The Gap Theorie

Hi, I had a conversation with a friend and he brought up this story where one has for example, a square of 1 m2, each side is 1 meter long, resulting in a 4 meter long border. Now, one takes a rigid string that does not stretch at all nor does it shrink, one lays it around the outside of the square and adds 10 cm or 0.1 m to its length, resulting in a 4.10 meter long string.
He now claims that there will be a gap between the 4 meter border of the square and the string of 1.25 cm all around at any point, this also is, according to him, true no matter how long the sides of a square, rectangle or the circumference of a circle are, he says that even if one lays a string around the Equator and then adds 10 cm, this would result in a gap of 1.25 cm all around the Globe between the surface and the string. Can that be true? :dubious:

c = pi*2r

for any 2 circles, the “gap” between the 2 will be equal to the difference between their radii.

r = c / 2*pi, so delta r = (c - c) / 2 * pi

Let’s call pi 3.14 for sake of example

delta r = (c - c) / 6.28

10 / 6.28 = 1.59

So, the idea is there - that the starting size of the body is irrelevant, since the relationship is linear, but the math is off a little because his example applies to rectangles and not circles, which have a different relationship between perimeter and radius than rectangles have betwen perimieter and the length of a side.

eta: I am not a mathemagician.

The story is usually told by adding a yard to a string laid around the earth’s equator. Then the string is six inches above the ground everywhere, because 3 feet divided by 2 pi is about 6 inches.

It’s the factor of 2 pi or 6.28 that’s the heart of this surprising result.

Now if you imagine a sphere with that kind of gap, you’ll get much different results and working through the math will let you understand the perimeter thing better.

The “string around the earth’s equator” is a classic problem. Here’s one online explanation.