Ant on a Rubber Rope

An ant crawling on a rubber rope can reach the end even if the rope stretches faster than the ant’s crawl.



The Ant on a Rubber Rope paradox is a mathematical puzzle with a counterintuitive solution. The puzzle asks,  “If an ant is crawling along a one-kilometer long, taut, rubber rope at one centimeter-per-second speed. If the rope stretches at a constant speed of one kilometer per second, would the ant ever be able to reach the end?”

Though it may seem impossible, the answer is “yes,” because as the rope stretches, it also moves the ant forward. So, the ant can reach the end of the rope, though it would take a very long time. However, if the rope stretches with an increasing speed, then the ant won’t be able to reach the end. A similar paradox is the Achilles and the Tortoise paradox in which Achilles gives the tortoise a head start but always finds himself running to cover the distance the tortoise had already moved.

The puzzle can be applied to the question of whether light from galaxies extremely far away would ever reach Earth considering the universe is expanding. If the expansion had been at a constant rate, then just like the ant, the light could reach us. However, since the expansion is accelerating, it might not.

2. Coastline Paradox

As the unit of measurement grows smaller the length of a coastline increases to infinity.



The Coastline Paradox is a counter-intuitive observation about how coastlines do not have a well-defined length. The phenomenon was first observed by Lewis Fry Richardson who was searching for a relationship between the probability of two countries going to war and the length of their common border. As he collected data, he found that there is a considerable difference between various published lengths of the international borders. Following the observation, Richardson demonstrated that the measured length of coastlines, or any other natural features, increases limitlessly as the unit of measurement becomes smaller.

Depending on the method used and how much cartographic generalization has been done, the measured length of the coastline changes. This is because of the fractal-like properties of coastlines. As any landmass has detail no matter how closely you look at it, at kilometer scale or at microscopic scale, there is no well-defined perimeter to measure the landmass.

3.  Grelling–Nelson Paradox

This paradox concerns the applicability of a word to itself. The word “heterological” means “inapplicable to itself.” So, is “heterological” a heterological word?



The Grelling–Nelson Paradox is an antinomy, referring to a real or apparent mutual incompatibility or a self-contradiction. It was proposed by Kurt Grelling and Leonard Nelson in 1908. If the answer to the above question is no, then the word “heterological” does describe itself which would mean the word “heterological” is heterological, making the answer a contradiction. If the answer is yes, then “heterological” does not describe itself, but that would mean “heterological” is not a heterological word, which again is a contradiction.

A variation of the Grelling-Nelson Paradox is the Barber Paradox. There is a barber who shaves all those, and only those, who do not shave themselves. So, does the barber shave himself? Answering the paradox either way leads to a contradiction. The paradox is based on Russell’s Paradox or Russell’s Antinomy formulated by British mathematician Bertrand Russell to show that the formalizations of naïve set theory by George Canto led to contradictions.

I'm expecting Twin Paradox

This one pass my mental capacity. I’ll come back later to try and make sense of it.