Summary
Bottle caps have been found in the ruins of Pompeii dating back to around 2000 BCE, but the modern crown-shaped bottle cap was invented in 1892 by William Painter and his wife in Baltimore. Painter once suffered from food poisoning after drinking a spoiled beverage and later discovered that the bottle cap was the cause. Following this incident, he decided to create a bottle cap that would prevent the contents of the bottle from spoiling.
The bottle caps used until then could not withstand the pressure of carbonated gas for long. To solve this problem, Painter spent five years collecting 3,000 bottle caps of 600 different types, studying their strengths and weaknesses, and eventually developed the screw-on cap.
However, this bottle cap had several issues. First, since it was not disposable, durability was a concern. Additionally, there was no way to tell whether the bottled beverage was new, which led some dishonest vendors to reuse bottle caps, filling expensive liquor bottles with cheap alcohol and selling them fraudulently. Moreover, the cap often failed to withstand the pressure of carbonated gas and would pop off unexpectedly.
To solve these issues, Painter decided to create a disposable bottle cap, but he repeatedly failed.
One day, his wife suggested, “What if you place a cap over the bottle like a hat and then tightly crimp the edges around it like a crown?” Inspired by this idea, he developed the crown-shaped bottle cap with serrated edges, similar to modern bottle caps.
However, there still was a problem: the number of serrations. If there were too few, the cap could not withstand the pressure, and if there were too many, it became difficult to open.
Use of Divisors
Painter considered the perimeter of the newly developed bottle cap as a circle and began sketching an appropriate number of serrations. Since a circle is 360 degrees, he calculated the divisors of 360 and spaced the serrations evenly. However, because there were too many divisors of 360, he decided to use equilateral triangles to determine the spacing.
Through this process, he discovered that by arranging eight equilateral triangles in a circular pattern, he could create 24 serrations, which provided both sufficient pressure resistance and ease of opening.
A circle has 360 degrees. Since 360=23×32×5, the total number of divisors of 360 is 24.
Since equilateral triangles had to be used, the number of triangles that could be evenly spaced while maintaining whole-number degree angles was narrowed down to 1, 2, 3, 4, 5, 6, 8, 12, 15, 20, 30, 40, 60, and 120.
Painter discovered that when using fewer than or just 6 equilateral triangles, the cap could not withstand the pressure, and when using 12 or more, opening the bottle became difficult. Thus, the optimal number of equilateral triangles, ensuring both sufficient pressure resistance and ease of opening while maintaining whole-number angles for easy manufacturing, was 8.
Since 360 ÷ 8 ÷ 3 = 15, the 24 serrations should be arranged at 15-degree intervals. This was how Painter determined the ideal number of serrations using the divisors of 360.
In 1892, the Painter couple developed the modern bottle cap, which they called the ‘Crown Cork’ due to its crown-like shape. Later, in 1894, they obtained a patent for it and earned $1,000 per day in royalties, a substantial sum at the time.
Implication
The Painter couple, using the properties of a circle and the concept of divisors, developed a bottle cap that doesn’t pop off unexpectedly, isn’t too hard to open, and is easy to manufacture. Thanks to this, we can store refreshing carbonated drinks for longer periods of time.
Underlying context – Background behind
No significant background knowledge needed
Notions – Key ideas
No significant new notions
Index – Source(s)
How to Think Like Mathematicians / 피타고라스 생각 수업
12 Angles Hidden in Bottle Caps | Divisors / 병뚜껑에 숨어 있는 각도 | 약수
Trajectory – Where I’m headed now
None
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