*Fractal structures* are also referred to as *self-similar structures*: Trees, river systems, lungs, circular systems in our body have fractal structures. When you zoom in on a fractal structure, its basic structure – at a different scale – is still similar. Fractal structures are *scale-free*.

**Figure 1:** This figure shows the fractal – self-similar – structure of a tree (above ground).

Research shows that distribution systems often have fractal structures: All above mentioned examples are distribution systems. These fractal structures are highly optimized – efficient – distribution systems, where an *optimum* is found between ‘competing’ forces.

The fractal structure of a tree – above and below ground – contributes to the optimization of processes like photosynthesis above ground, and the extraction of water and nutrition from the ground, and its subsequent distribution ‘in’ the tree to support its operation.

In case of a river system, water is extracted from a certain area, and its drainage to the sea is optimized: gravity ‘competes’ with friction, etc.

**Figure 2**: The fractal structure of a river system (source: International Rivers).

Fractal systems also have typical statistical properties: the distribution of branches in a fractal structure with a specific size – their respective numbers – can be best shown with a so-called *power-law* distribution.

When the branches of a tree for example are categorized by size, and their numbers are determined, the size distribution turns out to be a power law (a tree has many very small branches, a smaller number of medium sized branches, and an even smaller number of larger branches, etc.).

A power law is a straight line in a graph, when logarithmic scales are used; in this example: for the x-axis size of branches and y-axis their number.

The exact origin of power laws – they are quite common in nature and social systems – is not always clear.

However, research shows that systems at a critical point typically have scale-free (fractal) properties. It can be argued that these fractal structures at critical points, also point to the optimization of certain distribution processes in the system concerned.

Chaotic dynamics also produce power laws of various properties of the system concerned; chaotic systems have fractal attractors.

In my research, I show that the System also produces structures and processes with fractal – scale-free – properties.

During the unfolding of the finite-time singularity dynamic that was accompanied by four accelerating war cycles (1495-1945), several fractal structures can be identified, including: (1) the *sizes of wars* the System produced (size in terms of fraction, see study), (2) the *sizes of states* in Europe, shortly before the System’s collapse in 1939, and (3) the sizes of *military capabilities/organizations* shortly before the System’s collapse in 1939, can all be best described with power-law distributions.

This raises the question if these phenomena are related, and *what* exactly was optimized during the unfolding of the finite-time singularity dynamic?

It is important to keep in mind that Europe during the period 1495-1945, developed from a collection of circa 300 loosely connected and diverse ‘units’, with a total population of 83 million in 1495, into a tightly connected system of 25 highly standardized states with a total population of 544 million in 1939. I refer to this process – the outcome of the finite-time-singularity dynamic, that was accompanied by four accelerating war cycles (1495-1945) – as a *process of social integration and expansion*.

I argue that wars – and especially systemic wars – were instrumental in the crystallization of the System (Europe) into 25 states; this process was driven by the tensions the System produced through population growth, and (increasing) rivalries between states.

Tensions were released in the System through wars with fractal structures, that ‘*carved out*’ fractal state-structures. Armies – doing the ‘*carving*’ – were instrumental in that process.

I argue that the increasing optimization of the System and the finite-time singularity that unfolded (1495-1945), resulted in the fractal structures I mentioned. Increasing ‘*fractality*‘ of the System – its war activities, state-structures and the structures of armies – are related phenomena, pointing to its increasing optimization.

The question then is: ‘*what was optimized?*’. It is important to consider the System as a *process*, a process of *social integration and expansion*, as I mentioned. Because this process evolves at a (for us) relatively *slow time-scale*, and our ‘timespan’ (and also of historians) is often rather short, we tend to experience the System as more or less stable. But that is not the case.

With integration (and cooperation, this implies) social systems can achieve *economies of scale* and *scope* that enhance their survival changes, and the fulfillment of their basic requirements. The System is still evolving, and forces that push for *integration* continuously compete for forces that push for *fragmentation*. During the period 1495-1945, integration forces were stronger than fragmentation forces, and generated a self-reinforcing dynamic powered by population growth and rivalries between states (the *accelerating treadmill* I refer to in another article).

Population growth and rivalries between states produce(d) tensions that were transformed into destructive energy that was *put to work* – used – during wars, and powered the unfolding of the finite-time singularity dynamic (the four accelerating war cycles).

The finite-time singularity dynamic – the four accelerating war cycles, and ‘individual’ wars – is about the *accumulation and distribution of energy*, in efforts of the System to optimize its *performance*, that is, ensuring the balanced fulfillment of basic requirements of *uneven* states and their populations in the anarchistic System. With ‘*uneven*’ I refer to differences in power and influence (dominance) of states. While much divided these states and populations in the anarchistic System, at the same time they became increasingly dependent on each other, including for their security.

The fractal state-structures the System (Europe) eventually crystallized into, point to the optimization of this process: shortly before the System’s collapse in 1939, the state-structures in Europe had (‘finally’) become permanent (as historians also observe), and changes in their size and form had become ‘part of history’ (and had in fact become impossible from a system’s perspective). These fractal state-structures ensured that the tension *build-up* during relatively stable periods was *minimized* (as far as that was possible), while the *distributio*n of these tensions in the form of destructive energy during war-activity (especially systemic wars) is *maximized*.

Not coincidentally – as the crystallization of state-structures into fractal structures progressed during the period 1495-1945 – military capabilities also increasingly became fractal in nature. This phenomenon – ‘*fractalization*’ of military capabilities – ‘emerged’ in all states more or less simultaneously. The *fractalization* of military capabilities of (competing) states coevolved, but were ‘*imposed*‘ by physical laws that apply to the System.

**Figure 3**: This figure shows the typical ‘branching’ – fractal – organization structure of a military unit, of the 1st Cavalry Division of the US Army (1944–1945).

**Figure 4**: This figure shows the actual deployment of the 101st Airborne Division during various phases of operation Desert Storm in Iraq and Kuwait (1991). This shows how destructive energy was deployed on the battlefield. This was (for various) reasons a successful deployment of military capabilities – destructive energy – based on a ‘paradigma’ that has however become increasingly obsolete since 1991. In 1991 the System resumed chaotic war dynamics (see study) and the trend towards increasing ’empowerment’ has led to the introduction of networks, and network warfare. At this stage there is no optimization.

Military organizations are responsible for the *distribution of destructive energy* on the battlefield, in efforts to destroy (similar) enemy capabilities. Military organizations must also cope with competing forces: the destructive effect on the enemy must be maximized, while risks to their survival – had to be minimized, at the same time.

The need for an effective and efficient *span of control* of military units – which concerns the distribution of information – also contributed to the forming of fractal structures.

The optimization of this process, also reached an optimum, shortly before the collapse of the System in 1939.

These developments – the evolution of state structures and military capabilities, etc. – are closely related. Fractal military organizations are optimized structures to fight similar structures, and are products of the threat- and defense system of states.

Following the Second World War (the fourth systemic war, 1939-1945), a long-term trend of empowerment (see study) further continued. At this point in time, threats to states increasingly organize in networks.

States have difficulty coping with these networks: fractal military organizations (the outcome of a typical European context, 1495-1945) are not designed to fight these networks, but only their counter parts of other states.