Games for kids

Playing with modeling clay, recall the construction of generic multiplicative cascades. Start with a uniform bar cutting it into two pieces containing p = 70% and q = 30% of the mass, respectively. Then, mold the pieces uniformly, piling up the largest one and stretching the other, so that they form two contiguous rectangles having equal size. Repeat the process ad infinitum, as shown in Figure 1.

After n levels of the construction, the bar is split into 2n pieces of size 1/2n whose masses correspond to the expansion of (p + q)n, yielding (n + 1) clay layers arranged according to Pascal’s triangle. This divisive game produces a skewed multifractal object made of finely intercalated thorns supported by intertwined Cantor dusts[2]. The clay concentrates on a few isolated spikes, yielding the proverbial 80% of the mass in the tallest 20% thorns, just after 16 levels.

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 levels of this construction, the bar is split into 2n disjoint pieces of horizontal size 1/3n and mass 1/2n, yielding thorns that grow to infinity as (3/2)n, while supported by triadic Cantor dust.

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


Fig. 1: A generic multiplicative cascade propagating imbalances

From a bar, (a), to 212 pieces arranged into 13 clay layers, (e).


Fig. 2: A generic multiplicative cascade spreading disunity

From a bar, (a), to eight dispersed pieces, (d).

changed from 1/3 to a general size h, then the resulting dusts match the non-contiguous organization of equally-sized thorns on the first cascade, with denser layers requiring smaller hole sizes h and vice versa (Feder, 1988).

Accumulated resources

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 x.

The obtained boundaries below (Figure 3) reflect the dynamics of the games, as follows. The notches on the cloud associated with the first game correspond to the sequence of pile ups and stretches: 70% of the mass happens from 0 to x = 1/2, 49% up to 1/4, and so on. The plateaus found for the second game are related to the holes of the successive pile ups: 50% of the mass happens from the beginning to x = 1/3, 25% up to the ninth part, and so on. For comparison purposes, the graph also includes the straight cumulative distribution corresponding to the original bar, that is, the common level zero on both games.


Fig. 3: Accumulated clay (right) corresponding to objects (left) generated by:

(a) propagating imbalances, (b) spreading disunity, and (c) maintaining the initial bar.

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 imbalances, p ? 1/2, or holes, h ? 0, such rugged boundaries define devil’s staircases whose lengths, from their initial and final points, equal two units, that is, one unit horizontally plus one vertically (Feder, 1988).

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 maximal lengths of two, as they ‘travel’ by the legs of the right-angled triangle associated with level zero.

There is only one case that does not destroy the unity and true equilibrium of the original bar, and that corresponds to the ‘fifty-fifty’ adage (i.e., p = 1/2 and h = 0) that results in the aligned hypotenuse and the minimal distance of v2 ˜ 1.4142…


Fig. 4: The point of equilibrium within a sea of probabilities

The power of the air

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 universally consistent with a multiplicative cascade of eddies, that divide precisely according to the proportions shown: the ‘offspring’ eddies have half the size of their ‘parent’ and they transport respectively 70 and 30% of their parent’s energy (Screenivasan, 1991)[2].


Fig. 5: A multiplicative cascade of eddies in fully developed turbulence

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 violent thorns and explains the ultimate fate of division in the eventual dissipation of all energy in the form of heat.

The games and the world of politics

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 turbulence we experience in our modern world.

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 thorny game of globalization.


Fig. 6: Generic propagation of inequities and discriminations

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 n = 20 levels closely matches the distribution of wealth of the most powerful nation on earth, precisely when p = 70%. In fact, simple calculations, using Pascal’s triangle, reveal a close match for the wealth of the richest 5, 10, 20, and 40 percentiles in the United States, that is, in order, 59 (57), 71 (70), 84 (84), and 95% (95) of the resources, with the cascade values given in parenthesis[3].

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 n or p increases, or both[4]. For even if current conditions are termed ‘stable’ and ‘prosperous’ by experts, it is easy to foresee an ‘unhealthy’ future[5]. For as kids everywhere know, and as the laws of physics assure us, the game of imbalances ends as the game of disunity does: with ‘all biting the dust’.

The games and our quest for peace

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 balanced state where we all may experience friendship and peace.

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 (r = e-?), and the symbolic 2/3 = 0.666… that may be read on any divisive game[6], by an outward and loving one (r = e+?) that dreams a just future in order to restore unity. For in a rather graphical fashion, both arithmetically and geometrically, 1 = 0.999…, as may be appreciated when mountains are cut-down and valleys are filled-in on any (multifractal) object that gives rise to a devil’s staircase.

As with the arrow of time we may basically engender two distinct behaviors: either the shortest path that maintains equilibrium or an evil stair that through fragmentation augments entropy, that is, either v2 or 2, these observations remind us that it is wise to live life below the threshold[7]. For, it is thoughtful to augment our internal cohesion to repel the false lure of the superiority postures embedded in the pervasive cascades, which prevent us from truly standing together in solid and common high ground[8].


Fig. 7: Multiplicative reparation of unity via integration

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 rectitude, balance, forgiveness, friendship, and Love, while its antithesis is associated to indifference, vengeance, hate (even against our enemies) and with any type of ‘inequality’ or ‘discrimination’ that produces distrust, emptiness, and ample loneliness.

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 unique pathway, to one condition truly worthy for all, to one improbable but feasible organization that dismisses hypocrisy and terror, to one just state that ‘heals the wound’ and respects all life, that is, to the one and only situation that allows us to find our common root: the plain and simple case symbolized by the precious and imperturbable hypotenuse, Y = X [9].


(Carlos E. Puente)

By the wisdom of science

simply divides the air,

to dissipate all its heat

coding a subtle cascade.

Turbulence is selfish game

for it scatters the whole,

and its sequence is a frame

for the options of the soul.

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 no excuse,

let’s practice fair game:

it’s by the hypotenuse

or else by the legs.

There is no solution

but walking straight:

the spikes of disorder

insinuate the way. (2)

There is a best pathway:

the palpably smooth.

It’s by the hypotenuse

and walking in truth.

There is one solution,

I tell you the truth.

It’s by the hypotenuse

and walking in truth.

For any other pathway

will lead us astray.

It’s by the hypotenuse,

there is no other way.

Oh listen, you brother,

let’s brighten the day.

It’s by the hypotenuse,

there is no other way.

Otherwise, the devil

shall pull by the legs.

It’s by the hypotenuse

or else by the legs.

For such road is fractal:

as long as it gets.

It’s by the hypotenuse

or else by the legs.

Let’s end all disparities,

growing to the root.

It’s by the hypotenuse,

the one that yields fruit.

Let’s keep equilibrium,

avoiding dark soot.

It’s by the hypotenuse,

the one that yields fruit.

Oh listen, you sister,

a counsel from science.

It’s by the hypotenuse:

the simplest design.

I tell you integrating,

don’t leave it to chance.

It’s by the hypotenuse:

the simplest design.

It’s by the hypotenuse:

the simplest design…