Can the Pythagorean Theorem be extended like this?What is the significance, if any, of the fact that 3 3 + 4 3 + 5 3 = 6 3 ?How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.

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Can the Pythagorean Theorem be extended like this?What is the significance, if any, of the fact that      3    3   +       4    3   +       5    3   =       6    3  ?How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.

Josie123 · Accepted Answer

This is, in fact, the analogous expression to the distance between two points x and y in the space             R        4  . However, the &quot;euclidean&quot; (or &quot;pythagorean&quot;) distance in n dimensions is given by:            ∑              i        =        0                    n              (          x      i        −          y      i              )      2      There is a generalization, called Minkowski distance:            (                  ∑              i        =        0                    n                    |              x      i        −          y      i                      |                    1                  /                p                                )              p  And you recognise here your expression with p=1/3.In conclusion, this is an integer expression for the distance between two points in             R        4  , with the 1/3-Minkowski metric.

Can the Pythagorean Theorem be extended like this? What is the significance, if any, of the fact th

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