Re: Euclidean, Hyperbolic and Elliptic Geometry

[Brian]

Being the Brian that he is talking about, I did not ask that. However your questions are still valid. There is no universal rules that apply because there are no universal postulates that must be included a geometry. The main difference between Euclidean geometry and Hyperbolic and Elliptic Geometry is with parallel lines. In Euclidean geometry there is an axiom which states that if you take a line A and a point B not on that line you can draw one and only one line through B that does not intersect line A. In hyperbolic geometry you can draw an infinite amount of lines through point B that do not intersect line A. In elliptic geometry there is no such line though point B that does not intersect line A. Euclidean geometry is generally used on medium sized scales like for example our planet. On extremely large or small scales it get more and more inaccurate. This is all off the top of my head so please correct me if I am wrong.

Here is a Wikipedia URL which has information about Hyperbolic functions. http://en.wikipedia.org/wiki/Hyperbolic_function

[mathnerd4life]

can someone tell me some universal rules that apply all the time no matter what type of geometry?

My friend Brian asks: Is there a specific ratio between hyperbolic sines and euclidean sines?

[mather4ever]

There are no universal postulates. Parallel lines are different in hyperbolic and also euclidean, and also a hyperbolic triangle can have the sum of degrees as 210. As for ratio between euclidean sines and hyperbolic sines, i truly have no ides.

[Kirby Urner]

Right, no universal rules, just have to make up a game
that's self-consistent enough to have some appeal.

Whereas fiddling with the 5th postulate (about parallel
lines) has been a standard way to branch to non-Euclidean
geometry, another route was suggested by Karl Menger,
dimension theorist, in his The Theory of Relativity and
Geometry (1949).

Per this essay, we might define points, lines and planes
to be "lumps" (as in "of clay"), not distinguished by their
"dimension number" (standard ideas of "linear independence"
between height and breadth and width, per French res extensa
philosophy (ala Descartes)) is what's being undermined
(swapped out) in this picture (we keep an idea of energy
though, say "energy has shape").

http://en.wikipedia.org/wiki/Res_Extensa

By this time, we have a pretty coherent geometry built
up, including lots of input from Euler's topology,
sometimes share it with kids under the marketing label
of 'Claymation Station' (for obvious reasons).

This way of thinking about geometry doesn't really
interfere with most Euclidean proofs, so is hardly that
radical a departure, just a different way of thinking
and talking (alternative nomenclature, e.g. MITE for
"minimum tetrahedron"). At Saturday Academy, we think
of this as a sub-branch of Gnu Math, i.e. teach it
under that category, running FOSS on commodity hardware.
(FOSS = free and open source software).

More on math-thinking-l (open archive), also you can read
about my meeting with Dr. Livio on this topic at our
recent get-together @ Linus Pauling House in Portland,
Oregon.

http://coffeeshopsnet.blogspot.com/2009/02/glass-bead-game.html

Karl Menger's daughter Eve is a member of this group,
although she wasn't able to make this particular meeting.

Kirby Urner
Chief Marketing Officer
Coffee Shops Network

[K. E. Pledger]

> Can someone tell me some universal rules that apply

> all the time no matter what type of geometry?

Euclidean, hyperbolic and elliptic geometry have quite a lot in common. For example, the Euclidean criteria for congruent triangles also apply in the other two geometries, and from those you can prove many other things.

To give a more historical answer, Euclid I.1-15 apply to all three geometries. Euclid I.16 and various later propositions fail in the elliptic plane. Euclid I.29 and various later propositions fail in the hyperbolic plane.

> My friend Brian asks: Is there a specific ratio
> between hyperbolic sines and euclidean sines?

Are you willing to use complex numbers? If so, then
sinh(iz) = i.sin(z),
cosh(iz) = cos(z).

Ken Pledger.