A Belgian physicist has determined the maximum height of towers constructed from identical blocks stacked on top of each other. It turns out to be inversely proportional to the square of the standard deviation of the errors with which the blocks are positioned relative to each other. The work was published in the International Journal of Solids and Structures.
Sometimes scientists study phenomena that at first glance seem like child's play. For example, a group of scientists recently investigated the processes that occur when closing cardboard boxes. We encounter one such problem from childhood: if you stack blocks on top of each other, sooner or later the resulting tower will collapse. This problem is quite common in engineering and everyday life—from stacking books to building drywall fences or stacking shipping containers. Scientists study the processes that occur during such construction and propose optimal strategies for maximizing the height of such towers. However, the processes that occur when stacking blocks with randomly misaligned centers relative to each other have not yet been described or studied. Therefore, there is no understanding of the maximum height of such a tower.
Physicist and engineer Vincent Denoël of the University of Liège set out to correct this. In his work, the scientist established a relationship between the error in the installation of blocks relative to one another and the maximum height of a tower constructed from these blocks. To do this, the physicist considered an idealized case in which the blocks—perfect rectangular parallelepipeds—were installed one on top of the other with a random error distributed according to Gaussian law. In this case, the tower's height at the moment of collapse and the level at which it occurs are also random variables, and the scientist's task was to determine the distributions governing these variables and find the most probable values.
To solve this problem, the physicist proposed a mathematical description of the problem and modeled the behavior of the towers, including in the calculations the distribution of random errors in the placement of the blocks.
The average maximum height of the towers turned out to be inversely proportional to the square of the standard deviation of the errors. Moreover, the scientist was able to identify two most likely scenarios for the towers' collapse: they collapsed either at the base or at a position close to the top of the tower, but slightly lower.
Previously, mathematicians determined the required number of holes in a cube for it to fall apart.
