Currently, between eight thousand and twenty thousand planes plow through the sky at the same time worldwide. Each of these flights is possible thanks to mathematics, in aspects ranging from fuel consumption planning to aircraft design and improvement. The first vehicles that enabled humans to soar above the earth’s surface (hot air balloons, dirigibles, gliders, and airplanes) were based on various geometric, algebraic, or analytical concepts.
In the 9th century, the bird people became the forerunners of aviation. Without theoretical support and only hoping to fly, they spread the wings woven into their suits and plunged from great heights, each attempt failing again and again. After that, in the same century, the chemist, physicist and mathematician Abbas Ibn Firnás (810, Ronda; 887, Córdoba) performed the first more or less successful parachute flight, which would lay the foundation for future aircraft designers.
Trigonometry and classical geometry played an integral part in the study of aeronautics in this early period; They allowed pilots and navigators to calculate distances, angles and altitudes more precisely. Specifically, the trigonometric methods developed by the Arab mathematician Al-Biruni between the 10th and 11th centuries to solve astronomical and geodetic problems, such as measuring the radius of the earth, were decisive.
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The first airplanes to take to the skies were lighter-than-air airplanes, filled with helium or hydrogen so that their total weight was less than the weight of the air they displaced. Examples of these aircraft are the hot air balloon, first flown by the Montgolfier brothers; and the airship, designed in Germany by Ferdinand von Zeppelin. Studies of proportion and volume made by Leonardo Fibonacci in the 12th and 13th centuries were used to define its design and stability. His work on harmonious proportions has also been used to establish the dimensions of various aircraft components (such as wings, fuselage or other elements), resulting in harmonious and aesthetically pleasing designs.
In the 18th to 19th centuries, geometric research enabled the development of more efficient airfoils. An airfoil is a geometric, usually curved, shape present in airplane wings, helicopter blades, ailerons, and rudders that serves to optimize the behavior of a surface interacting with an airflow to ensure the least possible drag from the wind.
Mathematicians like Daniel Bernoulli studied the geometry of the wings and other structural aspects as well, trying to achieve the flight of a heavier-than-air airplane for the first time. Although he failed, Bernoulli’s work held the key to understanding the interaction between air and objects. In fact, one of the theorems he proposed is still used today in one of the explanations for why airplanes fly. This determines how the speed of the wind interacting with the wings creates a pressure distribution that lifts an aircraft.
Example of applying Bernoulli’s principle to an asymmetric airfoil in contact with air. Due to the high velocity above the profile, this area is low pressure while below it is low velocity and high pressure. This pressure difference creates a buoyancy force.YD
Another common approach to justifying the flight of airplanes uses Isaac Newton’s third law: Particles in the wind are directed downward, resulting in lift on the airplane. In fact, the laws of motion and universal gravitation (a precise mathematical description of how bodies move in space), formulated by Newton in the 17th century, provided the basis for establishing the principles of airplane flight shortly before they were invented. However, there are currently other, more accepted explanations based on the use of computational fluid dynamics.
In 1799, the Briton George Cayley developed the symmetric top and bottom airfoil, one of the first recognized airfoils. However, the real breakthrough in wing design came in the late 19th century with extensive new research into wing aerodynamics. At that time, Otto Lilienthal made numerous flights in gliders (aircraft designed for flight without a motor). Through his experiments, he collected data and refined airfoils that maximized lift and minimized drag.
Blades of a modern turbofan engine. A conical spiral can be seen in the center of the engine, which is used to scare off birds during flight. Jorge Lascar
A few years later, in 1903, the Wright brothers were the first to achieve controlled powered flight. His work, based primarily on wind tunnel experiments, is the basis for understanding the shape of the rotor blades in turbofan engines, the most common in commercial aircraft today. This type of engine contains a large fan that compresses incoming air mixed with fuel, creating high-velocity gases that are expelled to propel the aircraft. By optimally designing the geometry of these vehicles, it is possible to reduce drag, save fuel, reduce engine noise and eliminate turbulence during flight.
Since the beginning of the last century, the aviation industry has developed rapidly thanks to the use of theoretical and computational tools, various theorems, ideas and mathematical theories used in the analysis and optimization of various aspects of aviation. We dedicate the following article “Coffee and theorems” to this topic.
Yoshua Diaz Interian is a PhD student National Polytechnic Institute (Mexico).
coffee and theorems is a field dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT). In which researchers and members of the center describe the latest advances in this discipline and share points of contact between mathematics and other fields, social and cultural expressions and remember those who shaped its development and knew how to turn coffee into theorems. The name recalls the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that converts coffee into theorems.”
Edition and coordination: Agate Timon Garcia-Longoria. Is coordinator of Department of Mathematical Culture of the Institute of Mathematical Sciences (ICMAT)
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