Close Enough

 

The Long And Winding Road

The long and winding road
That leads to your door
Will never disappear
I've seen that road before
It always leads me here
Leads me to your door

How long are we taking about?

Pythagoras' Theorem is for a Euclidean, flat, surface.  The universe may be a hyperbolic surface.  This makes a difference for the solution of Pythagoras’ Theorem.  It does not really apply on spherical surfaces such as the earth, when the distances that are the sides of the triangle are large compared to the radius of the spherical surface (for example, the Earth).  On Earth, when the distances are small, the difference between the results of Pythagoras’ Theorem and the Great Circle Distance on the Earth is trivial. The same seems to be true for a hyperbolic universe.

For 1/3 of the size of the universe the difference between the hypotenuse on a hyperbolic surface of the universe and Pythagoras’ Theorem is less than 6%.  Since the size of the universe is approximately 14 billion light-years, until the sum of the squares of the distances exceeds 22 trillion billion miles the difference is less than 6%.  The distance from the Earth to the Sun is 93 million miles, 8.3 light-minutes, 1.6*10^-5 light-years.  The distance from our solar system to the Andomeda galaxy is 2.5 million light years, so we are talking about much more than intergalactic distances before there is an appreciable difference.  In fact for distances of a thousand miles on each side of a right triangle, the difference between a hypotenuse calculated with Pythagoras' Theorem and one for a hyperbolic surface is far less than 0.01%.  So don’t throw away Pythagoras' Theorem.  It is simpler and has very little error from the hyperbolic solution at most distances that you are likely to encounter.  So what is long to you, is insignificant to the universe.