Self Organizing of Mosaics and the Fifth Dimension

In a simplified view a mosaic can be considered "simply" as the combined action of environmental constraints that in turn create a complex system in which patterned sub-systems interact. But in reality, a mosaic is much more than a deterministic and "Euclidean" or "fractal" process of patch juxtaposition, or simply a three-dimensional space in which we recognize objects. In fact although we add another dimension (for volumes), resulting in a fourth dimension, this is not enough to justify the complexity of the system under evaluation (Fig. 3.18). We need to add another nonEuclidean dimension (a process dimension) that we call simply the "fifth" dimension (Fig. 3.19). This dimension probably contains the "overlap" of mosaics produced by different and/or independent processes. The fifth dimension is fundamental to the structure of the ecological complexity, and as in the niche theory, causes the emergent character of the biological complexity to appear. This dimension can be considered a meta-domain in which several "mosaic" domains

X,Y (first dimension) (point)

X&dX (second dimension) (line) X&Y (third dimension) (space)

Fig. 3.18 The first four

X,Y (first dimension) (point)

dimensions of a mosaic are represented respectively in Euclidean space (x, y, z) by

X&dX (second dimension) (line) X&Y (third dimension) (space)

point, line, space, and volume

X&Y&Z (fourth dimension) (volume)

Fig. 3.19 The fifth dimension (or mosaic meta-domain) is created by the contemporary presence of different mosaic-patterned processes inserting into the same topological space

Fig. 3.19 The fifth dimension (or mosaic meta-domain) is created by the contemporary presence of different mosaic-patterned processes inserting into the same topological space

interact producing emerging properties. The fifth dimension is a multiprocess space in which every process can act without affecting primarily all the other processes.

In the fifth dimension different mosaics are in action contemporarily exchanging information primarily inside the focal mosaic level but other emergent properties arise. We have to eliminate the misconception that all parts are linked to all other parts, substituting this assumption with another: some parts are more connected then others, and some connections are ephemeral. This assumption is quite important to understand how a complex system functions. We are used to considering an ecosystem as a full connection of elements that exchange information, energy, and material. This is true in principle, but, if we try to measure the strength of such connections, it is easy to verify that some connections are two or more orders stronger than others that can be considered marginal (Fig. 3.20). In conclusion, the overlapping of different mosaics into the fifth dimension is allowed by the fact that each mosaic moves scarce information from and to the other mosaics. One fundamental factor that allows such a dimension is represented by the time lag of the processes, the recurrence of phenomena and the interval by which a process is modulated (Fig. 3.21).

Fig. 3.20 The connections inside a mosaic are two or more orders stronger than the connection between the different mosaics of the fifth dimension

Fig. 3.20 The connections inside a mosaic are two or more orders stronger than the connection between the different mosaics of the fifth dimension

Fig. 3.21 Different mosaics can coexist because the focal processes have a distinct temporal (scale) frequency. In this case mosaic (A) has six recurrences, mosaic (B) three, and finally mosaic (C) two recurrences

A little more explanation of ecological time is necessary because there is a risk of confusion on terminology.

Time is a physical parameter, and we can't change this, at least in this universe, but in the same time lag the process can behave differently. The speed and frequency of a process is the way in which a process repeats a sequence in an interval of time.

Strong and weak connections between parts are widespread in the fifth dimension. The weak connections operate at the margin of the principal nodes that link the different patches.

There are in conclusion three ingredients of mosaic complexity: variety of internal and external constraint, weak connections between the overlapping mosaics, and different patterning in space and time. For instance if we use such a paradigm to shape a real mosaic in the field, we can produce many mosaics from the same image. What can be the meaning of this phenomenon? Probably some mosaics extracted by the aerial image (see f.i. Fig. 3.4) are not working in reality, but most of these can be coupled with a process or an organism. For instance, the mosaic represented only by two patches indicates the soil water content, the last with many patches the plant community.

The fifth dimension also incorporates the hierarchical organization of a mosaic. The hierarchy as later illustrated is another type of organization in which space is more relevant than time and in which elements are composing parts of a major system, and this sequence is repeated several times along the hierarchical chain.

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