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Tile growth cost is exponential so short of deliberately playing a cultural victory gameplan, you will rarely grow all 3rd ring tiles of any city without spending a lot of dough to purchase many and there's generally better uses for gold except early in a city existence.
There's very limited vertical yield growth in civ6 compared to civ5 or prior titles. You pretty much always need to go somewhat wide (say 10+ cities) except for cultural victory where you can get by at ~7 cities with some civs. So it's generally beneficial to expand min distance or min+1 (so 3-4 tiles between cities).
There can be exceptions for either a strategic defensive position or trying to claim high quality land before a neighbor and then settling in-between later, etc.
Loyalty pressure is up to 9 or 10 tiles but diminishing for each tile distance so clustering cities also helps you maintain control if you are in dark age while neighbors are in golden age.
IIRC, only t2 and t3 buildings from industrial zone and entertainment complex / water park have an area effect to also provide yield to other cities. The area effects are helping make the most out of more cities with fewer pop.
Like basides having a powerhouse capital or B2(B is build city hotkey so that just means 2nd city). you usually just want a couple of 10 or 13 pop cities, a couple of 7 or 10 pop cities, etc. You just want to hit breakpoints for the next districts since those scale your empire yields a lot more than centralized population in a single city.
Don’t overthink that, if you ́re a beginner. In 95% of the case, the closer is the better. 3 to 4 tiles between your cities allow more defensive ability, better adgency bonus, better tile swap, better loyalty pressure, more optimised range buildings. The "lack of space" won’t really mater.
https://steamcommunity.com/sharedfiles/filedetails/?id=3220078106
I tend to go for 5 as a preferred distance. Cities at distance 4 when that's the only way to grab some key tiles or settle on water, cities at distance 6 when that's a significantly better spot than distance 5, and anything more is a stretch to grab something important like a resource. Or I'm skipping over bad territory like desert or non-coastal tundra.
You want to build cities tightly against each other because you generally want as many cities as possible in your area because it doesn't matter if a city can build 10 different districts... It matters that you build the Campus or Theatre Square 10x
Note that each Civ game is different, Civ6 (this one) massively rewards as many cities as possible. Civ5 on the other hand massively rewards massive cities, so you want them as spread as possible (6 tile gap between cities)
So yea... 6 cities tightly packed will always be better in Civ6 than 3 perfectly spaced cities
Just think of it as a long-term profitability. Perfect late game cities with 40 populations and dozens of tiles to work would only be relevent for a very short amount of times. It's not worth to lose tempo early game for this.
Your cities have to be efficient way before that if you want to be relevent. If you make a good snowball, you won't have to care your cities lack spaces in lategame, because when late game comes, you'd virtualy already have win the game... Maybe not reach your victory condition yet, but assure to win.
Both the minimum distance between an edge or vertice of a shape and the nearest edge or vertice of another shape, and the distance between the centroid of a shape and the centroid of another shape are commonly used in geometry. Geospatial tools like google maps use and support both.
That's why communicating definitions is routinely necessary in most fields of work.
This is not an arbitrary definition by any stretch.
And "number of tiles between two tiles, not including those two" - what I was responding to - is not a distance at all. It doesn't satisfy the triangle inequality.
Alright this will be my last stop on this side discussion. Both are valid in this context. They're merely an application of euclidian distance on a plane with a loose concept of what a tile is in terms of unit of distance. The language just specifies how to select the point within a polygon to measure your euclidian distance on a 2D plane. Surely you can see how a game displaying a map trying to resemble a world map in some capacity might have users think about distances in much the same manner they would when working with widespread mapping tools.
The idea of the number of tiles between 2 tiles makes sense in practice to players and is the reason for the MP community tendency to count it in this manner. It makes some sense with the planar or euclidian geometry above.
You're implicitly using graph theory, laying all hexes as vertices whereby the distance metric to fit the triangle inequality is the number of edges in a shortest path between 2 vertices. The concept of vertices on a graph or the "number of moves", as you pointed out yourself, is problematic itself due to the number of exceptions.
However, even within this context, while it does not meet the mathematical definition of a "distance", the concept of the number of vertices between 2 vertices, i.e. the number of edges between 2 vertices minus 1, or the number of tiles between 2 tiles is still a very natural use of plain language to a player that doesn't want to have to do a math major to naviage definitions of distances, norms, graphy theory, etc.
Trying to say a language norm that a large player base for a videogame has been using for years is silly when it rose out of a functional language issue with the word "between" is just as silly.
- Try to keep a distance of 5-6
That usually allows for more optimal lategame layouts, as well as having a wider variety of terrain to work on and thus being more flexible in ressource gathering depending on what you need right now. Also having less cities means you can concentrate your trades more on fewer cities. Some people who've problems multitasking might also find this strategy easier
- Try to keep a distance of 3-4 tiles
This offers a more optimal early/mid game layout, as you can simply have more cities at the same space, thus being more flexible in building and unit production (especially when it comes to builders, wonders and some district with area effects)
In the early game in particular there's barely any difference in the efficiency of a city on its own between both of them, so having more cities is usually what allows you to be faster to build stuff and by the time the end game rolls around, there's barely any reason to have massive metropoles anymore, since someone most likely is about to win anyway, which is why i tend to go for the denser layout, given that the game doesn't punish you anymore for having too many pops like in Civ5.
The biggest disadvantage for having more cities early on however is, that you've to spend more time on building settlers, so it's way more important to get the right gouvernor and policies to make it possible out like 10 settlers in 20-30 turns and then get buch into closing the gap again through trade routes and districts. However i also had games, where i had problems closing the gap, since it is also dependant on the area you spawned in, so it's important to know, which side to go or if you maybe even mix those strategies.
As for the AI, they're usually not that great in planning ahead, so it's easier to have them simply take the easier strategy, which is rather the second one due to it being easier to move units, closing borders faster (thus them being harder to attack) and most importantly, bad decisions in city placement doesn't matter as much, if there are 3 other cities that are somewhat decent.