Diagonal Paradox and Coastline Paradox

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DiagonalParadox

Consider the length of the diagonal of a unit square as approximated by piecewise linear steps that may only be taken in the right and up directions. Obviously, the length so obtained is equal to half the perimeter, or 2. As the number of steps becomes large, the path visually appears to approach a diagonal line. However, no matter how small the steps, if they are constrained to be only to the right and up, their total length is always 2, despite the fact that the length of the diagonal is sqrt(2).

This apparent paradox arises in physics in the computation of Feynman diagrams, where it has implications for the types of paths that must be included in order to obtain a good approximation to physical quantities.

Determining the length of a country’s coastline is not as simple as it first appears, as first considered by L. F. Richardson (1881-1953) and sometimes known as the Richardson effect (Mandelbrot 1983, p. 28). In fact, the answer depends on the length of the ruler you use for the measurements. A shorter ruler measures more of the sinuosity of bays and inlets than a larger one, so the estimated length continues to increase as the ruler length decreases.

References : Feynman Diagrams

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