# Physics and Mathematics of Order in Twisted Beam Bundles

Geometry, not complex forces, determines how a group of compressed elastic beams will behave. When a group of thin elastic beams, such as toothbrush bristles or grass, is compressed vertically, the individual parts bend and collide, resulting in patterns. Now, experiments and computer models show how basic geometry regulates how order develops in these patterns. The findings could aid in the creation of flexible materials and in studying the interactions between flexible natural structures in living things, such as DNA strands.

The behavior of a single membrane, such as a thin disc of polystyrene fabric, a crumpled paper, or even a bell pepper, has often been central to studies of bending and twisting. However, few models have attempted to describe the dynamics of a collection of many elastic objects.

Ousmane Kodio, an applied mathematician at the Massachusetts Institute of Technology, was motivated to study the arrangement in elastic beams after observing how the gills of a dried mushroom bend and form patterns when compressed. According to Kodio, we were really interested in learning how a group of rays interact and in what order these interactions result.

To study the emergence of order, Kodio and his colleagues vertically fixed 54 flexible plastic beams, 1,6 mm tall and 26 mm thick, between two horizontal plates.

Ribbon-shaped rays could only move left or right. A small initial right or left bias was applied to each beam at the start of each experimental run to ensure randomness. This deviation was determined by tossing a coin. Then, as a result of the compression of the plates, the beams bent and came into contact with each other.

The number of beams bending in each direction was counted by the researchers to determine the order at any given moment during compression. Each beam was given a number; -1 for left bending and +1 for right bending.

By averaging these numbers and then taking their absolute value, they defined a measure of order that could range from 0, which corresponds to bending of beams in random directions, to 1, which corresponds to bending of all beams in the same direction.

In addition, Kodio and colleagues performed numerical simulations in which they changed a number of factors, including the coefficient of friction, the number of beams increased to 300, and the distances between the beams. Contrary to predictions, none of these changes had a significant impact on how the order emerged.

The ratio of uncompressed beam height to compressed beam height emerged as the primary determinant of ascending order with compression.

A mathematical model created by the experts also allowed them to predict how much order there would be at various compression levels. The model predicts, for example, that beams will have an order of 30 when compressed to about 0,6% of their height, meaning most will bend the same way.

The researchers noticed a number of phenomena that appeared to control the emergence of order in both tests and simulations. “Holes” are regions where beams create a gap between neighbors bending in opposite directions, as opposed to “clusters”, which are regions where many beams press against each other. Arman Guerra, a team member and PhD student at Boston University, explains that when a stack and a hole come into contact, the stack flows into the hole.

The researchers jokingly call these processes "stack-hole extinction," and they found that they can also be used to characterize the order of the system, as stacks and holes interfere with beam alignment.

The limitations of these studies are acknowledged by the researchers. For example, they did not consider situations involving extremely dense packaging where friction could become more important. Also, they did not examine more complex beam sorting scenarios, such as scalp hair, where only one end of each elastic beam is fixed and can move in more than one direction.

Harold Park, a professor of mechanical engineering at Boston University who was not involved in the study, suggests that future experiments include controllable friction between beams to further validate the predictions of numerical simulations. According to Park, the novelty of the method justifies the lack of adjustable friction in current experiments. Applied mathematician Dominic Vella from the University of Oxford in England was amazed how the group came up with such a simple plan. When you first saw the topic, Vella said, “God, how can you say anything useful about it?” He says you might think. Then you realize how important math is.

Source: physics.aps.org/articles/v16/54

Günceleme: 04/04/2023 17:01