A Belgian physicist has determined the maximum height of towers built 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 installed relative to each other. The work is 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 studied the processes that occur when closing cardboard boxes. We have encountered one of these problems since 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 dry fences or stacking shipping containers. Scientists study the processes that occur during such construction and offer optimal strategies for maximizing the height of such towers. However, the processes that occur when stacking blocks on top of each other with a random error in the position of their centers relative to each other have not yet been described and studied. Therefore, there is no understanding of what the maximum height of such a tower can be.
Physicist and engineer Vincent Denoël from the University of Liege decided to fix this. In his work, the scientist established a relationship between the error in installing blocks relative to each other and the maximum height of a tower made of these blocks. To do this, the physicist considered an idealized case in which the blocks - regular rectangular parallelepipeds - were installed one on top of the other with a random error distributed according to Gauss's law. In this case, the height of the tower at the moment of collapse and the level at which the collapse occurs are also random variables, and the scientist's task was to determine the distributions that these quantities obey and to 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 value of the maximum height of the towers turned out to be inversely proportional to the square of the standard deviation of the errors. Moreover, the scientist managed to identify two most probable scenarios for the collapse of the towers. They collapsed either at the base or at some position close to the top of the tower, but slightly below the top.
Previously, mathematicians figured out the required number of holes in a cube for it to fall apart.