The last few decades have witnessed the development of a host of ideas aimed at understanding and predicting nature’s ever present complexity[1]. It is shown that such a work provides, through its detailed study of order and disorder, a suitable framework for visualizing the dynamics and consequences of mankind’s ever present divisive traits. Specifically, this work explains how recent universal results pertaining to multiplicative cascades and fully developed turbulence entice all of us, in a logical way, to seek peace in a condition typified by the hypotenuse of a right-angled triangle.

Playing with modeling clay, recall the construction of generic multiplicative cascades. Start with a uniform bar cutting it into two pieces containing

After ^{n}^{n} whose masses correspond to the expansion of ^{n}

Recall now the construction of a uniform measure over a Cantor set, cutting a bar by the middle and piling up left and right, while leaving holes of proportion 1/3 as seen in Figure 2.

After ^{n}^{n}^{n}^{n}

This second cascade turns out to be intimately related to the first one, for if the hole size is

As the objects generated by both cascades grow without bound, it is convenient to portray their accumulated masses, from the beginning up to an arbitrary point

The obtained boundaries below (Figure 3) reflect the dynamics of the games, as follows. The

The accumulated sets for the two games turn out to be rather peculiar, for if one were to parachute on them from above, upon landing one would falsely believe that they are flat, that is, one would think to have landed on top of the uniform bar of clay. Irrespective of their

These accumulated sets and also those defined by more general multiplicative cascades, for example: combining arbitrarily the generic games (i.e., having constructs that use both imbalances and holes), splitting the bar of clay in more than two pieces, and using chance to define the associated partitions from level to level, all ultimately yield

There is

When the Reynolds number is high, the air’s internal cohesion is shattered and it flows in an intermittent and irregular fashion that is nicely captured by the progressive breaking (precisely by 70%) implied by the first game for kids (Frisch, 1995).

As depicted below, one-dimensional observations of natural turbulence, and also for other turbulent flows gathered in the laboratory, are

Although these results do not allow making predictions of this common condition, for the power of the air does not always use the 70% to the left but rather does it to both sides as guided by chance (Screenivasan, 1991), the process describes the formation, everywhere on earth of

As a simple repetitive cascade captures the complexity of natural turbulence and as various generalizations of the games herein have been found useful to represent other intricate geophysical processes, it is reasonable to employ such notions, certainly combining the games and adding randomness, in order to illustrate the possible pathways that give rise to the ample

In this spirit, the simple cascade games may be used to describe the inherent dynamics of the systems that have governed the world, as follows. While the first generic game helps us visualize the proliferation of economic imbalances and the resulting distrust of competition in capitalistic societies, the second game allows us to sense the futility of totalitarian regimes whose stubborn trapping of dissenting voices into ‘holes’ utterly destroys unity.

Although these observations may appear to be ‘too simplistic’, the cascades also allow us to view the ultimate and historic consequences of the world’s political and economic systems. On the one hand, the unexpected but predictable dissipation of communism due to its rather empty and false ‘equality’ and, on the other, the misery experienced by many in the world today and the further dangers posed by the

In this regard, the first game offers a rather poignant warning against current policies by the ‘civilized world’, as the very object produced by such a divisive cascade after

Although the wealth of the richest 1% in 1998 is underestimated by such a cascade, 38% (30), the simple game help us imagine what would happen if imbalances persist, either as

Clearly, the simple ideas herein also apply to us as individuals, as the cascades reflect and encompass our ever changing attitudes. But the notions also provide a positive image, for they allow us to contemplate the

As may be appreciated, to break or to mend equilibrium is our ultimate decision. For we, as kids that we are, may choose to ‘sail’ the natural cascade in reverse, exchanging a selfish and turbulent spiral (^{-?}^{+?}

As with the arrow of time we may basically engender two distinct behaviors: either the shortest path that maintains

These universal concepts reiterate that it is up to each one of us to find and to spread peace, for we may choose to break equilibrium using either cascade or their combinations, irrespective of the nation’s political system in which we reside. These images also reaffirm that our most desired state is intimately related to

For the common but anomalous states defined by our divisive traits are potently symbolized by the absurdity of ‘walking over thorns’, a feared condition that may not be escaped via hallucinogens or other worldly distractions, but by our daring to live in solidarity helping one another.

The evils we all face are, at the end, very similar. As our seemingly distinct ‘power games’ eventually yield ample dust and extreme separations (infinite in size if not corrected), some based on our relentless desire for wealth leading to obvious disparities (the first game), that is, $, which is the opposite of the desired integration, the slender letter S; and others caused by trying to impose a given life-style by force (the second game), these reflections point us to a

THE HYPOTENUSE

(Carlos E. Puente)

Two options before us

two pathways ahead,

the one is the longest

the other straight.

We journey directly

or go by the legs,

we follow intently

or end up in pain.

By walking the flatness

or hiking the spikes,

we travel in lightness

or take serious frights.

The incentive is unity

and the call proportion,

the key is forgiveness

and the goal true notion.

In wandering wickedness

there is never a fruit,

but in ample humbleness

one encounters the root.

There is a best pathway:

the palpably smooth.

There is one solution,

I tell you the truth.

For any other pathway

will lead us astray.

Oh listen, you brother,

let’s brighten the day.

Otherwise, the devil

shall pull by the legs.

For such road is fractal:

as long as it gets.

Let’s end all disparities,

growing to the root.

Let’s keep equilibrium,

avoiding dark soot.

Oh listen, you sister,

a counsel from science.

I tell you integrating,

don’t leave it to chance.

This work is dedicated to Silvio Rodríguez, a brother and a watchman who, dazzled by the light and despite the wind, has known how to sow, with “thousand littles of love,” the reason for equilibrium.