Mathematics of mattresses
The "Mathematics of Mattresses" explores the intricate relationship between mathematics and mattress design, highlighting how various materials and geometric configurations contribute to comfort and durability. Modern mattresses predominantly utilize artificial fibers or foam rubber, supported by coil springs that play a crucial role in their resilience. Historical advancements, such as the development of the innerspring mattress by Heinrich Westphal in 1871 and NASA's invention of soft memory foam in the 1960s, underscore the evolution of mattress technology influenced by mathematical principles.
Key factors like coil gauge and configuration affect the firmness and support of the mattress, with different designs—such as Bonnell coils and cylinder pocket springs—providing unique benefits and drawbacks. The document also discusses how wear and tear on mattresses can be mathematically analyzed, noting that uneven weight distribution leads to disproportionate degradation over time. To mitigate this, proper maintenance techniques, like rotating and flipping the mattress, are recommended. The mathematical concept of the "Klein 4-group" is introduced as a method for efficiently managing mattress orientations to optimize longevity. Overall, the synthesis of mathematics and mattress design illustrates how these principles enhance sleep quality and mattress performance.
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Mathematics of mattresses
Summary: Modern mattresses are superior to older designs because of the geometry and pressure distribution of the coil springs that define them. Mattresses last longer when rotated through four configurations.
The modern mattress is a cushion for sleeping and sits on top of a box spring that provides support and reduces wear and tear. While straw, coconut fiber, horsehair, feathers, pea shucks, and water have all been used to stuff mattresses in the past, in the twenty-first century most are filled with artificial fibers or foam rubber and derive much of their resilience and support from coil springs, which are either connected by interconnecting wires or encased in fabric. The first innerspring mattress is attributed to Heinrich Westphal in 1871, and its popularity may be because of hygiene and comfort considerations. The geometry of the coils impacts the durability and firmness. Manufacturers use calculations like the average load limit of a floor and the volume and weight of a waterbed. NASA attributes the 1960s invention of soft memory foam with high-energy absorption properties to aeronautical engineer Charles Yost, who was working under a NASA contract. Many studies employ statistics, such as those involving quality of sleep, amount of snoring, and the impact of sleeping positions. Mathematical techniques and models of mattresses have also been useful in studying factors such as pressure distribution, deformation, combustible behavior, and mattress flipping.
The gauge of the coils is one of the factors that impacts the mattress’s firmness and, therefore, its support and durability. Counterintuitively for the layman, lower gauges mean larger cross-sectional diameters—the number of passes through the drawing dies that are required to create a wire of a given thickness. Lower gauge means fewer passes, meaning thicker wire. Bonnell coils, named after the inventor, may have been adapted from the coils used for buggy seats in the nineteenth century. The configuration of adjacent hourglass coils connected by helical wire, called “helicals,” increases the spring’s resistance proportionally to the load. Cylinder pocket springs systems are individual cylinder pieces held together by clips. Continuous coils are rows formed from a single piece of wire. However, while the head-to-toe rows of continuous coils offer good support while still responding to shifts of position and weight, the movement of the coils in response to those shifts is noisier and more noticeable than in other mattresses.
Wear and tear on a mattress is disproportionate because of the fact that the sleeper and the mattress are not the same shape, and thus some coils will bear more load than others. Compressing and decompressing gradually weakens a mattress; it may show noticeable changes after a few years. Use of a firm box spring helps prevent the sagging that would set in quickly, and rotating and/or flipping a mattress twice a year helps to more equally distribute wear and tear over the course of the mattress’s life. The actual technique of mattress flipping has been the subject of some discussion for years because the interval is great enough that it is difficult to remember in which direction the mattress was last flipped or rotated without making some kind of mark on the mattress as a reminder.
A mattress can be rotated along three orthogonal axes (x, y, and z); or to compare a mattress to an airplane, roll, pitch, and yaw. The roll axis parallels the longest dimension, the pitch the next-longest, the yaw the shortest. Because a mattress has two sides suitable for sleeping on, and each of those sides has two possible orientations, this means that there are four possible mattress configurations. One mattress-flipping technique that cycles through these four configurations is called the “Klein 4-group,” named for mathematician Felix Klein, which is a group describing the symmetries of a rectangle in three-dimensional space. Absent a mnemonic device to remember the previous and next configuration, random selection may be the best choice to maximize the efficiency of mattress flipping. Over the course of 10 years of random selection every six months, for instance, the most-used orientation will be used about 31% of the time, and the least-used about 19%—a 6% deviation from perfectly distributed usage.
Bibliography
Hayes, Brian. Group Theory in the Bedroom, and Other Mathematical Diversions. New York: Hill and Wang, 2008.
Krasny, John, William J. Parker, and Vytenis Babrauskas. Fire Behavior of Upholstered Furniture and Mattresses. Norwich, NY: Noyes Publications, 2001.