Infantry operations (mathematics)

Summary: Mathematics has long played a significant role in infantry operations, including influencing cryptography, logistics, and military strategy.

The oldest military unit and still the backbone of most modern armies, infantry units consist of soldiers who engage the enemy face-to-face. Historically, infantry units marched from one location to another. In modern times, infantry units may be deployed in a variety of ways, including overland in trucks; by sea, such as the troops landing on Omaha Beach on D-Day; or by air, either from planes or helicopters. Paratroopers are often considered elite among infantry units. In general, infantry are distinct from other land-based mobile units, such as cavalry, employing different tactics and strategies.

Mathematics has always played a major role in warfare, including infantry movements. Early Babylonian clay tablets show evidence of sophisticated mathematical calculations of the volume of dirt that would be needed for siege ramps and what sort of minimum manpower would be required to accomplish the task. The sophistication of mathematics in ancient Greece was no doubt in part because of its usefulness to war—the Greeks may have left a legacy of philosophy and art but spent much of their time and resources at war among themselves and with their neighbors.

Napoleon Bonaparte is widely considered to be a military genius who revolutionized the use of light infantry and artillery. He was also an avid mathematics student and was often accompanied in the battlefield by mathematicians, including Joseph Fourier. He discussed his own solutions to mathematics problems with notable mathematicians, such as Lorenzo Mascheroni, Pierre Laplace, and Joseph Lagrange, including what is known as Napoleon’s Theorem. He was quoted as saying, “The advancement and perfection of mathematics are intimately connected to the prosperity of the state.” Many modern officers have been educated at the U.S. Military Academy at West Point and other military academies, which emphasize mathematics and engineering in their curriculums, and both military and civilian mathematicians continue to play critical roles in infantry tactics and deployment, especially in the modeling and simulation of twenty-first-century combat strategies.

History

Archimedes, one of the most famous ancient mathematicians, applied his knowledge of geometry, the estimation of weights and volumes, and three-dimensional rotations to defending the city of Syracuse from siege by Roman forces (214–212 b.c.e.). In addition to the standard trick of cutting holes into the walls for archers to fire arrows through, Archimedes helped to design the catapults used by the Syracuse artillery units, and he called for traps to be built in the walls to drop heavy stones on approaching ships. Cranes were even used to drop grappling hooks onto ships and capsize them. The siege took much longer than it otherwise would have, and the Roman commander reportedly ordered that Archimedes’s life be spared out of respect for his intellect—an order that was ignored, and Archimedes was killed when the siege finally succeeded.

The Renaissance was a time of flourishing mathematics, with applications in a wide variety of sciences, including cartography. While the Age of Discovery certainly was one cause for the demand for increasingly more precise maps, so too was the desire to accurately direct the movement of troops and ships while at war. Accurate chronometers were developed at the order of the military, which also called for more precise ways of determining latitude in order to increase the usefulness and accuracy of maps.

Modern Warfare

Eventually, mathematics would be used to more accurately determine the velocities and paths of projectiles, which in turn influenced not only the behavior of artillery units but also the design of infantry firearms, which became increasingly critical in conflicts like the U.S. Civil War and World War I.

World War II, because of its extraordinary size and resource consumption, put mathematicians to use in all areas of the military, a close relationship that has continued and been further assisted by the development of modern-day computers. The advent of paratroopers in World War II added a new level of complexity to the deployment of infantry troops, taking into account not only point-to-point movement on the ground but also precision insertion via parachute. Humans leaping from a moving plane do not fall straight down, so calculations had to be made to take altitude, speed, and other factors into account in order to determine when, where, at what altitude, and at what intervals paratroops should deploy to successfully land on a predetermined spot. A hybrid transportation algorithm that first mathematically computes an ideal solution, which is then used for stochastic simulations, has been successfully used to model deployment of troops and equipment.

Other investigations into this problem often use numerical methods, fluid dynamic equations, 3-dimensional flows, mesh resolution techniques, and simulation methods. The use of aircraft for combat reconnaissance was also largely pioneered during World War II, though it was hampered by their limited speed and at times by unreliable radio communications, which did not facilitate the rapid decisions infantry commanders in the field were required to make.

Modern communication methods allow for rapid computer modeling and real-time decision making, virtually as soon as the data are collected. Military radar was also in its infancy in World War II, though work by mathematicians and scientists such as physicist Luis Alvarez would improve its utility. For example, Alvarez helped create transponders, then known as Identification Friend or Foe (IFF) radar beacons, and improved antenna systems, which identified friendly aircraft without visual confirmation and facilitated precision delivery of troops and bombs even in poor weather.

Mathematics at War

The quantification of troops, inventory, and distances as well as the order of battle and the estimations of travel speeds and damage to fortifications have likely always played a role in warfare. The term “order of battle” originally referred to the order in which troops were positioned relative to the position of the commander but has come to refer to the composition of the forces involved in a field operation, including their command structure, personnel, disposition (the geographical locations of the headquarters of units and subunits), and equipment.

In U.S. Army practice, an order of battle prepared for an intelligence report also includes information on personalities (known enemy personnel and relevant information pertaining to them), unit history relevant to the current situation, a logistics report on how units obtain supplies, and a combat effectiveness section that is prepared using combat modeling applications based on sophisticated algorithms. Orders of battle are fundamental to a military commander’s situational awareness. Commanders depend more on combat effectiveness projections as modeling techniques have become more sophisticated and data from field operations have been applied in order to continually evaluate them.

In essence, the same mathematics responsible for governing the artificial intelligence of enemy forces in video games like Call of Duty is used—albeit with a great deal more data and more powerful processing—to evaluate enemy forces in real life. These models draw on a diverse array of mathematical methods. Game theory in general is concerned with modeling strategy. Statistical analysis, Andrey Markov chains, business logistics, and fluid dynamics have all played significant roles. During World War I, mathematician Frederick Lanchester devised Lanchester’s Laws, which use systems of ordinary differential equations to determine which of two sides will remain at the end of a battle, as functions of the defenders’ strengths and time, assuming neither side breaks off combat. They continue to be the basis for many modern simulations. Some models simplify problems or address only small portions of a vastly complex problem, including trying to quantify “soft” or qualitative aspects of combat, though hybrid modeling with both discrete and continuous components is a growing way to reliably model critical subsystems and also their interactions with one another. Mathematical analysis of satellite data and images is also used for detecting landmines and improvised explosive devices, which are some of the greatest threats to troops on the ground.

Perhaps the biggest impact of mathematics on the infantry is that the use of combat modeling means the ability to predict—if not always accurately, at least with a greater degree of accuracy than in the past—the outcome of various combat scenarios and, thus, to manage risk and reward when allocating troops. Military effectiveness can be maximized at multiple levels, from the allocation of funds at the budget stage to recruitment techniques to the command structure of the armed forces to troop movements.

Bibliography

Biddle, Stephen. Military Power: Explaining Victory and Defeat in Modern Battle. Princeton, NJ: Princeton University Press, 2004.

Booẞ-Bavnbek, B., and J. Høyrup. Mathematics and War. Basel, Switzerland: Birkhäuser, 2003.