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)

External Link

I would like to request the website http://www.taxicabgeometry.net be added to the external links. I am the website owner and did not want to instigate a conflict of interest by adding it myself. Thank you. -- Kevin Thompson

Done. I feel the same way about my web pages, some of which are linked to from WP by other editors. In general, let others decide if your pages are worthwhile to link to. — Loadmaster (talk) 16:40, 15 May 2012 (UTC)

What is the symmetry group for Taxicab geometry?

Felix Klein famously said that symmetry is geometry, so it would be good to have a section on the symmetry group for Taxicab geometry. In 2-D, the Taxicab circles for the continuous case are squares, hence the Taxicab symmetry group probably includes the symmetry group of a square. In higher dimensions the Taxicab symmetry group probably includes that of a cross-polytope. It is remarkable that these are discrete groups, in contrast to the Euclidean group. — Preceding unsigned comment added by 202.63.38.34 (talk) 06:29, 28 September 2012 (UTC)

Extended Taxicab Geometry

In 2006 in the paper entitled "Taxicab Geometry: some problems and solutions for square grid-based fire spread simulation" [1] I generalized the taxicab distance to an extended taxicab distance d that computes the distance between P and Q by using the two sides of a parallelogram that consists of a 45º diagonal side and either a horizontal or a vertical side. More specially the distance d(P (a, b), Q(x, y)) is given by the equation d(P, Q) = max(|a - x|,|b - y|) - min(|a - x|,|b - y|) + sqrt(2)* min(|a - x|,|b - y|). This was formulated to optimise forest fire simulation in grid-based cell automata algorithms. In 2012 Hope Sydner and Roman Wong (Mathematics Department, Washington & Jefferson College, Washington, PA 15301) [2] showed that that this generalized distance d is still a metric, thus satisfying the triangle inequality, and proceeded with a complete analysis of all the conics with graphs under this new metric. May I suggest to include this in the text. Thanks. David Caballero (gnomusy@gmail.com) talk • contribs) 07:32, 23 May 2013 (UTC) Gnomusy (talk) 07:35, 23 May 2013 (UTC)Gnomusy

Shouldn't the caption on the main picture say "All FOUR lines"

72.222.137.53 (talk) 16:45, 29 July 2015 (UTC)

Linked as a type of string metric, but why?

Taxicab Distance / L1 Distance / City block distance are all linked to from String metric, yet this article has no explanation of how or why this methodology applies to strings or is even a metric applicable to strings in a metric space. Looking for a better understanding here. — Preceding unsigned comment added by Tylerjharden (talk • contribs) 08:04, 2 March 2016 (UTC)

Please add information for more realistic distance

This article would be far more useful if it added a keyword, or link of a taxicab distance using roads with alternated directions. Taxicab distance is not realistic because most cities with rectangular grids have directed streets, so the taxicab distance is not the real distance. When searching information for real world taxicab distance, google directs here, but from here there are no information about what else to look for.

So at least a paragraph should say "for streets with directed orientations, the distance is named XXX" (so the reader can understand that he needs to search for XXX), or should give a link to the more general problem. — Preceding unsigned comment added by 206.132.109.103 (talk) 14:26, 8 August 2016 (UTC)

It’s an mathematical construct; despite the name it does not really have much to do with taxis or any other sort of vehicle. In a real city then yes, you have to take account of one way streets, of junction turn restrictions, of speed limits, of stop signs, of stop lights, etc., but none of that is really relevant here.--JohnBlackburne^words_deeds 16:56, 8 August 2016 (UTC)

Horrible page name

There's nothing intrinsic to taxicabs that confines them to rectilinear navigation.

Manhattan distance is at least excusable (because Manhattan really is built that way, to a solid cliche approximation).

Rectilinear distance wouldn't even need an excuse.

Alternatively, we could split the difference and call this page "Manhattan taxicab distance". Or—ooh ooh ooh—Rain Man taxicab distance (no way you're gonna drive him to the "Y" MCA).

Naked taxicab distance is the worst of all. — MaxEnt 18:15, 20 July 2017 (UTC)

+1 for revert to Manhattan distance, which is more widely used and makes more sense than the rather arbitrary "taxicab distance" which namesake doesn't make much sense. ExtremeHeat11 (talk) 06:36, 4 July 2020 (UTC)

Intro section - dating the analysis

The lead paragraphs currently say: "The geometry has been used in regression analysis since the 18th century [...] The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski." The Regression Analysis article dates regression analysis to 1805, which is still in the 19th century. And it appears that the bit about Minkowski could be folded in here a bit better too. I'm just throwing on a citation needed tag in case somebody knows something that's not linked and not reflected in the article. Edwin Herdman (talk) 02:56, 22 December 2021 (UTC)

Lines

What are lines in taxicab geometry? The "Properties" section mentions Hilbert's axioms, yet in general one can find two different shortest paths between a pair of points. Utricularia tubulata (talk) 22:16, 5 January 2022 (UTC)

Wiki Education assignment: Public Writing Fall 2022 E1

This article was the subject of a Wiki Education Foundation-supported course assignment, between 7 September 2022 and 12 December 2022. Further details are available on the course page. Student editor(s): Nitsujbrownie (article contribs). Peer reviewers: Rechhabra.

— Assignment last updated by Nitsujbrownie (talk) 16:54, 3 October 2022 (UTC)

Paper citation

For Note #6, the paper "The Nature of Length, Area, and Volume in Taxicab Geometry" was published in the International Electronic Journal of Geometry, Vol. 4, No. 2 (2011), pp. 193-207. Link: https://dergipark.org.tr/en/pub/iejg/issue/47488/599514 This should probably be used over (or in addition to) the arXiv link. - Kevin Thompson 47.218.30.79 (talk) 00:05, 26 November 2022 (UTC)

 Done —David Eppstein (talk) 00:34, 26 November 2022 (UTC)

The text of the first image has a factual error in the second sentence.

"In taxicab geometry, the red, yellow, blue, and green paths all have the same shortest path length of 12". The green line does not have a length of 12 but a length of 6*(2)^1/2, as the third sentence states. The green line is not a valid path in "taxicab geometry". No taxi cab could drive streets and avenues that way. The green line is only a valid a path in Euclidean geometry. Simply removing the word green from the second sentence would greatly clarify the point of the graphic. Mcrodgers2 (talk) 22:55, 27 December 2022 (UTC)

No, it's not an error. Taxicab geometry is a way of measuring length; this was explained reasonably clearly by Schaefer in the section "Biangles" above, for example, as well as in the article itself. (There is definitely something confusing about the image & its description, though.) --JBL (talk) 21:50, 1 January 2023 (UTC)

Probably the problem fundamentally is that "lengths of paths" is a confusing second-order thing to be thinking about -- the essential idea is about distances between points -- it happens that a consequence of the definition is that many kinds of paths have the same length (as in Taxicab_geometry#Arc_Length), but this is not the defining feature of the geometry, it is rather a nontrivial consequence. --JBL (talk) 21:55, 1 January 2023 (UTC)

I think the confusing nature of the image is somehow related to the staircase paradox, which has very similar looking images but where the (Euclidean) length of the diagonal does not equal the length of its staircase approximations. Here, it does equal the approximations. Instead, you would get a staircase paradox from diagonally-stepped approximations to an axis-parallel line. —David Eppstein (talk) 23:58, 1 January 2023 (UTC)

The caption needs to be changed. As is, it does not clearly state that the green line is also stepped, but instead, the reader assumes that it is a straight line. I suggest the reference to Euclidean length should be removed or clarified. I won't do it myself though, to avoid an edit war. Janke | Talk 10:32, 2 January 2023 (UTC)

The green line is a straight line. That's the point. --JBL (talk) 17:40, 2 January 2023 (UTC)

So, if it indeed is a straight line, how come it has a length of 12 (per the caption), and not 6*SQR(2)? --Janke | Talk 18:27, 2 January 2023 (UTC)

My responses to Mcrodgers2 above contain two links, one to a section of this talk-page and one to a section of the article. Before I answer, can you tell me whether you have clicked on either of those links and read the text there? --JBL (talk) 18:30, 2 January 2023 (UTC)

On the staircase paradox page, there's this: Taxicab geometry, in which the lengths of the staircases and of the diagonal are equal - guess I'm not very good at infinitesimals... Janke | Talk 18:31, 2 January 2023 (UTC)

I've gone ahead and swapped the previous lead image with one lower down in the body showing circles. I'm not entirely happy with it (like the previous lead image, it suggests strongly that the Manhattan metric is only for a discrete space) but at least I think it is less confusing. Improvements on the caption are very welcome. --JBL (talk) 18:00, 6 July 2023 (UTC)

@JayBeeEll – I found the "circle" image confusing, so I swapped the images back, but I rewrote the caption and the lead paragraph for clarity/accessibility. Is that any better? –jacobolus (t) 21:23, 14 December 2023 (UTC)

(While I was at it, I also removed "snake distance", a search for which in Google scholar only turned up a handful of sources that looked copy/pasted from Wikipedia.) –jacobolus (t) 21:28, 14 December 2023 (UTC)

Okay, @David Eppstein didn't like my caption, insultingly calling it "vague and evasive" in his edit summary. But in my opinion the new caption he replaced it with is confusing to readers and fundamentally misses the point of the picture. Maybe we can have a further discussion here.

In my impression, the purpose of this picture (which may be clearer to see from the original image added in special:diff/18349318, shown immediately above) is to demonstrate that irrespective of the path taken, any route following the street plan of a city can only travel along grid-aligned directions, meaning that any such route which zigzags its way from one point to another without ever going in the wrong direction has the same (Euclidean) length, the "taxicab distance" between the points. This distance can then be conveniently computed as the sum of the distance in the East–West direction and the distance in the North–South direction. This metaphor gives geometric motivation to the algebraic definition ${\textstyle d_{t}(p,q)=\sum _{i=1}^{n}|q_{i}-p_{i}|.}$

The replacement caption talks about the length "in taxicab geometry" of the various paths, but this is misleading to readers because elucidating taxicab-geometry paths isn't something we want to accomplish here where we're trying to demonstrate the definition of the taxicab distance between points, and is additionally confusing because lengths of taxicab-geometry paths, per se, are not defined anywhere in this article and defining them clearly requires significantly more explicit work than can possibly be done in the space of a caption.

The definition of taxicab distance between points is inherited from the definition of Euclidean distance in each separate coordinate, and the path lengths of the various colorful zigzags are Euclidean path lengths. The point of the taxicab metaphor is that acceptable (Euclidean) paths we can measure along to determine the taxicab distance are constrained to be oriented in line with the grid axes, just the way driving in a city is constrained to the streets. –jacobolus (t) 00:11, 15 December 2023 (UTC)

Your long message here, and the modified image File:Manhattan distance bgiu.png that you have included in it, illustrate a common misconception that perhaps you do not personally subscribe to, but has been seen time and time again in mistaken contributors to this article. To follow a shortest path in taxicab geometry, it is not necessary to follow a piecewise axis-parallel path. Any coordinatewise-monotone path is a shortest path. We should use a lead image that demonstrates the possibility of non-piecewise-axis-parallel shortest paths, and we should state clearly that the green path in the image is a shortest path of length equal to the other paths, to avoid leaving readers with the same mistaken impression. In particular, your statement "any route following the street plan of a city can only travel along grid-aligned directions" is not true of taxicab geometry, even though maybe it is true in the motivating example of city navigation, and we should not write our article as if it is true. —David Eppstein (talk) 01:43, 15 December 2023 (UTC)

Once again, we are talking about the definition of the taxicab distance, which is defined in terms of Euclidean distance of axis-aligned paths, we are not talking about paths in taxicab geometry. Making a caption all about the latter is therefore very confusing. (If we really want to elaborate further about this analogy somewhere, we can say that taxicab geometry is a continuous version of the discrete city grid geometry of taxi routes between grid points.) One good reason that the diagram showing something like city blocks is helpful in the very start is that without it the inspiration for the name/concept doesn't really make sense. –jacobolus (t) 03:31, 15 December 2023 (UTC)

No. The taxicab distance is defined in terms of sums of absolute differences of coordinates. The definition is not about paths. So if we illustrate it with an image about the lengths of paths, it is about lengths of paths, in taxicab geometry, and we should describe those lengths correctly. —David Eppstein (talk) 06:36, 15 December 2023 (UTC)

This algebraic/formulaic definition is precisely equivalent to a geometric definition in terms of (Euclidean) lengths of paths constrained to the coordinate axes, which is where the name and concept comes from. Your "No" declaration is a really pedantic nitpick, and strictly applying it gets in the way of non-technical readers getting an accessible definition expressible in plain English with an evocative picture based on the analogy which inspired the concept, instead leaving them with a pile of jargon and formulas. –jacobolus (t) 06:44, 15 December 2023 (UTC)

If you tell them that taxicab distance is the length of the stairstep paths, they will leave thinking the falsehood that the green straight line segment is shorter and that taxicab distance is not an actual distance on an actual metric space, just some weird way of measuring lengths of special paths in Euclidean spaces. That misunderstanding is why we have historically gotten so many newby editors trying to "correct" the old caption to give the Euclidean length of the green segment. You are fostering the same misunderstanding. —David Eppstein (talk) 06:49, 15 December 2023 (UTC)

If you don't tell them that, plenty of readers will hit the jargon wall and leave without ever understanding WTF is going on at all, and you'll mostly be serving readers who already knew what the subject was before they arrived, or else were already mathematical experts. –jacobolus (t) 07:15, 15 December 2023 (UTC)

Here are some examples from existing reliable sources (pretty much every source I looked at uses some variation of this kind of intuitive/geometric explanation):

Scheid (1961) "A taxicab travels on a network of roads, a typical part of which is shown in Figure 1. Using mathematical license (and following Euclid), we imagine these roads and taxicabs to have no thickness and to consist of Euclidean points. This network will be our plane. The existence of other "points" not on the roads is not recognized, not by taxicabs anyway. It is the lines of Figure 1, not the spaces, which concern us. As for distance, it is only common sense to use this word for a quantity measured along the roads as the taxicab goes, not as the crow flies. In skyscraper country even the crow may find our concept of distance quite useful."

Krause (1973): "Ituitively the taxicab distance from a point $(x_{1},y_{1})$ to a point $(x_{2},y_{2})$ is suggested by the route a taxicab might take (fig. 1)."

Gardner (1997) covers the discrete lattice case, Taxicab geometry is a metric system in which the points of the space correspond to the intersections of the horizontal and vertical lines of square-celled graph paper, or to the intersections of the streets in our idealized city. If two points, A and B, are at intersections on the same street, the distance between them is measured, as it is in Euclidean geometry, by counting the number of unit blocks from one to the other. If A and B are not on the same street, however, then instead of applying the Pythagorean theorem to calculate the distance between them we count the number of blocks a taxicab must travel as it goes from A to B (or vice versa) along a shortest-possible route." but at the end mentions "... taxicab geometry can be elegantly generalized to the entire Cartesian plane, where all points are represented by ordered pairs of real numbers from the two coordinate axes. The rule of measuring distance by the shortest path along line segments that parallel the axes must of course be preserved, so in this continuous form of taxicab geometry an infinite number of distinct paths, all of the same minimum length, connect any two points that are not on the same street."

Thompson & Dray (2000): Taxicab geometry, as its name might imply, is essentiallly the study of an ideal city with all roads running horizontal or vertical. The roads must be used to get from point A to point B; thus, the normal Euclidean distance function in the plane needs to be modified."

Kaya (2006): "Geometrically, $d_{T}(A,B)$ stands for the length of shortest path from A to Β composed of line segments parallel to the coordinate axes."

Ratliff (2010): "We think of this as the shortest driving distance between the two points where we are only allowed to travel along streets that run east-west or north-south."

Smith (2013): "The lattice points were street corners, and students needed to take a taxicab from corner A to either corner B or corner C. The distance would then be the number of city blocks covered during this taxicab trip along the most direct routes. [...] Although the idea of taxicabs and buildings gives the problem a charming physical context, we can, with some examination and discussion, extend this situation from its naturally discrete sense to a more continuous case. Removing the buildings but still maintaining the restriction that the taxicab can drive only parallel to the x- or y-axis allows it now to drive any real number of blocks. This created a greater sense of continuity and allowed us to draw line segments with greater conceptual confidence."

Cho (2014) "In taxicab geometry, distances are measured along paths of horizontal and vertical lines. Diagonal paths are not allowed. This measurement simulates the movement of taxicabs, which can travel only on streets, never through buildings."

Berger (2015): "The geometry measuring the distance between points using the shortest path traveled along a square grid is known as taxicab geometry. [...] In a real-world context, locations on a city grid would be associated with points having at least one integer coordinate. However, the following definition applies to all points in the plane."

etc. etc. etc. –jacobolus (t) 07:56, 15 December 2023 (UTC)

If we really want to go all-in on this "No", we might as well rename the page to L1 distance, relegate the "taxicab"/"Manhattan" names to a footnote, and maybe encourage third parties to adopt the wonderfully concise and obvious name "the sum-of-absolute-differences-of-coordinates distance". –jacobolus (t) 06:49, 15 December 2023 (UTC)

The conversation has advanced since I began drafting this (last night), but just to record my thoughts:

I find the circle image extremely straightforward and you haven't articulated what you find confusing about it, which is too bad. I agree that the concept "length of a curve in taxicab geometry" should not be in the caption of the lead (though you seem to be mistaken that it is not defined anywhere in the article: it's the subject of the section Taxicab geometry#Arc length). --JBL (talk) 21:36, 16 December 2023 (UTC)

I find the circle image extremely confusing out of context and at a glance, requiring readers to do a good bit of careful thought to make sense of. I don't think it does an adequate job illustrating the concepts of "taxicab geometry" or "taxicab distance". It also takes a ton of vertical space. (I am frankly not convinced it's an amazing demonstration of the concept of taxicab circles either; I tried to improve the text of that section to make it a bit clearer, but we'd maybe be better off with a picture showing just one "continuous" circle and showing distances from the center to various points on it.)

I think we could improve this article by adding a section before the formal definition section explaining taxicab geometry in a middle-school accessible way, with emphasis on the taxi metaphor, as is done in a pretty wide range of available sources aimed at students and the the general public. –jacobolus (t) 21:50, 16 December 2023 (UTC)

[path length is] the subject of the section Taxicab geometry#Arc length – the arc length section is (a) far below the lead section, and (b) currently not remotely accessible for the broadest intended audience of this article, to whom the lead section should be addressed. –jacobolus (t) 21:57, 16 December 2023 (UTC)

LASSO

“The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO” I don’t think that’s correct. As far as I know LASSO was developed at the end of the 20th century. It wouldn’t have been possible to use in regression earlier due to the computational complexity. Janshi (talk) 07:43, 28 September 2023 (UTC)

Please feel free to fix this up. I don't know precisely what the sentence was intending to imply, but the name LASSO is clearly much more recent. –jacobolus (t) 21:21, 14 December 2023 (UTC)