Talk:Taxicab geometry

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1 Paradox
2 Line
3 Circles vs. "Circles"
4 Biangles

[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 $D 0 (n)$ be the length of AC and $D t (n)$ 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, $Δ x$ and $Δ y$ 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,

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

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

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

$D t (n) = D 0$

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

$D t (n) > D 0$

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)

[edit] Circles vs. "Circles"

These circles are squares... It appears that we are claiming that circles are squares. Here "circles" refers to manhattan geometry, whereas "squares" refers to euclidean. Could we make this clearer, perhaps with scare-quotes, like this: A "circle" in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These "circles" are squares ...

I might also suggest using "diamond" instead of square, although that's hardly a mathematical term. -- Comment unsigned

I have added more description and an image, which should make the meaning clearer without resorting to scare-quotes. Circles in taxicab geometry are still circles, despite their appearance. -- Schaefer (talk) 23:04, 19 May 2007 (UTC)

Schaefer explained it well but to some this might still seem confusing. I suggest that we say that a circle in Taxicab geometry a circle, though still being a circle by definition, looks like a Euclidean-style square. Also, because this concept could be seemingly contradictory to some I suggest that we explain it like this or something like it: by definition, a circle is a shape in which all points are equidistant from a single fixed (centre) point. Since in Taxicab geometry one is restricted to a street grid, a circle must have it's lines avoid the blocks. Therefore, if you wish to create a circle five (5) units in all aspects from the centre point, one can first plot a point five units straight up from the centre point. But one can also plot a point up four (4) from the centre point and over one (1) coordinate to the right. Likewise, one can plot a point up three (3) units and over two (2) to the right, etc. All these coordinates lie on the radius of the circle creating a square (for Euclidean geometry) but still fulfilling the definition of a circle needing to be equidistant from the centre point.

Also, shouldn't we include the fact that even though circles in Euclidean geometry can only intersect at a maximum of two points without becoming the same circle, circles in Taxicab geometry can intersect an infinite number of times as long as they are infinitely large. Thanks for your opinions (in advance)! -76.188.26.92 20:02, 1 June 2007 (UTC)

How about this as a diagram of the above suggestion? ASprigOfFig 22:29, 1 June 2007 (UTC)

Why did it not upload correctly? It looks okay in maximum resolution. Could someoene please help? ASprigOfFig 22:36, 1 June 2007 (UTC)

[edit] Biangles

ASprigOfFig, I'm removing the section you added on biangles, more common known as digons, because as far as I can tell it is incorrect. The figures you depict in this image are not digons. Digons, like any other polygon, consist of points joined together by line segments, not arbitrary paths. The fact that the paths are the same length in your figures is irrelevant. Just as in Euclidean geometry, in taxicab geometry there is only one possible line segment joining two points—the difference between the two geometries is that in the Euclidean, that line segment is also the unique shortest path connecting the endpoints, whereas in taxicab geometry there are infinitely many paths with the same shortest possible length between the points. The figures you show each have two points connected by paths of equal length, but not connected by two line segments. A digon in taxicab geometry is degenerate (it necessarily encloses zero area) just as with Euclidean geometry.

Lines in taxicab geometry do not literally "go around the blocks" as you say in your description. Taxicab distance can be defined between points with non-discrete Cartesian coordinates (analogous to having a point in Manhattan at the intersection of 4.28th Avenue and Pi/2 Street). There are no actual "blocks", at least not any blocks of finite size. The idea of city streets laid out in a grid is more of a visualization aide: All of the shortest driving paths between any two points in a city with a grid layout are paths with a length equal to the length of a straight line segment connecting those points under taxicab geometry, but the path itself is not a single line segment by virtue of having the same length. In the figure to the right, the red, yellow, and blue paths consist of two, four, and twelve line segments, respectively. This is true in both Euclidean and taxicab geometry. The total length of the line segments of any one of these colors is twelve, again in both geometries. The green path consists of one line segment, once again in both geometries. The only difference between the two is the length of the green line: 6 * sqrt(2) in Euclidean and 12 in taxicab. You can visualize why the green line has length twelve by imagining that it zig-zags like the blue line, and then mentally decreasing the size of the zig-zags and seeing how it gets closer and closer to the path of the line without changing its length. However, the actual line doesn't zig-zag. It is a unique straight line connecting the points, but has a length defined on metric that behaves as if it were composed of microscopic zig-zags. I hope this makes things clearer. -- Schaefer (talk) 23:54, 1 June 2007 (UTC)

Okay, thanks. I was really just going of (but not as a copyright, mind) a book I have which briefly describes "biangles" and that was basically the definition they gave (and they had five of those nine pictures). Sorry. This now puts doubt into the following belief (fro the same book) though it makes sense. It is the belief that as long as a circle is sufficiently large, and has another circle inside it sufficiently large, the two circles can intersect (or meet, at least) an infinite number of times- just like the following picture shows: