Talk:Taxicab geometry

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, $\Delta x$ and $\Delta y$ 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)

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)

I don't see how the observation about Hilbert's axioms can be correct. Axiom II.2 says, "If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D." Is a point lying between A and C a point that lies on any path from A to C or a point that lies on all paths from A to C? If the latter, then there is no such point, contradicting the axiom. If the former, then take a look at Axiom I.2: "Any two distinct points of a straight line completely determine that line; ...." Which line do A and B determine? Certainly, the line with endpoints A and B. But we obtained B by selecting it as a point on line AC, so A and B must completely determine line AC. So we have A and B completely determining two distinct lines, a contradiction. —Largo Plazo (talk) 17:17, 5 September 2008 (UTC)

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)

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:

Is this corect? Thanks. In the most sincere manner, -A Sprig of Fig 00:41, 2 June 2007 (UTC)

Why Euclid Axioms Is Unacceptable

Hello, Jitse Niesen and the three editors that opposed my edit! I have received your message, Jitse Niesen, on my Talk Page. I will abstain from posting "Euclid axioms" into the Taxicab Geometry page, but I would like to take you up on your offer. Please let me know why Euclid is not a reliable source and why three editors have opposed my motion, Jitse Niesen and the three editors. Thank you for your time and effort.

(Rallybrendan2006 (talk) 05:19, 18 June 2008 (UTC))

Euclid is a reliable source, but that is not the problem. There are a few sets of axioms of geometry, and Euclid's is only one of them. David Hilbert's is more modern (see Hilbert's axioms). Your high school teacher may not be aware of that. Bubba73 (talk), 02:15, 19 June 2008 (UTC)

Euclid's axioms, as usually enumerated, don't even talk about distances. Euclid also assumes without stating as an axiom a lot of other properties of geometry (e.g. that any two circles will intersect in zero or two points). Hilbert's axioms were an attempt to reduce everything to fundamental principles without making such assumptions. I don't think it makes sense to attempt to interpret taxicab geometry using Euclid's axioms, because it contradicts some of Euclid's unstated assumptions — if you define a circle in taxicab geometry to be a set of points at equal distance from a center point, it won't have the properties Euclid expects circles to have, and Euclid's proofs won't go through, not because of a violation of one of the explicit axioms but because taxicab circles aren't really like circles (two of them might intersect in a line segment rather than a pair of points). Because everything in Hilbert's axioms is stated explicitly, one can examine the axioms and determine which ones do and which ones don't still hold in taxicab geometry without running into the same sort of difficulty. —David Eppstein (talk) 02:59, 19 June 2008 (UTC)

Well said. Can't a pair of circles intersect at one point? Bubba73 (talk), 03:08, 19 June 2008 (UTC)

Oh yeah, that too. But not more than two. —David Eppstein (talk) 03:46, 19 June 2008 (UTC)

Hello. Thank you for your explaination, Bubba73, and your extremely detailed paragraph, David Eppstein. Bubba73, since you said Euclid is a reliable source, why is it not possible to list both Hilbert's & Euclid's axioms. I see no problem in that solution. Also, you didn't really make the probelm of putting Euclid's axioms in very clear.

(Rallybrendan2006 (talk) 04:49, 19 June 2008 (UTC))

In Wikipedia, a "reliable source" is a published source of information about a topic that is considered authoritative. Euclid's book would be a reliable source for Euclid's axioms of geometry. But he is not an authority on geometry from a modern perspective. As David says, he makes several assumptions that should not be made. David explains very well about how Euclid's formulation doesn't work for Taxicab Geometry - the lack of a notion of a distance and the assumptions about circles that do not hold true in taxicab geometry. Bubba73 (talk), 04:58, 19 June 2008 (UTC)

One problem with the sentence "Taxicab geometry satisfies all of Hilbert's axioms and Euclid's axioms except for the side-angle-side axiom, as one can generate two triangles each with two sides and the angle between them the same, and have them not be congruent" is that side-angle-side is not an axiom in Euclid's Elements, but a theorem (Proposition I.4). The problem with Euclid's work, as David says, is that it is not considered rigorous nowadays. Taxicab geometry satisfies the list of ten axioms that I'd take to be "Euclid's axioms", but some of Euclid's theorems (like side-angle-side congruence) are wrong in taxicab geometry, because the proofs use unstated axioms that taxicab geometry does not satisfy.

I guess that some people may not be familiar with what Hilbert's axioms are. Perhaps we can make the text clearer by explaining this. For instance, we could write "Taxicab geometry satisfies all of Hilbert's axioms (a possible axiomatization of Euclidean geometry) except …" or perhaps even "Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclid's axioms) except …". -- Jitse Niesen (talk) 15:55, 19 June 2008 (UTC)

I am perfectly fine with Jitse Niesen's suggestion. I know the Euclid is not perfect, but just because a few things from Euclid doesn't apply to Taxicab geometry doesn't mean that we have to leave Euclid out completely. By the way, thank you Jitse Niesen, for telling me about the discussion page for each forum. I'm sorry for bothering and wasting everyone's time on Wikipedia fixing my posts; I'm new and I'm unaware of the discussion page. I think it is neat where you can debate about a topic and come to a soultion. Well, anyways, thanks for reply to my posts the past few days and I hope we can resolve this situation soon (hopefully with Jitse Niesen's suggestion). (Rallybrendan2006 (talk) 16:07, 19 June 2008 (UTC))

Well, Euclid's axioms don't really matter as far as taxicab geometry is concerned, so I don't see any point in mentioning them in this article. Bubba73 (talk), 17:37, 19 June 2008 (UTC)

IF some of Euclid's axioms do apply and work in taxicab geometry, you should at least list his name the way Jitse Niesen did in his fabulous example, not leave him completely out of the picture.

(Rallybrendan2006 (talk) 22:40, 14 July 2008 (UTC))

I have gotten PRIOR APPROVAL (see posts above) and I am TIRED OF IT GETTING SWITCHED BACK EVERYDAY, so if you would leave it the way I have changed it, that would be great. THANKS!

Rallybrendan2006 (talk) 23:28, 1 May 2009 (UTC)

Hilbert's Axioms are a formalization of Euclidean geometry, not of Euclid's Axioms. Bubba73 (talk), 23:32, 1 May 2009 (UTC)

Bubba73, would it be fine if I posted the same thing, except change axioms to geometry?

Rallybrendan2006 (talk) 23:37, 1 May 2009 (UTC)

I believe it would be correct to say that Hilbert's axioms are a formalization of Eucildian geometry, but for months you have insisted on Euclid's axioms, which is not correct. Bubba73 (talk), 23:40, 1 May 2009 (UTC)

It doesn't even seem to make sense to say that Hilbert's axioms are a formalization of Euclid's axioms. At least I don't understand what that would mean. Bubba73 (talk), 23:44, 1 May 2009 (UTC)

It was Jitse Niesen who proposed that. But I guess Euclidian geometry makes sense, compared to Euclidian axioms. Can I change it to Euclidian geometry, with your permission?

Rallybrendan2006 (talk) 23:46, 1 May 2009 (UTC)

That is OK with me. This is what I think is the misunderstanding, and it came to me after you mentioned Hilbert formalizing Euclid's axioms. Hilbert's axioms are not simply a restatement of Euclid's axioms in more formal language or more formal terms. That would be the common use of the word. Hilbert's axioms are a formalization of what we call Euclidian geometry in the mathematical/technical sense of a formal system. Bubba73 (talk), 00:26, 2 May 2009 (UTC)

Yes, this has became clear to me now. I have already changed it. Thank you for your time and efforts. FINALLY, after 9 months....haha!

Rallybrendan2006 (talk) 00:31, 2 May 2009 (UTC)

Quasimetric

If a given taxicab geometry has one-way streets, it then has a quasimetric distance function. In other words, the minimal-distance path from point A to point B comprises a different set of street blocks than the minimal path from B to A, so d(A,B) ≠ d(B,A). Should this be mentioned in the article, perhaps as an "extended taxicab geometry"? — Loadmaster (talk) 16:46, 11 June 2009 (UTC)

Anyone know what an angle is for L1?

I have a small problem with this section:

Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry) except for the side-angle-side axiom, as one can generate two triangles each with two sides and the angle between them the same, and have them not be congruent.

As far as I know, angle is only defined for Rn + euclidean distance. What is angle for Rn + manhattan distance?

Also, why are we even mentioning that it doesn't satisfy a Hilbert's axioms, if we make the point of stating it is a formalization of Euclidean geometry? Isn't it a bit unsurprising? Don't you only get Euclidean geometry when you're working with the Euclidean distance? 141.214.17.5 (talk) 19:59, 27 July 2009 (UTC)

Hexagonal tiles

What about taxicab geometry on hexagonal grids? --77.56.90.38 (talk) 08:25, 23 August 2009 (UTC)

Naming

There appear to be many names for this concept (taxicab, Manhattan, etc.) - and their use is mixed throughout the article. Does anyone know the 'correct' term (is it taxicab, as the article name suggests? Which name came first? Why did the others emerge? Which is more used in academic journals?)? Whichever it is, it should become consistent throughout. --129.234.252.67 (talk) 11:39, 13 November 2009 (UTC)

Isnt Taxicab Distance just a special case of finite network/graph distance?

It just counts the number edges from one node to another, in the special case of a checkerboard grid? 03:28, 11 February 2010 (UTC)

For points with integer coordinates, yes. But the taxicab distance can be defined for any two points in the plane, not just those with integer coordinates. —David Eppstein (talk) 03:46, 11 February 2010 (UTC)

error

The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two points in the borough to have length equal to the points' distance in taxicab geometry.

This is not true for all possible combinations of points, actually the Manhattan distance can be shorter then the shortest path a car could take. Examples for this:

In both cases, the streets are represented by the thin black lines, Manhattan distance is represented by the red line and one of the shortest paths for a car is represented by the thick black line. --MrBurns (talk) 19:26, 2 December 2011 (UTC)

I've changed 'points' to 'intersections' which I think is clearer and more accurate.--JohnBlackburne^words_deeds 19:34, 2 December 2011 (UTC)