# Are there names for the different types of adjacency in a square grid?

See the image. How do you call the blue cells in relation to the black cell?

I've seen "orthogonally adjacent" for the first picture and "diagonally adjacent" for the second picture, but that leaves a void for the third picture (other than "orthogonally or diagonally adjacent", which is really long-winded).

If I were feeling inventive, I could make up "edge-adjacent", "corner-adjacent", and "edge- or corner-adjacent". That last one's still long-winded, but less so. The problem is, well (a) it's long-winded, and (b) "corner-adjacent" is a bit ambiguous. Technically, even the edge-adjacent cells share a corner with the black cell! So it might refer to the third picture.

(If I could enforce nomenclature, I would actually make people say "edge-adjacent" for the first, "corner-adjacent" for the third, and leave a void for the second (we don't need it anyway).)

Another option I can make up is "weakly adjacent" and "strongly adjacent", but again it's not clear if "weakly adjacent" refers to the second picture or the third.

So, what are the names for these types of adjacency? I mean this both in the "What's the official name" sense and "What would you call them" sense. I'm most interested in a name for the third picture; you can leave a void for the second picture for all I care. An example usage would be "The king can attack all [name] squares."

(I'm asking this in the board games forum because it seems like it'd important for talking about game rules.)

(Notice, by the way, that the Wikipedia page on Minesweeper just gives up, leaving the words "neighboring" and "adjacent" ambiguous.)

• One possible solution is to say "at least corner-adjacent" (or "at least weakly adjacent") for the third picture. Still long-winded - fewer letters than "edge- or corner-adjacent" but the same number of syllables. – Akiva Weinberger May 18 at 3:11
• Ooh - someone suggested "omni-adjacent" for the third picture. I like that. So (1) orthogonal(ly) adjacent, (2) diagonal(ly) adjacent, and (3) omni-adjacent. – Akiva Weinberger May 18 at 4:09
• Adjacent Orthogonally/Diagonally and Surrounding are the most direct/simple and clear ways to describe the three cases. Anything else is just description over terms (or require you to specify the definitions of the terms you'll be using anyhow, which is the same). – L. Scott Johnson May 18 at 11:49
• Great question! – gameaddict May 18 at 23:53
• If you really need all three and need to distinguish between them a lot, you could define "+-adjacent", "x-adjacent" and "✷-adjacent" (or, as StartPlayer suggests, just 'surrounding'). – CompuChip May 19 at 13:36

A common way to describe this in game rules is surrounded or 'surrounding spaces`.

As an example from the rules of Carcassonne

A monastery is completed when it is surrounded by 8 tiles. Each of the monastery’s tiles (the 8 surrounding tiles and the one with the monastery itself) is worth 1 point.

• Even so, note that Carcassonne specifies the "8 surrounding spaces", to distinguish it from games like Go where a piece is considered surrounded by only the four orthogonal adjacencies. – Arcanist Lupus May 21 at 0:41
• Not a board game but I noticed after I read this that Marvel's Puzzle Quest also uses surround. Reminded me of this=) – joedragons May 21 at 20:51

In mathematics, specifically in the field of cellular automata, picture one is known as the "Von Neumann Neighborhood" and picture three is known as the "Moore neighborhood" or the "Conway neighborhood" (referring to Conway's Game of Life ). (Sometimes these terms include the center cell as part of the neighborhood, i.e. the center cell is considered to be in its own neighborhood.) Unfortunately there isn't a named term for picture two.

For even more math jargon, you can say that picture one has all cells at a Manhattan Distance of 1, and picture three has all cells at a Chebyshev distance of 1. Again there's no short way to describe picture two.

• If you are involving metrics, Euclidian distance equal to root of 2 is what describes the second image. Constant 1-norm for the first, contant 2-norm for the second and constant ∞-norm for the third. – Džuris May 18 at 23:01
• The cells in picture two could be said to have a "Manhattan−Chebyshev distance" of 1, but you have to use a minus sign instead of a hyphen. – Ryan Veeder May 21 at 0:43

For the third situation, a term I see quite frequently (perhaps because I'm a chess player) is that the blue squares are a King's move away from the black one. This is similar to your "a King can attack all squares" but it's shorter and can be used as a compound modifier.

• The term is also used by variants sudoku, kings move constraint, knights move constraint. And you can also have a bishop and a rook constraint to complete the set. – Toon Krijthe May 18 at 9:26
• @ToonKrijthe Bishop and rook would refer to the entire row and column or diagonals, because they can move more than one space at a time. – Akiva Weinberger May 18 at 16:13
• @ToonKrijthe Rook constraint doesn't make sense to me because that's already part of normal Sudoku rules. – GendoIkari May 18 at 17:03
• @GendoIkari point but I mentioned it for completeness. – Toon Krijthe May 18 at 18:44

This may depend a bit on the context, and if you can define the phrases used beforehand, or if you need to use something that's immediately clear to everyone.

If we think about, say, writing the rules of a game, the easiest case would be where all parts of the game only care about one sort of adjacency. Then you could mention at the very start of the rules that when you say "adjacent", you mean something laying in the four orthogonally neighboring squares and there would be no need for the qualifier later. That would also be a good place to post a describing image. (Or the eight orthogonally or diagonally adjacent squares, or rarely, just the diagonally adjacent ones.)

If you have different cases, then "orthogonally adjacent", "diagonally adjacent" and "orthogonally or diagonally adjacent" would likely be the phrases easiest to understand.

If you mostly mean all eight adjacent squares, then you might avoid the awkward long phrase by just saying "adjacent" in that case, and reserve the qualifiers to the exceptional cases. This may also warrant some introductory explanation.

(For example, the Freeciv wiki seems to mostly use this custom, though I didn't check if they have a proper definition. It's a computer game, not a board game but the concepts are the same.)

You could also use "the eight surrounding/adjacent squares" and "the four surrounding/adjacent/connected" squares, if you only need one of the two cases with just four squares in question. That may also require a definition somewhere in the rules, but should be clear enough.

Those of a somewhat technically-oriented mind might like the phrases "8-connected squares" or "8-neighborhood", and the similar "4-connected squares" or "4-neighborhood", where the latter two would always mean the four orthogonally connected squares. But these are usually written with the number as a digit, and that may be at risk of mistyping, or at least suspicions of mistyping more than the written out words.

For the "one step orthogonally and one diagonally away" case, calling it a knight's move seems to be rather common, it's used even by go players. (There's also the concept of "a large knight's move", which is two steps orthogonally and one diagonally. (The step is large, not the knight.)). But I haven't seen the reference to "a king's move" that often, and probably wouldn't use it, unless it would be helpful to also refer to other chess pieces.

I considered asking this on english.SE, as it's not related to only board games, but there actually already was a similar question: What word describes something that can move orthogonally and diagonally?.

The highest-voted answer there suggests the word octilinear for all eight directions, but at least to my non-native English eye, it appears somewhat technical. The simple contracted combination orthodiagonally is also mentioned there.

• "Orthogonal" is how I've seen "adjacent but not diagonal" referred to, but rarely. And even when it's used I believe it comes with a more simplistic explanation since it's not commonly used. – Captain Man May 18 at 17:19
• @CaptainMan, "orthogonally" seemed immediately familiar to me, but you're right in that it might also often be written out as "vertically or horizontally". Amusingly, the first google hit I got for "orthogonally adjacent", was a question on BGG asking what it means. – ilkkachu May 18 at 17:30
• That's actually pretty funny :) The main reason I say to explain it somewhere in the rules is because apart from them needing to be very clear, but there may be younger players who don't know what it means (depending on the audience). – Captain Man May 18 at 17:35
• Catering for younger players who might not know words such as "orthogonal" or "adjacent" is fair enough. But a picture is worth a thousand words. Refer to a diagram and use the words (e.g. "orthogonally adjacent"). Then the reader will learn the words, and just the words will be enough after you used them the first time. – Rosie F May 19 at 5:46
• @RosieF In fact, I wrote a bunch of drafts of this question before finally deciding to get out markers and a piece of paper – Akiva Weinberger May 19 at 13:53

Simply "adjacent" can naturally be used to mean a square is either "diagonally adjacent" or "directly adjacent".

Another phrase, if it's mentioned before that distance is counted as 1 per straight or diagonal move, would be to say "within one square" or "one square away" or "one distant".

• Your first para is a good warning. "Adjacent to" literally means "lying next to". Only four squares lie next to a specific square. So using "adjacent" also for those four squares that touch it at a corner is to use the word a bit loosely. And you can't rely on readers guessing that that's what you mean. So that makes "adjacent" on its own ambiguous. So you should be more specific. – Rosie F May 19 at 5:52

You could describe the first example as 'sharing an edge', and the third example as 'sharing a corner'. If you need the second example, you could call it 'sharing an edge but not a corner'.

• For the second, I think you mean "sharing a corner but not an edge". And for the third, I'd want more than just "sharing a corner", which definitely suggests the second to me. – Rosie F May 20 at 15:48

The first one is called Von Neumann Neighbohood, the second one is Rotated Von Neumann Neighbohood and the third one is Moore Neighbohood.

• Where do these names come from? (What was Von Neumann doing?) – Akiva Weinberger May 19 at 17:17

I've seen neighbouring squares used to refer to all eight.

It's as much an issue of writing as of terminology though, and you need to be sure to introduce your terms clearly as in technical writing. In the case of writing rules, you may wish to recap at points where you feel people may jump in.

I'd use "the eight neighbouring squares" or "all eight neighbouring squares", "the four adjacent squares", which you could even expand as "the four adjacent squares north, south, east, and west of the starting square" and "the four diagonal neighbours". Yes adjacent means "next to", but we often think in terms of NSEW rather than left/right/up/down and a neighbour to the N is just as much a next door neighbour as one to the W. Later you can refer to "all neighbours", "diagonal neighbours", and "adjacent squares".

Of course, this doesn't lead to a choice of 3 simple adjectives to apply to one noun, but in rules given as prose that may even be a good thing; it would be a disadvantage in a table but there you could use a graphical approach (four/eight headed arrows), or a column "neighbour" with "diagonal"/"adjacent"/"all". The arrows are also useful as a shorthand (if of course it's about moves/attacks/similar interactions). Unfortunately suitable arrows don't seem to exist as unicode symbols, so here's a sketch.

Adjacent Orthogonally or diagonally adjacent; share at least one common vertex; has a Chebyshev distance of 1

Orthogonally adjacent, neighboring Share an edge (or two common vertices); has a Manhattan distance of 1

Diagonally adjacent Share exactly one common vertex; has a Chebyshev distance of 1 but a Manhattan distance of 2