Why is hypotenuse path shorter than sum of legs?

[valentin tihomirov]

The question may be dumb but it intrigues me from the early childhood. The
shortest distance between points A(xa, ya) and B(xb, yb) is hepotenuse
d=sqrt(xd^2+dy^2) but if we can move only strightly (up and right) we are
doomed to dx+dy distance which is up to sqrt(2) longer. Is there any
understanding of the automata cells arrangement (lattice) in space and how
the signal travels?

[Gerard Westendorp]

valentin tihomirov wrote:

There is a self consistent geometry called taxicab geometry in which
length is dx + dy instead of sqrt(dx^2 +dy^2). So there is no
mathematical necessity for Pythagora's theorem. Euclidean space needs
to be introduced by postulating the Euclidean metric.

One cellular-like process that mimics approximately circular
symmetry is a random walk on a square lattice.
Start somewhere, and throw a dice to go up, down, left, right.
If you repeat this many times, the probability of getting
in a place (x,y) is proportional to exp(-(x^2+y^2).
You can do this experiment on a computer.

So if a taxicab sometimes takes a wrong turn, than
taxicab geometry becomes more and more Euclidean as then
taxi drive gets more "drunk".

Gerard

[exama...@gmail.com]

So the l2 norm comes off as some sort of a statistical property of
small random computations in this system? Interesting! Do you think
there could be some yet unknown theorem as it relates to such
geometries?

Regards,

--
Eray

[valentin tihomirov]

Very nice explanation and in accordance to minimal action principle. Light
travels from sourse to destination using the fastest path. It would be
interesting to hear if anybody explains the implulse of a roaming light beam
in the same manner (why probability to move in some direction is higher).
May be the special relativity is the result of such a movement? In the
Plamn's/Zuse thesis it was clamed that finity of light speed can be
explained in DP but I did not found the mechanics.