Firework mathematics
Firework mathematics is the study of the underlying principles that govern the behavior and design of fireworks, integrating aspects of physics and chemistry to enhance their visual and auditory effects. Originating in ancient China, fireworks have evolved into complex displays characterized by explosive patterns of light and sound. At the core of firework mathematics is the understanding of timing, trajectory, and the chemical reactions that produce vibrant colors. Mathematicians and engineers have formulated models to predict the flight paths and maximum heights of firework shells, which follow predictable parabolic trajectories governed by gravitational forces.
The colors produced in fireworks stem from two primary methods: blackbody radiation and atomic emission, each contributing to the visual spectacle depending on the temperature and materials used. Fireworks are meticulously crafted, with careful calculations ensuring that explosions coincide perfectly with their ascent. In competitive environments, fireworks are judged not only on artistic merit but also on quantitative measures such as color purity and explosion height, reflecting a blend of technical skill and creativity. This intersection of artistry and science makes firework mathematics a fascinating field that enhances the enjoyment of these celebratory displays.
Firework mathematics
Summary: Firework mathematics involves the timing and rhythm of burning, rocket flight, and explosions.
Fireworks are explosions for entertainment with design elements of light, sound, and smoke. Chemical additives are used to color fireworks, which originated in ancient China. The province of Liuyang is known as the home of fireworks. Fireworks as an art are temporal, like dance or animation; therefore, much of firework mathematics has to do with the timing and rhythm of burning, rocket flight, and explosions. Mathematicians around the world have modeled and quantified various aspects related to fireworks, like the path and maximum height. In the seventeenth century, Claude Dechales published what became a popular textbook on mathematics that included pyrotechnics. Engineer Amédée-Françis Frézier, whom some also refer to as a mathematician, worked on the theory of fireworks in the eighteenth century. The process of mathematical induction has been likened to a sequence of connected fireworks. In the United States, the Bureau of Alcohol, Tobacco, Firearms and Explosives classifies and regulates fireworks.
Patterns of Explosions
Most fireworks shot into the air explode in spherical patterns. By modifying the composition of fireworks, it is possible to add or remove tail effects, change the speed of individual parts, and produce delayed explosions to parts, filling spheres with radial lines or creating expanding spheres. Less frequent are fireworks that burn sustained, extending, two-dimensional shapes, such as rings or hearts.
Ratios and Proportions of Shells and Mortars
Many fireworks are packed into shells and fired out of special mortars, or small cannons. Larger shells are fired out of larger mortars with higher speeds and also fly higher. As with any projectile, the path and height of the firework shell, until the explosion, obey the quadratic equation of gravitational deceleration and the shell flies following the path of a parabola. On the other hand, because of the physics of the black powder or pyrex used to propel the shells out of the mortars, the relationship between the size of shells, mortars, and their initial speed is linear. The relationship between the size of the shells and the maximum height they fly is also linear. Pyrotechnician formulas approximate 100 feet of the shell’s maximum flight height per every inch of its diameter. The explosion of the firework has to be timed so it happens when the shell is high up in the air, which is achieved through solving the height equation and matching the time of chemical reactions in the shell to that height.
Fireworks Color and Temperature Gradients
There are two distinct ways to color fireworks. The first method is based on the same physical process used in incandescent light bulbs and the second on that used in neon lights. The first method uses blackbody radiation—the property of objects to emit more light with higher temperature. Blackbody radiation emits light over a broad spectrum. As metals heat, they start to become red to the human eye because the majority of the spectrum is light at infrared wavelengths human beings cannot see. When the temperature rises, the emission of the light in the visible spectrum increases and the object becomes first yellowish and then white, the mixture of all visible-light wavelengths. Thus, fireworks that depend on blackbody radiation for their color can only be dull red, pale yellow, or white.
The second method of firework coloring is based on the so-called atomic emission. Atoms in the firework material, before the firework is fired, are in a stable state, corresponding to particular orbits of electrons. If atoms are electronically excited, they emit photons to return to that stable state. When photons are in the visible spectrum, the human eye sees a color as the atomic emission takes place. Some elements have a narrow spectral band in their atomic emissions, allowing particular pure colors to be pinpointed. For example, sodium emits bright yellow and barium emits green when electronically excited. Copper salts emit pure blue but they are so unstable at high temperature that people only recently learned to use them safely in fireworks.
If the firework material burns too hot, the blackbody radiation process takes over. Therefore, to produce pure colors of the atomic emission process, pyrotechnicians create mixtures that burn relatively cool. The chemistry breakthrough allowing this to happen was the substitution of potassium chlorate, which burns at around 120 degrees Celsius, for potassium nitrate, which burns at 560 degrees Celsius. Fireworks contain coolants that prevent burning from reaching higher temperatures, for example, by releasing some water and carbon dioxide, as sodium bicarbonate does.
Pyrotechnic Competition and Measurements
At competitive events, fireworks are measured based on several criteria, mostly qualitative and artistic. The quantitative criteria include purity and brightness of color and the appropriate explosion height. The timing of the intended fireworks effects, such as the change of shape and color, is also taken into consideration—it has to follow a recognizable temporal pattern and to form a pleasing rhythm.
Competition judges add points for technical difficulty, celebrating innovations in fireworks. For example, when strobe effects were first discovered, fireworks using them were awarded technical difficulty points at competitions. After a few years, as strobe effects became well researched, judges stopped awarding points for them.
Bibliography
Danby, J. M. A. “Fireworks.” The College Mathematics Journal 23, no. 3 (1992).
Lancaster, Ronald. Fireworks, Principles and Practice. 4th ed. Gloucester, MA: Chemical Publishing, 2005.
Shimizu, Takeo. Fireworks: The Art, Science, and Technique. 3rd ed. Post Falls, ID: Pyrotechnics Publications, 1996.