What wrapping rope around the Earth reveals about the limits of human intuition

So, my intuition was off, like the article describes. The article doesn't mention it, but the rope would be nearly 6 inches off the ground. After I did the math, this makes trivial sense, and I feel rather moronic for both knowing the equations and not intuiting it. π is the ratio between circumference and diameter. The planet is a red herring.

If you're still not seeing it, I'll try to explain:

Let's say the circumference of the earth in yards, c, is 43,825,760:

  >>> c = 43_825_760

Remember C = πd, so the diameter of the earth is d = C/π:

  >>> import math
  >>> d = c / math.pi

Add 1 to c (our 1 yard); this is the circumference of the rope plus slack, then divide by π to get the diameter of the rope. Subtract earth's diameter above.

  >>> ((c+1) / math.pi) - d
  0.3183098863810301  # yards. ~11.4 inches

Remember that this is the change in diameter. From the perspective of a person looking at this rope, you'll only see half of that (the other half is on the other side of the earth), so it's really only ~5.7 inches (The "nearly 6 inches" above).

Here's the thing that I missed thinking this through. That 0.318 number? If we take the "1 yard" we added in slack (circumference) and we make a ratio between it and the change in diameter (0.318) — the ratio between circumference and diameter is called π. The math:

  >>> 1/0.3183098863810301
  3.1415926516431183

Adding 1 yard slack will make that much difference on any planet. You could be on Jupiter, or the Sun, and this would work.

A different way to think about this is if you pulled an inelastic chain tight to the earth, and closed the loop you would have the starting condition. Then, standing where you are you cut the chain, grabbed an extra yard of the chain and spliced it to the touching ends of the starting chain lying on the ground, laying that extra yard folded perpendicular to the starting chain wrapped around the earth. That extra chain added is now sticking out half a yard from the starting chain. When you think about it that way, and I ask you “would that extra yard make a difference?”, where does your intuition lead you? With that visual, I would say, that is a ton of slack, I could loop it around my body. I think this is really more about how you frame the question than some flaw in intuition, although I’m not trying to say humans have flaws with our intuition.

Pretty clever.

My instinctual reaction was to say, well, no, it's trivial, of course not, it's a tiny length compared to the Earth's size.

But mathematically, well, c = 2πr, right? It's another 15cm or so on the radius, if it were perfectly rigid.

In practice though it'd flop down (perpendicular to the plane of the circle) and if that were fairly well distributed I'd assume the 15cm wouldn't be noticeable, and friction+gravity would make it difficult to pull up?

My brain went the other direction, but my intuition still went to a correct answer. A rope wrapped around the Equator, with the exact length required to be around it, would rest on the ground and be tight. Adding an extra yard would make it loose. Assuming it was frictionless and completely inelastic, anyone could pull it and amass a yard length of length of the rope at their location, and that yard could be distributed to any point along the Equator, which would be noticeable.