A recent posting on another of my favourite 'go to' blogs Battle Game of the Month has once again provoked some further thoughts in my mind concerning the use of square grids. What Ross raises here are some problematic features associated with the 'orthogonal only' convention. I mean, it works, but one feels that there ought to be some method of dealing with diagonal orientations. Would it not be wonderful if we could create a tessellation of regular octagons on a flat, two-dimensional surface?
Here was my immediate response.
In the matter of diagonals, I seem to recall discussing this with you about 18 months ago:
I didn't really follow up on that as I was not then thinking a going down the gridded wargames track. I've had reason to revisit the notion since.
I've been looking at Bob Cordery's Portable Wargames (I now have a copy of both his PW books) and looking to adapt my own Byzantine Army of c950-1050AD and its opponents (Bulgars and Georgians{Abasgians}) to the system. Your current posting has got me rethinking about this.
The PW square-grid system measures moves and shooting by orthogonals only. It would not add very much complexity in my view to add diagonals. Talking specifically of 'Ancients', movement varies from 1 'step' (artillery), 2 (heavy infantry) to 3 and 4 for more mobile troops. Weapon ranges are 2 or 3 except for the artillery, which is 6 squares.
The effect of this is that that the movement allowances and ranges themselves still form a square, but set at 45-degrees from the grid orientation. A square lozenge if you like.
It occurs to me that you could add 'and a half' to diagonal moves and ranges, with the fraction dropped when you reach the target square. For example, my heavy horse, movement allowed 3 squares, moves one orthogonal (1), one diagonal (2 'and a half'). It can move one more orthogonal (3 'and a half') but not one more diagonal (4).
I'd also suggest that the unit ends the move facing the direction it was moving. Otherwise a unit
1. Can begin the move with a 45-degree turn or 180-degrees;
2. Turn 45-degrees when stepping INTO a cell at the beginning of the step;
3. Count a 90degree turn as a full 1-square step
Where things get really tricky is shooting. But that can be resolved in the Cordery system by ignoring the adjacent squares for shooting - that's close combat country - and limiting shooting to the orthogonals adjacent to the orthogonal line of facing, or the diagonals CORNER-TO-CORNER adjacent to the diagonal line of facing. In the latter case I'd be inclined to exclude the nearest two outside diagonal squares.
It seemed to me a posting on my own blog was called for.
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| Left: 'Orthogonal-only' movement allowances: Right: Movement allowance with diagonal 'steps' permitted. |
The first of this two diagrams compares the system of movement used in Bob Cordery's Portable Wargame, for any units with up to 4 'steps' allowed per move, all steps to be orthogonal - that is to say, through the sides of the squares - and my proposal that does allow diagonal movement.
If we count a move from one square orthogonally to the nest a 'step', let us count a move from one square diagonally to the next one-and-a-half steps. Two such moves would therefore be 3 steps; and three, 4-and-a-half. And so on. When counting off steps, at the end of the turn, you drop the 'and a half', and the remaining integer must be not greater than the unit's movement allowance in 'steps'. In the diagram above, I show the blue unit's allowance under my suggested scheme. Except for the unit with an allowance of 1 only (artillery), the movement range more nearly approximates a circle, which is my aim.
This leads us to facing. I suggest that the at the end of the turn, the unit remains facing the direction it was moving when it entered the final square. If the moving unit comes adjacent to an enemy, then it must turn to face that enemy (or at least one of them). The Portable Wargame rule set is a lot more flexible in regard to turning.
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| Top row: weapons ranges with orthogonal facing: Bottom row: weapons ranges with diagonal facing. |
The problem Ross saw had to do with manoeuvring and facing threats from 'Northwest', so to speak, rather than 'due North' or 'due West'. It seemed to me that shooting arcs and ranges ought to be possible with a diagonal facing as much as an orthogonal one. For ranges, I use the same 'x-and-a-half' to denote distances through diagonals.
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| Some quick pix to liven up a dry posting. Byzantine cavalry in pursuit of a Bulgar raiding force run up against a reargurad of light horse and spearmen. |
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| Battle is joined! The Bulgars take heavy losses early - down 4 strength points to no loss to the Byzantines/ |
The diagonal equivalent of this orthogonal convention was not so easy to nut out. Eventually I arrived at restricting the arc to the diagonals corner-to-corner adjacent to the diagonal line of facing, excluding the squares not contained within the 90-degree angle formed by the square sides converging in front of the shooting unit. Those squares are indicated by the asterisks in the diagrams above. If you check out the artillery angles and ranges you will find the 'kill zone' for the artillery is equal in size (15 squares) for either orientation.
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| The spearmen look menacing, overlapping the Byxantine flank... |
It so happens this topic is of considerable interest to me as I am thinking of adapting the Cordery rule set for 'Ancients' for my c.1000AD armies and enemies of Byzantium. Many of the units, especially the Byzantines, comprise troops of disparate arms. The Byzantine Horse comprise lance-armed and bow-armed riders. Even the elite cataphracts (kataphraktoi: fully armoured and with barded horses, not to be confused with the standard Tagmatic and Thematic heavy horse: kavallarioi) carries a small contingent of horse archers within its trapezoidal formation. The heavy spearmen (skutatoi) are complemented by bowmen (toxotai) in the middle ranks. All the Bulgar horse, light and heavy, were armed with javelins and bows (as well as swords).
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| Indecisive, back-and-forth, combat ensues... The Byzantines began with 29 SP, the Bulgars with 34. |
For Byzantine units that can not shoot you would be looking to the Varangian Guard (elite heavy spearmen in my army - the axe-toting Englishmen came later); or the rather poorer quality peltastoi spearmen (I don't really believe they were rough terrain troops, but could easily be wrong. I can see them doing OK in urban fighting, say).
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| ... the Byzantines taking hard knocks themselves 3 SP lost to the Bulgars' 6... This part game was played using the standard PW conventions, but all horsed troops could shoot. |
I am also inclined to place a close combat premium on lances. The Bulgars didn't use them, but the Abasgians (early Georgians) did, as of course did the Byzantines.
Tell you what, this will make the Byzantines a formidable army! More on Portable wargames Armies and Enemies of Byzantium (c950-1050AD) another time.
In some of Peter Pig's rules the measurement is orthogonal only, but you are allowed a single diagonal as 'one'. For maximum distances of 6 or so the result is close to the 1.5 rounded down method but is also convenient and easy to remember or explain.
ReplyDeleteThat single-diagonal 'step' crossed my mind as a possibility too. A reasonable compromise.
DeleteI must admit, I find the diagonals thing impossible to get excited about. They are what they are. I think it's because I associate the 'diagonal problem' as something the anti-grid people rally around and I think it harks back to the days when many of us ran campaigns on pieces of squared paper and the prospect (nay - Horror) of something that didn't seem mathematically accurate and therefore not historically accurate was something to huff and puff about.
ReplyDeleteI think it is an easy thing to let go once it is accepted that our soldiers and dice, try as hard as they might, do not reflect realism as much as we might fancy.
If one throws out the diagonal conundrum and just plays 'a square is a square', accepting that any advantage / disadvantage affects both players equally, then a simplicity of that, soothes the mind and play just becomes a bit less complicated.
I just think it is too easy to get drawn into the diagonals dilemma.
I like hexes, they don't so much have the diagonal issue, but they do have that strangeness of compressing the battlefield in one polarity. So with the hex vertex pointing forward, you would need twelve 4 inch hexes on a four foot wide table ...... but you only need 42" of depth to get 12 inches deeps, so somewhere on the depth, you have gained one and a half hexes over that distance. This of course is directly related to what 'non-believers' call the zig-zag effect. But in truth, it is a total irrelevance to play and there are no advantages or disadvantages to either player, except they have of course just gained an extra 6" worth of table for free!
There is a problem of having single units spread over 1 or more squares and hexes from the point of view that one in real terms has halved the width of the table, while the depth remains the same, so the square and hex have in real terms become distorted over distance.... but it looks OK, so what's the problem, well none unless you want your 'game' to be 'real' and life is too short for that one.
Squares to look very useful when two sides are lined up against each other (assuming we a playing a linear based period). The beauty of the square unravels as we think of real space' and how flank attacks develop and how fire arcs are managed so that blind spots for artillery are not created.
Fire arcs against enemy infantry start to look strange on a square grid because our eye expects to see units that engage each other to be roughly in front of each other, not sat out to the side on a diagonal that would be impossible to fire at unless the unit turned to face the enemy ..... creating a need to have 8 facings in a square that would include all four corners. Suddenly the advantages of the easy to manage square start to look a bit clunky, which is why it seems essential to not worry about the maths of the thing, but just do what is simple, straight forward and looks right.
I wonder whether or problem is that too often we try to make the grid accord with what we think is the real world, instead of letting a grid just accord with the limits of the grid and then step back and while losing some 'realism', we get some more playable play.
I seem to have tinkered and designed and pondered all my wargaming life and on reflection, I wonder whether I have simply lost too much actual game time by thinking about how I should play.
You have some good points Norm. My concerns weren't so much on an individual level but with situations where opposing armies are not opposite to each other or where they are opposite but not aligned with a grid. Obviously one can normally avoid this by reorienting the battlefield but with converging forces this doesn't always help. I find it is also an issue primarily with close order linear warfare where individual units were slightly less flexible.
DeleteIt is one of those things, likely units of varyung strength where one can shrug them off for a generic game but it can really distort a refight of a particular historical engagement unless well thought out. I think simetime that thinking about these things is a sub-hobby in itself!
Thanks for your comments Norm. You raise several points that I probably could have spared myself the trouble of thinking about by staying with the 'free board' system to which I ave become accustomed.
DeleteBut I have had occasions in the past to think about this. As a sometimes chess player, I am acutely conscious of the geometry of the chessboard that enables a king to cross the board diagonally in the same number of moves as orthogonally, yet the physical distance is roughly 1.4 times as great (Interestingly enough, the knight move does form an octogon, so that 3 knight moves takes the knight to one square from a8 (b7) but THREE squares from h8 (e6 or f5).
Over twenty-five years ago I began looking into offset squares as a substitute for hexagon fields. Actually they were rectangles with an aspect ratio of roughly 0.87:1. Apart from the aspect ratio thing, I never took the thing as far as, say, Bob Cordery has. I fact it never occurred to me anyone else would come up with the same idea and develop it.
For the rest, I thought square grid has conveniences of their own. I am starting to see they might have (by me) unconsidered inconveniences as well. But I have always thought the 'diagonal problem' solvable.
There is another motivation here, and I see that Ross Mac touches upon it. I find thinking about this sort of thing rather fun. But I always was rather fond of mathematics...
I'll have to think about that fire arc, I've always felt that the 45 was too generous but inevitable to avoid dead zones.
ReplyDeleteMy main issue was not with individual units though but with opposing armies not aligned with the grid, say in a situation where they are converging or where the armies are approaching by various roads from edge and corner so that the lines naturally form at an angle to the grid no matter which way you twist it. Tricky in life or off grid or on a hex by really tricky on a square grid.
I think you're right about the 'dead zone' thing. I have to admit that it is only partially - not entirely - eliminated by the method I suggest here. It's OK for ranges of 2 or 3 squares (though I can see a potential problem for javelins shooting diagonally) but for greater ranges (artillery) they can creep in. I'm inclined, though, to think that might not be so unrealistic.
DeleteAnd here was I thinking I might be on to something. Something reasonably simple, easily understood, and workable. Maybe not.
ReplyDeleteSometimes the problem with grid war games has to less with designing armies to fit the grid system - that's (relatively) easy. It is designing the grid system to fit my armies.
There is no way I can fit my 30YW battalia and tercios onto a single cell of a sensibly sized square grid (6" or less) So they have to be spread over several grid cells (shot-pike-shot - a microcosm of Ross's problem). Allowing only for orthogonal facing means that to respond to a threat from, say 'right front' the unit may be forced into a 'stepped' formation. But that looks more like an echelonned formation than a proper well-ordered line.
I guess for myself I'll simply have to 'suck it and see' with some play testing of my own. I'm about to post once more on this topic, looking into the 30YW formations on square grids.
This has been a very interesting discussion although I am chiming in a bit late, perhaps.
DeleteNorm makes an excellent point when is says,
"If one throws out the diagonal conundrum and just plays 'a square is a square', accepting that any advantage / disadvantage affects both players equally, then a simplicity of that, soothes the mind and play just becomes a bit less complicated."
True if the goal of the model is to represent the premise that a "square is a square." But, is that the objective?
To me, the discussion of grid type and which measurement heuristic to use is putting the cart before the horse. Figure out the game design first and then apply or overlay the appropriate grid mechanism to conform to the chosen design philosophy.
Grid systems are avenues to a means. Each has advantages and disadvantages. In the chosen grid, should distance be measured as Euclidean, Chebyshev, Manhattan, or something else? Is facing important? Does a BMU span more than one grid?
Once the type of game has been formulated then the most appropriate grid can be superimposed to allow the game designer to accomplish the desired objective. I say, "stop forcing a square peg into a hex hole."
Jonathan - Now we are entering arcane country! I had to look up 'Chebyshev' and 'Manhattan' in this context. Fortunately they were easy to find, and turned out to be names to concepts already familiar to me. As I mentioned to someone else (unaware of the nomenclature), the chessboard distances are 'Chebyshev' to the king but 'Euclidean' to the knight (and have no real meaning to any of the other pieces, nor to the pawns).
DeleteThe Bob Cordery system I now know as using Manhattan distance conventions (which, by the way, the computer game 'Civilization' uses - at least in its IIIrd incarnation). It is my belief, though, that the Manhattan and Chebyshev distances have (for our purposes) identical effects, simply reversing the orthogonal/diagonal effective distances. Possibly the one advantage of the Manhattan is in effect to make the playing board 'larger' - no bad thing!
The fact is I was looking for ways to to adapt the grid system to be as near to Euclidean as could be practically achieved.
The irony of all this is that I had it in mind that if one could indeed practically to make a square-grid area resemble Euclidean space (E-space) more closely than 'M-space' or 'Ch-space', then one would be spared the effort of drawing up a hex-grid field.
I might have spared myself the effort. Fun though this is to think about, I do want the thing to have some practical effect. It is the more ironic, as not long ago I discovered how to make a pretty decent hex-field quite easily.
A sudden though has occurred to me, thinking about the knight move in chess. New posting in order methinks...
DeleteMy own motivations in favour of the grid has two points of origin. The first is that I have a loooong history in boardgaming, so the hex is a very natural 'device' for both my eye and brain. secondly, I have a bad back and can't lean across tables to do precise millimetre measuring and placement ..... much easier to visually eyeball distance and then plonk a unit down in a space.
ReplyDeleteOutside of that convenience, I think the open table is generally more attractive and troop lines can be more pleasing on the eye.
To be honest, I tend to prefer the open table myself. My first look into grids - especially the offset rectangles, was actually to carry out operations-level manoeuvres on a map, with the actual fighting going on table. It never really got off the ground, though.
DeleteEven then, the stretcher bond 'brick' system with the rectangles having an aspect ratio of roughly 100:87 was simply to replicate a hex field without having to scribe a hex field.
See also my blogpost here: https://wargaming-mechanics.blogspot.be/2017/06/square-grids.html
ReplyDeleteAnyway, I am also a big fan of gridded wargames, although I use a hexgrid (my permanent gaming table has a permanent Kallistra surface ...).
ReplyDeleteWhen designing wargaming rules for any sort of grid, I see the following issues popping up all the time. They are seperate issues, but all related to the grid.
* How to measure distances when you have different types of connectivity (e.g. edge-edge and point-point)?
* How to deal with facing and flanking in close combat situations?
* How to deal with orientation and fire arcs?
* Does melee happen when two units are in adjacent cells, or in the same cell?
* Units per grid cell ratio?
* how to maintain battle line coherency for those periods where this is wanted feature?
In a good game design, all these issues merge into one single coherent gaming engine, but it's often not that easy.
Sometimes, I would like to design a set of ruels for an irregular tiling, e.g. a pentagonal tiling of the plane: https://en.wikipedia.org/wiki/Pentagonal_tiling
There is nothing weird about this - after all - both hexes and squares are simply a discretization of the gaming surface in cells, so why not have irregular patterns? Thinking about rules might work on such a surface, might make it easier to think why you need hexes or squares.
(I don;t have all the answers, I'm just giving everyone food for thought ;-))
Just as an example why you might want to take a step back now and then.
ReplyDeleteIn my Napoleonic rules (also on a hexgrid), we mad a big deal about orientation of the units (6 different directions, or 12), because we felt orientation was needed for firing arcs and flanking manouvres.
However, by rethinking the rules, we did away with orientations, and baked limitations into the firing and melee rules:
- units do not have a specific orientation in a grid cell
- when firing at multiple targets, targets should be within the cone of a firing angle from the hex
- flanking bonuses in melee apply when 2 units are attacking target unit, entering from different hexes.
By formulating our rules this way, orientation was never made explicit, but implicit. I am not saying this might work in your preferred ruleset or would be compatible with your scale of game, but I'm just offering it as an example that issues with the grid can sometimes be solved in more creative ways compared to when you would stick to the classic no-grid mechanisms.
I have just begun encountering these problems. My reaction right now is that it is probably not worth the effort to try and resolve them! I think it can be done, but the formulation of the rules sets would be no simple matter.
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