Can you tie your shoes? This question seems to be asked of kindergarten children. But sometimes, tying shoelaces is also a highly technical job, and it is not easy to choose the best shoelace.
What’s the best way to tie shoelaces? Literary Xiaoqing must like the most beautiful and handsome one. But the dead rationalists see this problem as an optimization problem: Which method of tying shoelaces saves the most shoelaces and requires the shortest length of shoelaces.
Common ways to tie shoelaces
There are thousands of ways for shoelaces to pass through the shoe holes. Let’s only consider the more common ones, that is, the situation where the shoelaces shuttle back and forth between the left and right shoe holes. .
After reviewing the information, we found that there are three conventional shoelacing methods that meet the above instructions (as shown in the figure below): American tying method, European tying method, and shoe lacing method. Store-style tying method (that is, a simple tying method for shoe stores selling shoes in order to save trouble).
American System
European system
Shoe Store Quick Tie
Which of the three methods is better? In other words, which method is the most economical for shoelaces?
Let’s first determine what variables are involved in this “shoelace project”:
How many pairs of holes are there in the shoe: n
The distance between two adjacent shoe holes: d
The distance between the left row of shoe holes and the right row of shoe holes: g
According to the Pythagorean theorem and some simple geometric knowledge, we can easily calculate the needed shoelace length for these three shoelacing methods (only considering The length through the shoe hole, not considering the length required to tie the bow) are:
Which of the three functions is the largest when the values of n, d, and g are different? Pick a few numbers and bring them in to do the math. Assuming that the shoe has 8 pairs of shoe holes ( n = 8 ), the distance between the shoe holes is 1 cm ( d = 1 ), and the distance between the left and right shoe holes is 2 cm ( g = 2 ), the shoes required for these three tie methods The belt lengths are:
American style: 38 cm
European system: 40 cm
Shoe store tying method: 42 cm
If a 60cm shoelace costs 1 yuan, Using the American tie method can save you a lot of money compared to the 3rd tie method: 0.07 yuan Yuan. Try another model of shoe, almost every time the American style can beat the other two styles, the American style seems to be the best of the 3, Then Is it the best of all systems?
Shoelacing and light transmission
In fact, this problem has a very clever connection with the problem of light reflection and refraction. Light has a “quirk” in its propagation, always “cutting the corner” and taking the shortest path. We know that the angle of incidence of light is equal to the angle of reflection, but few people may have thought about the deeper reason behind this phenomenon: because only in this way, can the light always reach the destination by the shortest path. For example, in the figure below, point A is on the incident light, and point B is on the reflected light. The two points AB are connected by a straight line, “between the two points, the line segment is the shortest”, if the incident angle is not equal to the reflection angle, The light passing through two points AB may not necessarily guarantee the shortest path.
Back to the question of shoelaces, the shoelaces traveling back and forth between two rows of shoe holes are equivalent to light bouncing back and forth between two opposing mirrors. The American-style tying method is exactly equivalent to the back and forth reflection of light in a way that the incident angle is equal to the reflection angle (except for the nth shoe hole), which naturally guarantees that it is the most economical shoelace among the three methods, and it is also one of all possible methods. The shortest way to tie shoelaces.
In the picture below, each horizontal line alternately represents a row of shoe holes on the left ( A ) and a row of shoe holes on the right ( B ), and the vertical lines represent the 1st, 2nd, and 3rd rows. , …n shoe holes. We can clearly see how these three shoelacing methods pass through all the shoe holes on a flat surface. The American system is almost a straight line, while the other two are twists and turns, and the distance traveled is unsurprisingly long.
Non-mainstream tying method that saves shoelaces
Seeing this, you may ask, who is so idle to study such an unexpected problem? In fact, the above guide to tying shoelaces from a paper, the picture below is the author of this peculiar paper, Uncle John H. Halton of the University of North Carolina.
Although we have briefly introduced the content of this paper, the story is not over yet. The shortest way of tying just mentioned is only when the shoelaces are shuttled back and forth between the left and right shoe holes every time. under the assumption. In fact, after John published the paper, some people began to object that if this assumption was thrown away, there were shorter faculties than the American faculties, such as the following faculties:
If n is an even number, only ( n C 1 )( g + 2d ) is required for the length of the shoelace. It is said that the Royal Canadian Navy and Royal Air Force used this method to tie shoelaces, not to save shoelaces, but because shoelaces can be easily cut through the middle with a knife, so that in critical situations, such as When drowning, it is easy to take off your shoes to escape.
So, tying shoelaces is not easy at all!
Author: Albert_JIAO
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