Liu 1
Ziyuan Liu
Math 07
04/04/2013
Rope Around the Earth Puzzle
The first time that I heard of the “rope around the earth” puzzle was during middle school when the properties of circles were introduced. After learning about the circumference of a circle, we were given a famous math puzzle from William Whiston, a 16th Century mathematician.
If a person were to wrap a piece of rope around the equator of a model-sized globe, what would the length of the rope be if one wanted the rope to be exactly one foot,perpendicularly away from the equator? Better yet, think of wrapping a piece of rope around the equator of the rope around the equator of the earth. Now considering the size of the Earth, what is the length of the rope, exactly one foot from the surface, needed to circle the Earth, the globe?
Fig 1. A graphical representation of the problem[1]
The graphical representation gives a very good idea what the rope should look like. The question mark in the space between the Earth and the rope represents the distance between the two entities. Obviously, the image isn’t up to scale with a real life model. For the sake of simplicity, let the circumference of the Earth be 25,000 miles. In this case, before even contemplating a solution, one of the harder ideas to process is the fact that when the equator on the globe expands outwards, radially, by a foot, not much rope should be added; whereas, the rope circling the Earth should have a significant addition to its length. However, the result is 2π for both cases: conceptually hard to justify visually.
The approach is simple geometry shown by the following:
1.
2.
3.
4.
The first step states the circumference for both the Earth and the globe. In this puzzle, we will not worry about the units. For the second step, calculate theradii. Next, add one foot to both radii. Finally, with the calculated radii of the longer rope, calculate the new circumferences.
Both situations required an additional 2π to be added to length of the respective ropes. This puzzle is straightforward, however, the answer astounds most people. The puzzle is sometimes asked in reverse where the length of the rope is elongated and the distance of the rope from the surface is asked to be calculated.
Further consideration:
If the same puzzle were to be applied to another shape (a square), would the answer differ in any respects? Surprisingly, the answer is exactly the same. If there existed a square with a rope round around it exactly 1 unit away at every corresponding points, the length of the rope would still be 2π longer than the rope circumscribing the square. The difference between the square and the circle, in this situation, is the existence of edges. The square shown in figure 2 has four edges and corners. The corner in this case will correspond to a quarter-circle edge where each point on the quarter-circle is one unit away from the corner point. These four corners will require four quarter-circles will result in a circle of one unit radius. The resulting will have 2π circumference. As we can see from figure 2, the rounded edges of the outer box are the only addition to the original shape of the square.
Fig 2. The same puzzle is applied to a square. The outer square with rounded edges is the rope that is exactly one unit away from each corresponding point on the square.
[1]Rope around the Earth. Digital image.Math Forum. N.p., n.d. Web. 2 Apr. 2013.