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I'm writing up an activity where students are looking at pathlengths that follow along gridlines. enter image description here

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Is there a word or phrase that is commonly used to describe those paths, but doesn't include diagonals?

I'll probably call them 'grid line paths', but if there is a common term, I'd like to align with that.

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    $\begingroup$ I don't think there's a common term for such paths but I'm not confident enough to post that as an answer. I think "grid line path" is by far the best option but "lattice path" and "taxicab path" are decent alternatives. I think "rook path" is a poor choice because some people like me don't play chess and have no idea how a rook moves! $\endgroup$ – Thierry Nov 2 '20 at 23:41
  • $\begingroup$ Does it need to be a proper math term? If they're young enough I almost want to call that a "snake path" after the game Snake. $\endgroup$ – Owen Reynolds Nov 3 '20 at 5:11
  • $\begingroup$ I would use longitudinal and latitudinal, even though those are generally for when cardinal points are involved. $\endgroup$ – Aaron F Nov 3 '20 at 9:24
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    $\begingroup$ @OwenReynolds Also, a rook moves from the centre of one square to the centre of another; it doesn't sit on or move along the dividing lines, so ‘rook path’ could be confusing to people who do know chess too! $\endgroup$ – gidds Nov 3 '20 at 14:35
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    $\begingroup$ In the field of combinatorics I come from these are called "lattice paths", and one finds a lot more hits on Google Scholar for "lattice path" than for "grid line path". In this community, the Manhattan lattice is a lattice in which a direction is assigned to the each horizontal and vertical line. In real-world Manhattan, most streets are one-way. (In real-world Manhattan it is also the case that the avenues are separated by a much longer distance than the streets are, but I don't think I've seen that feature incorporated into the mathematics.) $\endgroup$ – Will Orrick Nov 3 '20 at 18:26
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Generally, this math falls under the scope of what is commonly called Taxicab Geometry.

I would use taxicab path as a noun to describe the specific paths illustrated in the original question; whereas taxicab geometry would be a term I'd use for the subset of mathematics covering these types of scenarios.

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    $\begingroup$ Or Manhattan geometry. $\endgroup$ – J W Nov 2 '20 at 18:38
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    $\begingroup$ I think adding that to your answer would improve it so it directly responds to the question (about what the paths are called). $\endgroup$ – Nick C Nov 3 '20 at 3:38
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    $\begingroup$ Also directly related is the L1 norm, or manhattan distance, which some people might still remember from school. $\endgroup$ – Polygnome Nov 3 '20 at 9:16
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    $\begingroup$ Doesn't taxicab geometry allow motion on all horizontal and vertical lines, not just the marked gridlines? $\endgroup$ – gidds Nov 3 '20 at 14:36
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    $\begingroup$ I suspect that it's partly that you are in NYC that you think taxis are restricted to such a grid; the rest of the world wouldn't understand the reference. $\endgroup$ – Pete Kirkham Nov 3 '20 at 16:14
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In the field of micro/nano-lithography, such geometry would be called Manhattan geometry; containing only two directions of edges orthogonal to each other. If diagonals were included, it would be classified as having skew edges. Finally, if it were truly freeform, it would be called curvilinear.

Example: https://doi.org/10.1117/12.2243030

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These are pretty much universally called lattice paths, which Will Orrick comments on the question.

A quick search on the arXiv reveals a large number of papers from combinatorics and computer science using the term in this way. (The main competition is from papers about path integrals on lattices, which gets unfortunately concatenated to "lattice path integral.") For examples from published literature, this Google Scholar search is a good start.

The jargon is not exactly localized to combinatorics and comp sci either; here, for example, is a probability paper using the same terminology.

Finally, Wikipedia's article on lattice paths has picture with a path moving diagonally; this generalization is an unusual use of the term.

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This paper calls a horizontal/vertical path a rook path, for the movement of a rook on a chess board.

If the paths connect the lower-left corner to the upper-right corner, this would be a North-East Lattice Path.

Edit: If the path didn't need to go through points with integer coordinates, I would submit the name "Cardinal Paths" in reference to the cardinal directions of North, East, South and West.

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The term that I have always used for this is “rectilinear”, that is, following the lines of a rectangular grid.

Most of the definitions of rectilinear found on the internet are commonly non-mathematical and IMHO, not correct (they essentially define it the same as “linear” which is clearly not what it means in math). However the Wikipedia article on Rectilinear Polygons provides the best explanations.

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