. 

After all this time, for those to whom the discussion is still relevant and a text book or paper reference is needed: The car example outlined by Jorge Morales Pedraza above is provided and well explained by Espejo & Reyes (2011, p.35) who look at Human Complex Systems in Chapter On Complexity: How to Measure It?

However, they don't address complicatedness directly nor can't I remember a transition to be discussed. They focus on complexity in terms of low/high and options for measurement. They greatly include the aspect of perception.

A transition of complicated to simple or complex is directly tackled by Dave Snowden's Cynefin Framework especially for social systems representing managed business organizations. As a model it provides clear characteristics and strategies given per each domain - whether it leads to razor-sharp distinctions for a use case at practice I doubt but that isn't the goal of the model.

The requested example of a transition is a bit tricky since social systems are often described as being complex by definition. But one can imagine an army with thousands of soldiers at peace times following prescribed rules and standards all the time - that system would be predominantly complicated in terms of knowable cause-effect relationships and certainty in prediction of behavior yet confusing in full understanding due to innumerous people and units and rules involved. The transition obviously takes place when they start interacting with another army at less peaceful times.

If you don't like military examples, Remington & Pollack in their book "Tools for Complex Projects" (2007, p.20) describe the phase transition of a project from an orderly situation (yet confusing in terms of its many tasks involved in a detail that one can't easily grasp - which makes it complicated) to a more complex situation (multiple inter-dependencies and ambiguity/uncertainty arising through new contractors that can't commit to completion dates)