Talk:Taxicab geometry

From Wikipedia, the free encyclopedia

You have new messages (last change).

[edit] Paradox

A friend of mine told me of an interesting paradox contradicting the Pythagorean theorem, based in Taxicab geometry.

Let the vertices of right triangle ABC be on grid points in taxicab space, with AC being the hypotenuse of the triangle, and edges AB and BC following grid lines. Let D0(n) be the length of AC and Dt(n) be the best approximation of the length of AC in taxicab space with n grid divisions between the endpoints of the hypotenuse. Note that there will be multiple such approximations, but they will all have equal lengths.

For the sake of notation, Δx and Δy refer to the horizontal and vertical distances between end points of the hypotenuse.

D_0 = \sqrt {\Delta x^2 + \Delta y^2}

D_t(n) = \left| \Delta x \right| + \left| \Delta y \right|

Logically, as the number of subdivisions increases, the best approximation should approach the Euclidean distance. That is,

\lim_{n \to \infty}D_t(n) = D_0

\lim_{n \to \infty}\left(\left| \Delta x \right| + \left| \Delta y \right|\right) = \sqrt {\Delta x^2 + \Delta y^2}

\left| \Delta x \right| + \left| \Delta y \right| = \sqrt {\Delta x^2 + \Delta y^2}

Dt(n) = D0

However, it is clear from looking at simple cases that,

Dt(n) > D0

This is an interesting paradox, since it essentially puts the validity of the Pythagorean theorem in jeopardy. --CoderGnome 7 July 2005 18:52 (UTC)

You are wrong in assuming that taxicab distance along the hypotenuse should converge to Euclidean distance. It doesn't. It just stays constant (and much longer than Euclidean distance) no matter how much you subdivide. --345Kai 02:20, 20 April 2006 (UTC)
By my understanding, the Pythagorean theorem, as with many other triangle-related theorums, only applies in Euclidean space. A more fundamental example is that the angles of a triangle drawn on the surface of a sphere will not add up to 180°. --me_and 8 July 2005 03:37 (UTC)
Very true, but Taxicab geometry is just a special case of Euclidean geometry. The only restriction is how to move within that plane. As the number of subdivisions increases, Taxicab geometry approximates Euclidean geometry with increasing accuracy. Thus, if you take the limit as n \to \infty, it should be equivilent to Euclidean geometry. This paradox shows that this is not the case. --CoderGnome 15:00, 11 July 2005 (UTC)
Wrong. Taxicab geometry is essentially different from Euclidean geometry. SAS congruence criterion holds in Euclidean geometry, but not in Taxicab geometry. That's the whole point! --345Kai 02:16, 20 April 2006 (UTC)
The article states: "A circle in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These circles are squares whose sides are parallel to the coordinate axes." Is this really the case? I'm picturing it and I'm imagining that the "circle" would be a diamond offset by 45 degrees from the coordinate axes. Ed Sanville 03:58, 14 January 2006 (UTC)
You're right, that was my mistake, now fixed. Thank for noticing it. -- Jitse Niesen (talk) 04:13, 14 January 2006 (UTC)
As for a stated 'paradox': there is a general statement in differential geometry that the length of a curve is not greater than a lower limit of lengths of it's approximations, L(C) \leq \underline{\lim} L(C_n); this statement is clearly fulfilled in the discussed case. Note that the lengths of approximations aren't obliged to converge to the length of the original curve. Elenthel 21:58, 5 October 2006 (UTC)

I find the following two statements not understandable right now:

Taxicab geometry satisfies all of Euclid's axioms except for the side-angle-side axiom, as one can generate two triangles with two sides and the angle between the same and have them not be congruent. In particular, the parallel postulate holds.

and

A circle in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These circles are squares whose sides make a 45° angle with the coordinate axes.

--Abdull 13:37, 21 February 2006 (UTC)

In fact, the first statement is wrong: it should read Hilbert's axioms not Euclid's axioms. Euclid claimed to be able to prove the SAS property. The taxicab geometry proves that Euclid was wrong, and SAS in independent of the rest of his geometry. Someone should fix this. --345Kai 02:14, 20 April 2006 (UTC)

[edit] Line

What is a taxi-cab line, for geometry purposes? If a line is simply a geodesic, I would fear for the uniqueness of lines between a given pair of points. 128.135.96.222 00:45, 17 August 2006 (UTC)

You probably meant straight line? "If" is a good word. Many mathematical notions change their appearance or disappear altogether, if you change some underlying definitions. `'mikka (t) 01:09, 17 August 2006 (UTC)
Retrieved from "http://en.wikipedia.org../../../t/a/x/Talk%7ETaxicab_geometry_7fce.html"