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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 be the length of AC and be the best approximation of the length of AC in taxicab space with 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, and refer to the horizontal and vertical distances between end points of the hypotenuse.
Logically, as the number of subdivisions increases, the best approximation should approach the Euclidean distance. That is,
However, it is clear from looking at simple cases that,
This is an interesting paradox, since it essentially puts the validity of the Pythagorean theorem in jeopardy. -- CoderGnome 7 July 2005 18:52 (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)
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)
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
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)
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)
Is this corect? Thanks. In the most sincere manner, - A Sprig of Fig 00:41, 2 June 2007 (UTC)
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))
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))
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))
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))
Rallybrendan2006 ( talk) 23:28, 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)
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)
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)
What about taxicab geometry on hexagonal grids? -- 77.56.90.38 ( talk) 08:25, 23 August 2009 (UTC)
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)
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)
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 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
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)
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
72.222.137.53 ( talk) 16:45, 29 July 2015 (UTC)
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)
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)
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)
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)
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)
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:
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— Assignment last updated by Nitsujbrownie ( talk) 16:54, 3 October 2022 (UTC)
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)
"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)
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)
"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."
"Ituitively the taxicab distance from a point to a point is suggested by the route a taxicab might take (fig. 1)."
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."
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."
"Geometrically, stands for the length of shortest path from A to Β composed of line segments parallel to the coordinate axes."
"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."
"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."
"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."
"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."
[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)
“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)
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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 be the length of AC and be the best approximation of the length of AC in taxicab space with 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, and refer to the horizontal and vertical distances between end points of the hypotenuse.
Logically, as the number of subdivisions increases, the best approximation should approach the Euclidean distance. That is,
However, it is clear from looking at simple cases that,
This is an interesting paradox, since it essentially puts the validity of the Pythagorean theorem in jeopardy. -- CoderGnome 7 July 2005 18:52 (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)
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)
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
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)
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)
Is this corect? Thanks. In the most sincere manner, - A Sprig of Fig 00:41, 2 June 2007 (UTC)
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))
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))
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))
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))
Rallybrendan2006 ( talk) 23:28, 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)
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)
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)
What about taxicab geometry on hexagonal grids? -- 77.56.90.38 ( talk) 08:25, 23 August 2009 (UTC)
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)
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)
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 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
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)
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
72.222.137.53 ( talk) 16:45, 29 July 2015 (UTC)
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)
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)
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)
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)
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)
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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: 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)
"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)
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)
"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."
"Ituitively the taxicab distance from a point to a point is suggested by the route a taxicab might take (fig. 1)."
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."
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."
"Geometrically, stands for the length of shortest path from A to Β composed of line segments parallel to the coordinate axes."
"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."
"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."
"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."
"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."
[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)
“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)