Riemann space

[Ya Sam]

Thanks!

I'm green with envy, why didn't I study maths properly!


>From: "Anville Azote" <anville.azote at gmail.com>
>To: "Ya Sam" <takoitov at hotmail.com>
>CC: pynchon-l at waste.org
>Subject: Re: Riemann space
>Date: Tue, 28 Nov 2006 22:11:53 -0500
>
>On 11/28/06, Ya Sam <takoitov at hotmail.com> wrote:
>>Another appeal to the Renaissance men (and women) on the list, who dig 
>>both
>>sciences and humanities. I read a dozen of articles trying to get an
>>approximate layman's notion of what it is and I have to confess that I 
>>don't
>>get it. Is there a simple or even simplistic, but nonetheless digestible 
>>way
>>to explain it using some easily perceptible images? I'm helpless with
>>maths...
>>
>
>First, start with the notion of "space".  What does "space" mean to a
>mathematician?  Fundamentally, in order to behave in a reasonable way,
>a space must incorporate some notion of distance.  I can think of at
>least two ways to assign distances on the Earth's surface, for
>example:  for any points A and B, the distance between them can be
>either the straight-line separation through the body of the planet or
>the "great circle" course along the surface.  (A great circle is a
>circle which divides a sphere into two equal halves; segments of great
>circles are the shortest possible paths between points on the sphere
>if you don't leave the sphere.)
>
>For any two points A and B, we'd like a rule for calculating the
>distance between them.  Think of it as a machine, call it "g", which
>takes two points and spits out a number.  We write this process as
>"g(A,B)" -- feed points A and B to g and get out a number which
>measures the separation between them.  Mathematicians like to require
>such rules, called "metrics", to have a few key properties:
>
>First, the value of a metric must always be greater than or equal to
>zero.  (Negative distance?  Outside of Negativland, what could that
>possibly be. . . .)  Second, the metric function must only work out to
>zero if the two points are identical.  In symbols, one would say that
>g(A,B) = 0 if and only if A = B.  Also, it shouldn't matter in which
>order we feed the points to the metric machine (the distance from
>Chicago to Tunguska is the same as that from Tunguska to Chicago).  We
>write this as g(A,B) = g(B,A).
>
>Finally, we require that the metric g obey a rule called the "triangle
>inequality".  Given any three points A, B and C anywhere in our space,
>g must satisfy the relationship
>
>g(A,B) + g(B,C) >= g(A,C).
>
>This says that the distance directly between A and C can only be less
>than or equal to the total distance you travel if you go from A to
>some other point B and then from B to C.  In our ordinary, familiar
>Euclidean space, we can see this apply:  the only way the journey from
>A to B and thence to C can be the same length as the straight shot
>from A to C is if B is on the line between A and C.  Otherwise, g(A,B)
>+ g(B,C) will be bigger than g(A,C).  Imposing the triangle inequality
>is a way to get some of the properties of familiar space to apply to
>much more esoteric mathematical constructions.
>
>It might be a fun exercise to show that the "Manhattan distance"
>qualifies as a valid metric.  Consider a two-dimensional plane with x
>and y coordinates, x measuring east-west (say) and y measuring
>north-south.  The Manhattan distance between A and B is
>
>g(A,B) = |Ax - Bx| + |Ay - By|
>
>(where "Ax" means the x-coordinate of point A, etc.).  Also known as
>the "taxicab metric", this represents the distance you have to travel
>to get from A to B if you can only move along the Manhattan street
>grid, with no diagonals.  Another, equally valid metric is the
>Pythagorean,
>
>g(A,B) = sqrt((Ax - Bx)^2 + (Ay - By)^2),
>
>which we can visualize by making the line segment AB the hypotenuse of
>a right triangle and applying the Pythagorean theorem.
>
>A "manifold" is, loosely speaking, an object made of pieces which look
>on the small scale like Euclidean space.  Imagine stitching together
>several sheets of rubber, which you can flex and twist however you
>like.  If we can define coordinates (latitude and longitude, say) on
>each piece and have the coordinate grids on adjoining pieces mesh
>together in a nice way (which technically has to do with derivatives
>being well-behaved), then we have a manifold.
>
>Assorted readings:
>
>http://scienceblogs.com/goodmath/2006/08/metric_spaces.php
>
>http://scienceblogs.com/goodmath/2006/10/meet_the_manifolds.php
>
>And in fact, the rest of Mark Chu-Carroll's topology series:
>
>http://scienceblogs.com/goodmath/goodmath/topology/

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