Talk:Taxicab geometry

1639:"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 1767:

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. —

3431:"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." 3254:"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." 1672: 3002:, 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 1865:, 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. 243: 222: 342: 1628:

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.

332: 311: 2767:"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. 3363:"... 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." 3181:. 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. – 2991: 1632: 2667: 3359:

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."

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each forum. I'm sorry for bothering and wasting everyone's time on Knowledge 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). (

3143:. 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 " 437: 416: 2740: 447: 3443:"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." 1539:

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 (

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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)

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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.)

3555:“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. 3504:

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

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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"

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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?)?

2830:, 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. — 1538:

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

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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

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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.

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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

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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

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In Knowledge, 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

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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

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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.

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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)

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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

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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.

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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.

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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?

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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.

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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

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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

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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.

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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:

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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

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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.

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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°.

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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.

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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

3437:"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." 2892:

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?

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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. —

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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.

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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. –

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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.

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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.

1559:) 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. 153: 810: 2109:. 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 1507:

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:

3467:, 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". – 735: 3527:– 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. – 2342:

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. --

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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.

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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.

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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. --

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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:

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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

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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

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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.

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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! --

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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!

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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

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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,

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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. —

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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?

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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. —

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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.

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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.

670: 647: 601: 565: 2967:(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 Knowledge.) – 2955:– 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? – 531:

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

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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:

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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):

3425:"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." 1543:) 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 ( 2383:

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

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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. —

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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

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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.

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Probably the problem fundamentally is that "lengths of paths" is a confusing second-order thing to be thinking about -- the essential idea is about

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grid divisions between the endpoints of the hypotenuse. Note that there will be multiple such approximations, but they will all have equal lengths.

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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. –

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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.

3602: 1456:? "If" is a good word. Many mathematical notions change their appearance or disappear altogether, if you change some underlying definitions. `' 503: 364: 3607: 2790:, for example, as well as in the article itself. (There is definitely something confusing about the image & its description, though.) -- 2687: 2538: 85: 1510:

A "circle" in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These "circles" are squares ...

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information for real world taxicab distance, google directs here, but from here there are no information about what else to look for.

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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!

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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

3151:, 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. — 2286:

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)

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

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In 2006 in the paper entitled "Taxicab Geometry: some problems and solutions for square grid-based fire spread simulation"

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are constrained to be oriented in line with the grid axes, just the way driving in a city is constrained to the streets. –

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

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

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

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

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This is an interesting paradox, since it essentially puts the validity of the Pythagorean theorem in jeopardy. --

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No, it's not an error. Taxicab geometry is a way of measuring length; this was explained reasonably clearly by

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stands for the length of shortest path from A to Β composed of line segments parallel to the coordinate axes."

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Alternatively, we could split the difference and call this page "Manhattan taxicab distance". Or—ooh ooh ooh—

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In my impression, the purpose of this picture (which may be clearer to see from the original image added in

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So, if it indeed is a straight line, how come it has a length of 12 (per the caption), and not 6*SQR(2)? --

2808:-- it happens that a consequence of the definition is that many kinds of paths have the same length (as in 2252: 3118:

path lengths. The point of the taxicab metaphor is that acceptable (Euclidean) paths we can measure along

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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This article was the subject of a Wiki Education Foundation-supported course assignment, between

2628: 2605: 1959: 1654: 1524: 1460: 1258:, it should be equivilent to Euclidean geometry. This paradox shows that this is not the case. -- 1235: 347: 70: 2062:

It doesn't even seem to make sense to say that Hilbert's axioms are a formalization of Euclid's

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any route following the street plan of a city can only travel along grid-aligned directions

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The caption needs to be changed. As is, it does not clearly state that the green line is

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This should probably be used over (or in addition to) the arXiv link. - Kevin Thompson

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

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Taxicab geometry, in which the lengths of the staircases and of the diagonal are equal

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I've changed 'points' to 'intersections' which I think is clearer and more accurate.--

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

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refer to the horizontal and vertical distances between end points of the hypotenuse.

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I believe it would be correct to say that Hilbert's axioms are a formalization of

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If we really want to go all-in on this "No", we might as well rename the page to

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it were composed of microscopic zig-zags. I hope this makes things clearer. --

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Taxicab Distance / L1 Distance / City block distance are all linked to from

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line connecting the points, but has a length defined on metric that behaves

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

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distance function. In other words, the minimal-distance path from point

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You're right, that was my mistake, now fixed. Thank for noticing it. --

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

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comprises a different set of street blocks than the minimal path from

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I think the confusing nature of the image is somehow related to the

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be the best approximation of the length of AC in taxicab space with

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taxicab distance (no way you're gonna drive him to the "Y" MCA).

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I find the following two statements not understandable right now:

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to have length equal to the points' distance in taxicab geometry.

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One problem with the sentence "Taxicab geometry satisfies all of

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

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Well said. Can't a pair of circles intersect at one point?

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Whichever it is, it should become consistent throughout. --

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Why did it not upload correctly? It looks okay in maximum

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is suggested by the route a taxicab might take (fig. 1)."

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

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https://dergipark.org.tr/en/pub/iejg/issue/47488/599514

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I guess that some people may not be familiar with what

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diamond offset by 45 degrees from the coordinate axes.

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of the taxicab distance, which is defined in terms of

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not necessary to follow a piecewise axis-parallel path

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Wiki Education assignment: Public Writing Fall 2022 E1

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Is this corect? Thanks. In the most sincere manner, -

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than can possibly be done in the space of a caption.

464:, a collaborative effort to improve the coverage of 359:, a collaborative effort to improve the coverage of 254:, a collaborative effort to improve the coverage of 1686: 1679: 3411: 3344: 3298: 3096: 2522:Please add information for more realistic distance 2247:What about taxicab geometry on hexagonal grids? -- 1381: 1250: 1181: 1127: 1077: 988: 874:{\displaystyle \lim _{n\to \infty }D_{t}(n)=D_{0}} 873: 804: 729: 664: 641: 615: 595: 559: 3356:Gardner (1997) covers the discrete lattice case, 2403:What is the symmetry group for Taxicab geometry? 890: 824: 33:for general discussion of the article's subject. 3134:Your long message here, and the modified image 2915:- guess I'm not very good at infinitesimals... 1735:, but that is not the problem. There are a few 3260:"Ituitively the taxicab distance from a point 2592:Naked taxicab distance is the worst of all. — 174: 8: 3628:B-Class chess articles of Bottom-importance 2486:Linked as a type of string metric, but why? 1798:Oh yeah, that too. But not more than two. — 2989: 2532: 2215:I have a small problem with this section: 1818:of putting Euclid's axioms in very clear. 410: 305: 216: 3388: 3382: 3333: 3320: 3311: 3287: 3274: 3265: 3086: 3080: 3067: 3058: 3052: 3041: 3013: 3007: 2113:in the mathematical/technical sense of a 1477:path from A to C or a point that lies on 1370: 1350: 1333: 1237: 1173: 1151: 1145: 1119: 1097: 1091: 1067: 1051: 1042: 1002: 978: 962: 953: 893: 887: 865: 843: 827: 821: 749: 743: 719: 703: 694: 685: 679: 654: 631: 608: 578: 572: 542: 536: 1958:Hilbert's Axioms are a formalization of 2865:a straight line. That's the point. -- 1586:as a diagram of the above suggestion? 412: 307: 218: 188: 3521: 3442: 3436: 3430: 3424: 3376: 3368: 3362: 3357: 3259: 3253: 3144: 2029:, but for months you have insisted on 1547:) from the centre point and over one ( 1468:I don't see how the observation about 3165:Once again, we are talking about the 1743:, and Euclid's is only one of them. 7: 2379:I would like to request the website 2211:Anyone know what an angle is for L1? 458:This article is within the scope of 353:This article is within the scope of 248:This article is within the scope of 2622:Intro section - dating the analysis 207:It is of interest to the following 23:for discussing improvements to the 2675: 2671: 1245: 1060: 1044: 1028: 1009: 971: 955: 934: 915: 900: 834: 791: 772: 712: 696: 656: 633: 14: 3613:Mid-priority mathematics articles 3120:to determine the taxicab distance 1700:Why Euclid Axioms Is Unacceptable 373:Knowledge:WikiProject Mathematics 3623:Bottom-importance chess articles 3598:Low-importance Robotics articles 3136:File:Manhattan distance bgiu.png 2937:the caption are very welcome. -- 2738: 2678:. Further details are available 2665: 2335: 2330: 1670: 1599:. Could someoene please help? 445: 435: 414: 376:Template:WikiProject Mathematics 340: 330: 309: 241: 220: 189: 45:Click here to start a new topic. 2311:The latter names allude to the 498:This article has been rated as 393:This article has been rated as 288:This article has been rated as 3565:07:43, 28 September 2023 (UTC) 3406: 3394: 3339: 3313: 3293: 3267: 3173:of axis-aligned paths, we are 3087: 3059: 3031: 3019: 2424:06:29, 28 September 2012 (UTC) 2381:http://www.taxicabgeometry.net 1376: 1363: 1344: 1338: 1242: 1163: 1157: 1128:{\displaystyle D_{t}(n)=D_{0}} 1109: 1103: 897: 855: 849: 831: 761: 755: 590: 584: 554: 548: 268:Knowledge:WikiProject Robotics 1: 3603:WikiProject Robotics articles 3578:21:21, 14 December 2023 (UTC) 3535:21:57, 16 December 2023 (UTC) 3517:21:50, 16 December 2023 (UTC) 3500:21:36, 16 December 2023 (UTC) 3475:06:49, 15 December 2023 (UTC) 3455:07:56, 15 December 2023 (UTC) 3345:{\displaystyle (x_{2},y_{2})} 3299:{\displaystyle (x_{1},y_{1})} 3245:07:15, 15 December 2023 (UTC) 3232:06:49, 15 December 2023 (UTC) 3217:06:44, 15 December 2023 (UTC) 3204:06:36, 15 December 2023 (UTC) 3189:03:31, 15 December 2023 (UTC) 3161:01:43, 15 December 2023 (UTC) 3130:00:11, 15 December 2023 (UTC) 2975:21:28, 14 December 2023 (UTC) 2963:21:23, 14 December 2023 (UTC) 2787: 2777:22:55, 27 December 2022 (UTC) 2757:00:34, 26 November 2022 (UTC) 2732:00:05, 26 November 2022 (UTC) 2698:— Assignment last updated by 2637:02:56, 22 December 2021 (UTC) 2300:03:46, 11 February 2010 (UTC) 2277:11:39, 13 November 2009 (UTC) 1492:17:17, 5 September 2008 (UTC) 1421:13:37, 21 February 2006 (UTC) 472:and see a list of open tasks. 367:and see a list of open tasks. 271:Template:WikiProject Robotics 262:and see a list of open tasks. 42:Put new text under old text. 3608:B-Class mathematics articles 3367:Thompson & Dray (2000): 3149:not true of taxicab geometry 2450:) 07:32, 23 May 2013 (UTC) 2369:19:34, 2 December 2011 (UTC) 2352:19:26, 2 December 2011 (UTC) 1568: 1565: 1534: 1505:These circles are squares... 1320:04:13, 14 January 2006 (UTC) 1302:03:58, 14 January 2006 (UTC) 1251:{\displaystyle n\to \infty } 1182:{\displaystyle D_{t}(n): --> 3524:Taxicab geometry#Arc length 3522:the subject of the section 3488:Taxicab geometry#Arc length 2924:18:31, 2 January 2023 (UTC) 2903:18:30, 2 January 2023 (UTC) 2888:18:27, 2 January 2023 (UTC) 2875:17:40, 2 January 2023 (UTC) 2857:10:32, 2 January 2023 (UTC) 2840:23:58, 1 January 2023 (UTC) 2822:21:55, 1 January 2023 (UTC) 2810:Taxicab_geometry#Arc_Length 2800:21:50, 1 January 2023 (UTC) 2708:16:54, 3 October 2022 (UTC) 2656:22:16, 5 January 2022 (UTC) 2313:grid layout of most streets 2257:08:25, 23 August 2009 (UTC) 1464:01:09, 17 August 2006 (UTC) 1447:00:45, 17 August 2006 (UTC) 1394:21:58, 5 October 2006 (UTC) 1352: 478:Knowledge:WikiProject Chess 50:New to Knowledge? Welcome! 3649: 3633:WikiProject Chess articles 3412:{\displaystyle d_{T}(A,B)} 2566:16:56, 8 August 2016 (UTC) 2547:14:26, 8 August 2016 (UTC) 1432:02:14, 20 April 2006 (UTC) 1284:02:16, 20 April 2006 (UTC) 1209:02:20, 20 April 2006 (UTC) 626:For the sake of notation, 504:project's importance scale 481:Template:WikiProject Chess 294:project's importance scale 3593:B-Class Robotics articles 3179:paths in taxicab geometry 2597:18:15, 20 July 2017 (UTC) 2515:08:04, 2 March 2016 (UTC) 2481:16:45, 29 July 2015 (UTC) 2430:Extended Taxicab Geometry 2238:19:59, 27 July 2009 (UTC) 2206:16:46, 11 June 2009 (UTC) 2033:, which is not correct. 1932:22:40, 14 July 2008 (UTC) 1914:17:37, 19 June 2008 (UTC) 1899:16:07, 19 June 2008 (UTC) 1883:15:55, 19 June 2008 (UTC) 1847:04:58, 19 June 2008 (UTC) 1831:04:49, 19 June 2008 (UTC) 1808:03:46, 19 June 2008 (UTC) 1794:03:08, 19 June 2008 (UTC) 1777:02:59, 19 June 2008 (UTC) 1760:02:15, 19 June 2008 (UTC) 1722:05:19, 18 June 2008 (UTC) 1263:15:00, 11 July 2005 (UTC) 497: 430: 392: 325: 287: 236: 215: 80:Be welcoming to newcomers 3361:but at the end mentions 2947:18:00, 6 July 2023 (UTC) 2806:distances between points 2788:section "Biangles" above 2614:06:36, 4 July 2020 (UTC) 2460:07:35, 23 May 2013 (UTC) 2397:16:40, 15 May 2012 (UTC) 1694:00:41, 2 June 2007 (UTC) 1660:23:54, 1 June 2007 (UTC) 1604:22:36, 1 June 2007 (UTC) 1591:22:29, 1 June 2007 (UTC) 1577:20:02, 1 June 2007 (UTC) 1530:23:04, 19 May 2007 (UTC) 1197:7 July 2005 18:52 (UTC) 665:{\displaystyle \Delta y} 642:{\displaystyle \Delta x} 596:{\displaystyle D_{t}(n)} 567:be the length of AC and 560:{\displaystyle D_{0}(n)} 399:project's priority scale 2554:really relevant here.-- 2152:00:31, 2 May 2009 (UTC) 2126:00:26, 2 May 2009 (UTC) 2100:23:46, 1 May 2009 (UTC) 2075:23:44, 1 May 2009 (UTC) 2042:23:40, 1 May 2009 (UTC) 2009:23:37, 1 May 2009 (UTC) 1975:23:32, 1 May 2009 (UTC) 1951:23:28, 1 May 2009 (UTC) 1747:'s is more modern (see 1222:8 July 2005 03:37 (UTC) 356:WikiProject Mathematics 3618:B-Class chess articles 3413: 3346: 3300: 3098: 3057: 2222: 1873:axioms) except …". -- 1635: 1555:) units and over two ( 1383: 1252: 1184: 1129: 1079: 990: 875: 806: 731: 666: 643: 617: 597: 561: 197:This article is rated 75:avoid personal attacks 3414: 3347: 3301: 3099: 3037: 3000:special:diff/18349318 2682:. 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943: 939: 936: 932: 928: 924: 920: 917: 913: 908: 902: 899: 896: 892: 868: 864: 860: 857: 854: 851: 846: 842: 836: 833: 830: 826: 800: 796: 793: 789: 785: 781: 777: 774: 770: 766: 763: 760: 757: 752: 748: 722: 718: 714: 711: 706: 702: 698: 693: 688: 684: 661: 658: 638: 635: 612: 592: 589: 586: 581: 577: 556: 553: 550: 545: 541: 525: 522: 520: 516: 515: 512: 511: 508: 507: 496: 490: 489: 487: 484:chess articles 470:the discussion 457: 456: 440: 428: 427: 419: 407: 406: 403: 402: 391: 385: 384: 382: 365:the discussion 352: 351: 335: 323: 322: 314: 302: 301: 298: 297: 290:Low-importance 286: 280: 279: 277: 260:the discussion 246: 234: 233: 231:Low‑importance 225: 213: 212: 206: 195: 181: 180: 118: 117: 113: 112: 107: 102: 93: 92: 90: 89: 82: 77: 68: 62: 60: 59: 48: 39: 38: 35: 34: 28: 13: 10: 9: 6: 4: 3: 2: 3645: 3634: 3631: 3629: 3626: 3624: 3621: 3619: 3616: 3614: 3611: 3609: 3606: 3604: 3601: 3599: 3596: 3594: 3591: 3590: 3588: 3579: 3576: 3573: 3569: 3568: 3567: 3566: 3562: 3558: 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2876: 2872: 2868: 2864: 2860: 2859: 2858: 2855: 2851: 2847: 2843: 2842: 2841: 2837: 2833: 2829: 2825: 2824: 2823: 2819: 2815: 2811: 2807: 2803: 2802: 2801: 2797: 2793: 2789: 2785: 2781: 2780: 2779: 2778: 2774: 2770: 2762: 2758: 2754: 2750: 2745: 2736: 2735: 2734: 2733: 2729: 2725: 2721: 2713: 2711: 2709: 2705: 2701: 2700:Nitsujbrownie 2695: 2693: 2689: 2685: 2684:Nitsujbrownie 2681: 2668: 2660: 2658: 2657: 2653: 2649: 2641: 2639: 2638: 2634: 2630: 2629:Edwin Herdman 2621: 2615: 2611: 2607: 2606:ExtremeHeat11 2603: 2602: 2601: 2600: 2599: 2598: 2595: 2590: 2588: 2583: 2580: 2577: 2571: 2567: 2563: 2558: 2552: 2551: 2550: 2548: 2544: 2540: 2536: 2528: 2521: 2519: 2516: 2512: 2508: 2504: 2500: 2493: 2492:String metric 2485: 2483: 2482: 2478: 2474: 2473:72.222.137.53 2465: 2464: 2461: 2457: 2453: 2449: 2445: 2440: 2436: 2429: 2427: 2425: 2421: 2417: 2413: 2402: 2398: 2394: 2390: 2386: 2385: 2384: 2382: 2375:External Link 2374: 2370: 2366: 2361: 2356: 2355: 2354: 2353: 2349: 2345: 2340: 2338: 2333: 2328: 2325: 2324: 2322: 2318: 2314: 2305: 2301: 2297: 2293: 2289: 2288: 2287: 2281: 2279: 2278: 2274: 2270: 2261: 2259: 2258: 2254: 2250: 2242: 2240: 2239: 2235: 2231: 2225: 2221: 2216: 2210: 2208: 2207: 2203: 2199: 2195: 2191: 2187: 2183: 2179: 2175: 2171: 2167: 2159: 2153: 2149: 2145: 2142: 2141: 2140: 2139: 2133: 2132: 2131: 2130: 2127: 2123: 2120: 2116: 2115:formal system 2112: 2108: 2104: 2103: 2102: 2101: 2097: 2093: 2089: 2076: 2072: 2069: 2065: 2061: 2060: 2059: 2058: 2057: 2056: 2055: 2054: 2053: 2052: 2043: 2039: 2036: 2032: 2028: 2024: 2023: 2022: 2021: 2020: 2019: 2018: 2017: 2010: 2006: 2002: 1999: 1998: 1997: 1996: 1995: 1994: 1986: 1985: 1984: 1983: 1982: 1981: 1976: 1972: 1969: 1965: 1961: 1957: 1956: 1955: 1954: 1953: 1952: 1948: 1944: 1937: 1936: 1935: 1933: 1929: 1925: 1920: 1915: 1911: 1908: 1904: 1903: 1902: 1900: 1896: 1892: 1884: 1880: 1876: 1871: 1867: 1864: 1860: 1856: 1852: 1851: 1848: 1844: 1841: 1836: 1835: 1834: 1832: 1828: 1824: 1819: 1809: 1805: 1801: 1797: 1796: 1795: 1791: 1788: 1784: 1783: 1782: 1781: 1778: 1774: 1770: 1765: 1764: 1761: 1757: 1754: 1750: 1746: 1745:David Hilbert 1742: 1738: 1734: 1730: 1727: 1726: 1725: 1723: 1719: 1715: 1710: 1707: 1699: 1697: 1695: 1691: 1690: 1689: 1682: 1675: 1673: 1664: 1663: 1662: 1661: 1656: 1652: 1647: 1642: 1633: 1629: 1626: 1622: 1618: 1611: 1605: 1602: 1598: 1594: 1593: 1592: 1589: 1585: 1581: 1580: 1579: 1578: 1575: 1558: 1554: 1550: 1546: 1542: 1537: 1533: 1532: 1531: 1526: 1522: 1517: 1516: 1515: 1512: 1511: 1506: 1499: 1493: 1489: 1485: 1480: 1476: 1471: 1467: 1466: 1465: 1462: 1459: 1455: 1454:straight line 1451: 1450: 1449: 1448: 1445: 1437: 1433: 1430: 1425: 1424: 1423: 1422: 1419: 1414: 1413: 1409: 1406: 1405: 1400: 1395: 1392: 1371: 1367: 1360: 1355: 1347: 1341: 1335: 1327: 1326: 1321: 1317: 1313: 1309: 1308: 1307: 1306: 1303: 1300: 1295: 1294: 1285: 1282: 1278: 1277: 1276: 1275: 1274: 1273: 1272: 1271: 1264: 1261: 1239: 1230: 1229: 1228: 1227: 1226: 1225: 1221: 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119: 111: 110:Verifiability 108: 106: 103: 101: 98: 97: 96: 87: 83: 81: 78: 76: 72: 69: 67: 64: 63: 57: 53: 52:Learn to edit 49: 46: 41: 40: 37: 36: 32: 26: 22: 18: 17: 3554: 3178: 3174: 3170: 3166: 3148: 3140: 3119: 3115: 3107: 2912: 2862: 2845: 2805: 2766: 2743: 2724:47.218.30.79 2717: 2696: 2664: 2645: 2625: 2591: 2586: 2584: 2581: 2578: 2575: 2533:— Preceding 2529: 2525: 2503:Tylerjharden 2497:— Preceding 2489: 2471: 2433: 2416:202.63.38.34 2410:— Preceding 2406: 2378: 2341: 2329: 2326: 2310: 2309: 2285: 2265: 2246: 2230:141.214.17.5 2226: 2223: 2218: 2214: 2193: 2189: 2185: 2181: 2177: 2163: 2110: 2106: 2090: 2087: 2063: 2030: 2026: 1941: 1921: 1918: 1887: 1875:Jitse Niesen 1820: 1816: 1711: 1703: 1685: 1684: 1676: 1669: 1645: 1640: 1637: 1615: 1574:76.188.26.92 1567: 1513: 1509: 1504: 1503: 1478: 1474: 1441: 1415: 1411: 1410: 1407: 1402: 1401: 1398: 1312:Jitse Niesen 1192: 1143:D_{0}}": --> 1139: 1136: 1086: 997: 882: 816: 813: 738: 674: 625: 530: 527: 519: 499: 459: 453:Chess portal 395:Mid-priority 394: 354: 320:Mid‑priority 289: 249: 209:WikiProjects 171: 165: 157: 150: 144: 138: 132: 122: 94: 19:This is the 3465:L1 distance 3435:Cho (2014) 3306:to a point 2249:77.56.90.38 2174:quasimetric 2164:If a given 2160:Quasimetric 1617:ASprigOfFig 1601:ASprigOfFig 1588:ASprigOfFig 1484:Largo Plazo 1299:Ed Sanville 370:Mathematics 361:mathematics 317:Mathematics 148:free images 31:not a forum 3587:Categories 3167:definition 2769:Mcrodgers2 2389:Loadmaster 2198:Loadmaster 1625:this image 1597:resolution 1582:How about 1260:CoderGnome 1195:CoderGnome 3572:jacobolus 3529:jacobolus 3511:jacobolus 3469:jacobolus 3449:jacobolus 3370:modified. 3239:jacobolus 3211:jacobolus 3183:jacobolus 3124:jacobolus 3116:Euclidean 2969:jacobolus 2957:jacobolus 2953:JayBeeEll 2692:Rechhabra 2317:Manhattan 2180:to point 1962:, not of 1641:visualize 1570:advance)! 88:if needed 71:Be polite 21:talk page 2784:Schaefer 2587:Rain Man 2535:unsigned 2511:contribs 2499:unsigned 2448:contribs 2412:unsigned 1741:geometry 1651:Schaefer 1612:Biangles 1536:Schaefer 1521:Schaefer 1391:Elenthel 265:Robotics 256:Robotics 228:Robotics 56:get help 29:This is 27:article. 2981:Okay, @ 2907:On the 2786:in the 2463:Gnomusy 2452:Gnomusy 2344:MrBurns 2321:borough 2119:Bubba73 2068:Bubba73 2035:Bubba73 1968:Bubba73 1907:Bubba73 1840:Bubba73 1787:Bubba73 1753:Bubba73 1680:A Sprig 524:Paradox 502:on the 397:on the 292:on the 199:B-class 154:WP refs 142:scholar 3557:Janshi 2594:MaxEnt 2262:Naming 2122:(talk) 2107:axioms 2071:(talk) 2064:axioms 2038:(talk) 1971:(talk) 1910:(talk) 1843:(talk) 1790:(talk) 1756:(talk) 1729:Euclid 1687:of Fig 1621:digons 1429:345Kai 1418:Abdull 1404:holds. 1281:345Kai 1220:me_and 1206:345Kai 1183:D_{0}} 494:Bottom 205:scale. 126:Google 3551:LASSO 3490:). -- 3147:" is 2917:Janke 2881:Janke 2850:Janke 2642:Lines 2562:deeds 2365:deeds 2306:error 2192:, so 1731:is a 1646:as if 1458:mikka 1167:: --> 475:Chess 466:Chess 422:Chess 169:JSTOR 130:books 84:Seek 3561:talk 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