Talk:Taxicab geometry

message: 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)
replies: text: You probably meant straight line? "If" is a good word. Many mathematical notions change their appearance or disappear altogether, if you change some underlying definitions. `'mikka (t) 01:09, 17 August 2006 (UTC)
replies: text: I don't see how the observation about Hilbert's axioms can be correct. Axiom II.2 says, "If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D." Is a point lying between A and C a point that lies on any path from A to C or a point that lies on all paths from A to C? If the latter, then there is no such point, contradicting the axiom. If the former, then take a look at Axiom I.2: "Any two distinct points of a straight line completely determine that line; ...." Which line do A and B determine? Certainly, the line with endpoints A and B. But we obtained B by selecting it as a point on line AC, so A and B must completely determine line AC. So we have A and B completely determining two distinct lines, a contradiction. —Largo Plazo (talk) 17:17, 5 September 2008 (UTC)
thread_title: Line

message: These circles are squares...
It appears that we are claiming that circles are squares.  Here "circles" refers to manhattan geometry, whereas "squares" refers to euclidean.  Could we make this clearer, perhaps with scare-quotes, like this:
A "circle" in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These "circles" are squares ...
I might also suggest using "diamond" instead of square, although that's hardly a mathematical term. -- Comment unsigned
:I have added more description and an image, which should make the meaning clearer without resorting to scare-quotes. Circles in taxicab geometry are still circles, despite their appearance. -- Schaefer (talk) 23:04, 19 May 2007 (UTC)
replies: text: Schaefer explained it well but to some this might still seem confusing.  I suggest that we say that a circle in Taxicab geometry a circle, though still being a circle by definition, looks like a Euclidean-style square.  Also, because this concept could be seemingly contradictory to some I suggest that we explain it like this or something like it: by definition, a circle is a shape in which all points are equidistant from a single fixed (centre) point.  Since in Taxicab geometry one is restricted to a street grid, a circle must have it's lines avoid the blocks.  Therefore, if you wish to create a circle five (5) units in all aspects from the centre point, one can first plot a point five units straight up from the centre point.  But one can also plot a point up four (4) from the centre point and over one (1) coordinate to the right.  Likewise, one can plot a point up three (3) units and over two (2) to the right, etc.  All these coordinates lie on the radius of the circle creating a square (for Euclidean geometry) but still fulfilling the definition of a circle needing to be equidistant from the centre point.
Also, shouldn't we include the fact that even though circles in Euclidean geometry can only intersect at a maximum of two points without becoming the same circle, circles in Taxicab geometry can intersect an infinite number of times as long as they are infinitely large.  Thanks for your opinions (in advance)! -76.188.26.92 20:02, 1 June 2007 (UTC)

text: How about this as a diagram of the above suggestion?  ASprigOfFig 22:29, 1 June 2007 (UTC)
replies: text: Why did it not upload correctly?  It looks okay in maximum resolution.  Could someoene please help?  ASprigOfFig 22:36, 1 June 2007 (UTC)
thread_title:  Circles vs. "Circles" 

message: 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)
replies: text: Okay, thanks.  I was really just going of (but not as a copyright, mind) a book I have which briefly describes "biangles" and that was basically the definition they gave (and they had five of those nine pictures).  Sorry.  This now puts doubt into the following belief (fro the same book) though it makes sense.  It is the belief that as long as a circle is sufficiently large, and has another circle inside it sufficiently large, the two circles can intersect (or meet, at least) an infinite number of times- just like the following picture shows:
200px
Is this corect?  Thanks. In the most sincere manner, -</font>A Sprig of Fig 00:41, 2 June 2007 (UTC)
thread_title:  Biangles 

message: 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)
thread_title:  Quasimetric 

message: 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)
thread_title:  Anyone know what an angle is for L1? 

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

message: 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)
thread_title:  Naming 

message: 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)
replies: text: For points with integer coordinates, yes. But the taxicab distance can be defined for any two points in the plane, not just those with integer coordinates. —David Eppstein (talk) 03:46, 11 February 2010 (UTC)
thread_title: Isnt Taxicab Distance just a special case of finite network/graph distance?

message: 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:
File:Manhattan 1.PNG File:Manhattan 2.PNG
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)
replies: text: I've changed 'points' to 'intersections' which I think is clearer and more accurate.--JohnBlackburnewordsdeeds 19:34, 2 December 2011 (UTC)
thread_title:  error 

message: I would like to request the website http://www.taxicabgeometry.net be added to the external links. I am the website owner and did not want to instigate a conflict of interest by adding it myself. Thank you. -- Kevin Thompson
:Done. I feel the same way about my web pages, some of which are linked to from WP by other editors. In general, let others decide if your pages are worthwhile to link to. — Loadmaster (talk) 16:40, 15 May 2012 (UTC)
thread_title:  External Link 

message: 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)
thread_title:  What is the symmetry group for Taxicab geometry? 

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

message: 72.222.137.53 (talk) 16:45, 29 July 2015 (UTC)
thread_title:  Shouldn't the caption on the main picture say "All FOUR lines" 

message: 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)
thread_title:  Linked as a type of string metric, but why? 

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

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

message: 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)
thread_title:  Intro section - dating the analysis 

message: 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)
thread_title:  Lines 

message: — Assignment last updated by Nitsujbrownie (talk) 16:54, 3 October 2022 (UTC)
thread_title: Wiki Education assignment: Public Writing Fall 2022 E1

message: 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)
replies: text: —David Eppstein (talk) 00:34, 26 November 2022 (UTC)
thread_title:  Paper citation 

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