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The inhabitants of the planet Pilas enjoy stacking things. Some believe that the small size and childish appearance of the pylons correspond to an equally childish mentality, which leads them to play happily for much of the day, while others believe that the ubiquitous stacks have a secret purpose or hidden meaning.
The Pilans show a clear predilection for cubes, the most regular of stackable objects, and the great stability of the planet (in which there are never earthquakes or strong winds), together with the rare skill of conscientious stackers, allows the elevation of rows that are truly amazing, both for their height and for their audacious exploration of the limits of balance.
The most stable way to put a cube on top of another of equal size is, obviously, to make the faces in contact coincide perfectly. The elevation and the side view of this basic stack will be two dominoes, and its plan will be a square.
If we move the upper cube laterally and we don’t want it to fall, we will have to stop when half of it protrudes from the lower cube (if we do the maximum displacement diagonally, three quarters of the upper cube will be overhanging).
In turn, this minimum stacking at the limit of equilibrium can be placed on a third cube as long as it does not protrude more than a quarter of the middle cube, as it is easy to calculate without more than taking into account that, for there to be equilibrium, the vertical that passes through the center of gravity of the assembly formed by the two upper cubes cannot fall outside the base of support, which is the upper face of the cube below. And, generalizing, we see that the stack of cubes will remain standing as long as the “steps” are not greater, successively and in descending order, than 1/2, 1/4, 1/6, 1/8… of the edge of the cubes: this simple series (which is none other than the well-known harmonic series with its terms divided by 2) establishes the limit of equilibrium.
A first piece of counterintuitive data is that, with only five cubes, a stack can be formed such that the top cube is completely cantilevered, that is, outside the vertical of the bottom one: it is supported by it, ultimately instance, and yet it is not even partially on top of him, thus giving the impression of defying the laws of gravity.
But there is another even more surprising fact. What is the maximum displacement that the top cube can reach with respect to the base in one of these sliding stacks? In other words: if we have an unlimited number of cubes and we lift the stack under a zenithal sun, how long can its shadow be? The incredible answer is that it can be as long as we want. The series 1/2 + 1/4 + 1/6 + 1/8… grows very slowly, but it grows indefinitely.
Of course, reality imposes limits that theory does not, and it is extremely difficult to lift extremely tall slippery piles. However, the meticulous Pilans have achieved the amazing feat of stacking nine two-meter stone cubes on the equator of their planet, achieving, by means of maximum diagonal slip, that at noon the curved tower casts a shadow of almost four meters.
The texts in this series are brief narrative approaches to that “great game” of science, technique and technology -three inseparable threads of the same braid- that is transforming the world faster and faster and in which we all must participate as players, if we do not want to be mere toys.
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