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The mathematical beauty of bubbles: How to use soap bubbles to unlock the minimal surface theory?
Whenever we see colorful soap bubbles floating in the sun, have we ever thought that there are profound mathematical principles hidden behind this seemingly simple entertainment activity? Soap bubbles are more than just children's play; their structure and behavior have important implications for mathematics, physics, and even art. This article will lead readers to explore the beauty of mathematics behind these light structures and how it allows us to unlock the mysteries of the minimal surface theory.
A soap bubble is an extremely thin film of soap or detergent and water surrounded by air, forming a hollow spherical structure with a rainbow-like surface.
Soap bubbles have existed for more than 400 years. The earliest Flemish paintings in the 17th century showed children blowing bubbles with clay pipes. Over time, this form of entertainment evolved into the medium of expression for countless artists, such as the famous "Soap Bubble Magician" Tom Noddy and "The Amazing Bubble Man" Fan Yang, who used soap bubbles to create charming visual effects.
Mathematically speaking, soap bubbles are a perfect object of study. It surrounds a given volume with minimal surface area, a property that makes it a concrete example of minimal surface theory. Back in 1884, mathematician H.A. Schwarz proved that a round soap bubble is the smallest surface way to surround a given volume of air. By 2000, researchers proved for the first time that two merged soap bubbles are the best way to surround two different volumes with the least surface area. This is called the "double bubble conjecture."
When bubbles merge, they adopt a shape that minimizes the sum of their surface areas, holding constant the volume of air they surround.
This unique property allows soap bubbles to find applications in many engineering and structural designs. Structural engineer Frei Otto exploited the properties of soap bubble membranes to determine optimal surface shapes and applied this geometric principle to his revolutionary tension roof structure designs, such as the West German Pavilion at the 1967 Montreal Exposition.
The structure of the bubble is not limited to spherical shape, it can be designed into various shapes through the frame. In fact, it is sometimes more intuitive to actually physically generate these shapes than to calculate them using mathematical models. Therefore, soap bubbles are often seen as an analog computer that can surpass traditional computers.
Only when three or more bubbles meet, the bubble wall can intersect along a line, and the angle between these three angles must be equal to 120°.
In physics, the merging process of bubbles follows specific laws, which are called Plateau's laws. When the sizes of the two bubbles are inconsistent, the merged partition wall will bulge toward the larger side to accommodate the difference in internal pressure between the two bubbles; and where three bubbles meet, there will only be three walls. Connect along a line.
Although soap bubbles are constantly changing, there is a scientific basis for their stability. This extremely thin liquid film (about one micron thick) is very fragile and prone to rupture. The longevity of a bubble is affected by a variety of factors, including drainage caused by gravity in the liquid, evaporation rate, and even contact with objects.
The study found that using a solution of 85.9% water, 10% glycerin and 4% dishwashing liquid made the bubbles last longer.
Soap bubbles not only show their charming side in the entertainment field, but also play an important role in education. From two-year-olds to college students, bubbles can be used to teach concepts ranging from fluidity to color formation to reflection and refraction. A Swiss university professor has found that exposing children to bubbles can improve their motor skills.
With the advancement of science and technology, the application scope of soap bubbles has been extended to many aspects, including artistic creation, scientific experiments and entertainment performances. The artistry and skill of soap bubble performances have made it a part of modern art. The appeal of soap bubbles continues to attract this popular market.
Soap bubbles are not only a magical toy in the eyes of children, but also an artistic practice intertwined with mathematics and science. Their elusive nature fascinates every observer, leaving us wondering: Are there more mysteries hidden within these light bubbles?
