![String_girdling_Earth](https://upload.wikimedia.org/wikipedia/commons/8/87/String_girdling_Earth.svg)
String girdling Earth is a mathematical puzzle with a
counterintuitive
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
solution. In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. If the string should be raised off the ground, all the way along the equator, how much longer would the string be?
Alternatively, of string is spliced into the original string, and the extended string rearranged so that it is at a uniform height above the equator. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a thin knife blade.
Solution
![String_girdling_square](https://upload.wikimedia.org/wikipedia/commons/d/d2/String_girdling_square.svg)
As the string must be raised all along the entire circumference, one might expect several kilometres of additional string. Surprisingly, the answer is 2 or around .
In the second phrasing, considering that is almost negligible compared with the circumference, the first response may be that the new position of the string will be no different from the original surface-hugging position. The answer is that a cat will easily pass through the gap, the size of which will be metres or about .
Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the
Milky Way
The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
— the result depends only on the amount it is raised. Moreover, as in the
coin-rolling problem, the shape the string girdles need not be a circle: 2 times the offset is added when it is any
simple polygon
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...
or closed curve which does not intersect itself. If the shape is
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
, 2 times the offset, times the absolute value of its
turning number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...
must be added.
p. 2436
This diagram gives a visual analogue using a square: regardless of the size of the square, the added perimeter is the sum of the four blue arcs, a circle with the same radius as the offset.
More formally, let ''c'' be the Earth's circumference, ''r'' its radius, ''Δc'' the added string length and ''Δr'' the added radius. has a circumference of 2''R'' ,
:
regardless of the value of ''c'' .
This observation also means that an athletics track has the same offset between starting lines on each lane, equal to 2 times the width of the lane, whether the circumference of the stadium is the standard or the size of a galaxy.
See also
*
Visual calculus
Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculatio ...
, an intuitive way to solve this type of problem, originally applied to finding the area of an
annulus, given only its
chord length
*
Napkin ring problem
In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a co ...
, another problem where the radius of a sphere is counter-intuitively irrelevant
References
{{reflist
Length
Mathematical paradoxes