Non-Euclidean geometry

I'll raise you one by suggesting that all of geometry is part of our culture because we are Greek.

The very idea of thinking about geometry, whether it's Euclidean or non-Euclidean, is a Western idea.

Show me one evidence of geometry in any culture other than one that came through the Middle East and Greece.

And I'll give you a dollar.

Why? Because the cube has nine sides.

It's a non-Euclidean cube.

Or non-Pythagorean, I guess.

Interesting.

For all we know, an ephemeral insect that dies in a day may live a longer life than a Galapagos turtle that exists for two centuries.

what mark twain said about classical music applies also to science it is not so bad as it sounds the thing that the chemist calls sodium chloride other folks call salt and so does he when he is off duty don't let the scientist bluff you by his polysyllabic propensity just try to see what he means by such language Now, what these new-fashioned non-Euclidean geometricians call the four-dimensional space-time continuum is essentially the same system of reference as you have used ever since you could toddle.

Minkowski did not invent it.

Everybody thinks that way, unless he is an idiot.

CHAPTER VII.

NON-EUCLEDIAN GEOMETRY.

the idea that space itself may be curved and that the axioms and assumptions on which our geometry since the time of euclid have been based may not be absolutely and exactly and eternally and universally true has been diligently studied during the last fifty years the russian lobaczewski the hungarian bolyai and the german riemann have developed systems of geometry by starting from premises the opposite of those of euclid and these systems are just as logical and consistent with themselves as the ordinary or euclidean geometry These non-Euclidean geometries were at first commonly regarded as mere freaks of the mathematical imagination but they have already proved valuable in leading to a reconsideration of the fundamental principles of our thinking and, if Einstein is right, they may be necessary to explain physical phenomena.

it is hard for the mathematician to discover anything useless a distinguished american mathematician in announcing a new theorem exclaimed and thank heaven no possible use can ever be found for it but whatever it was he made a rash boast for nowadays the mechanic trades on the heels of the mathematician and uses imaginary quantities actual only in the fourth dimension, like square root of minus one, in figuring out the winding of his dynamo.

End of section 30 Section 31 The Possibility of a Finite and Yet Unbounded Universe But speculations on the structure of the universe also move in quite another direction.

The development of non-Euclidean geometry led to the recognition of the fact that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience.

Raymond Helmholtz These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

In the first place, we imagine an existence in two-dimensional space.

We can imagine innumerable laws which would serve the same purpose, without our being able to state a reason why one of them is to be preferred to the others, for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton.

End of section 30 Section 31 The Possibility of a Finite and Yet Unbounded Universe but speculations on the structure of the universe also move in quite another direction the development of non-euclidean geometry led to the recognition of the fact that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought, or with experience? Raymond Helmholtz These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

In the first place, we imagine an existence in two-dimensional space.

Flat beings with flat implements, and in particular, flat rigid measuring rods, are free to move in a plane.