Saturday, November 22, 2014

Big Battles for Small Tables - a digression into Kleiber's Law...

Work in progress on recent Bring-and-Buy acquisitions.  I've made
the Spanish into a single composite, probably anachronistic
20-figure unit.  The stove-pipe officer I have added is Minifigs;
the rest might be Hinchliffe, but I don't actually know.
Let me tell you about Kleiber's Law.  It comes from the field of biology, and the observations made in the 1930s by one Max Kleiber.  It states that the metabolism rate of most animals is proportional to the fourth root of the cube of their mass. Symbolically:
...where R is the animal's metabolic rate, and M is its mass.  Now the thing about  M3/4 is that the ratio M/M3/4 is the same as  M3/4/M.

A French flag bearer has been removed (he'll probably end up on
on logistics element as a horse holder); the one blue-coated grenadier
will have a white coat; A British stove-pipe shako officer has been
added, and a bicorne guy with the damaged musket became a flag bearer.
Now, how does this relate to war games and rule set design?  You will recall from way back (link) that I discussed the relationship between ground and time scale, that the latter was (i.e. should approximate) the square root of the former.
Ground scale = G = 1:3600
Time scale = T √G = 1:60.
Then I found that the movement rates according to ground scale alone and according to time alone just would not work.  Intuitively I came up with the notion that movement rates 'm' ought to be such that
G/m = m/T;
and, therefore, as it has transpired: 
m = G3/4 = 1:√216,000~ 1:465.

British Fusiliers. who might yet be elevated to Guards status.
This is an 18-figure unit; further off you see a 16-figure unit that
includes10 'genuine' guards figures. Minifigs.
Quite how 'mass' and 'metabolic rate' fits with ground and time scales is a bit obscure  - I have my doubts about it myself.  But it does go towards scaling.  It works.

Here is a 'thought experiment' I carried out whilst pondering this back in July. Imagine a small insect a half-millimeter in length (1:3600 my height in fact).  You watch for one minute it walking across a flat surface. It would certainly travel farther than 1.2 inches (Sorry about mixing measuring scales, here, but I'm going for measures easiest to imagine. Can you imagine 2/100ths of an inch?).  Will the little wight travel as far as as 6 ft though? I don't think so.

Too lazy to strip the original paint job, I'm planning to keep
 as much of the original, but add the white belts,straps, wings and
 flounders etc.  The quick white dry brush identifies highlights
as a visual aid to me in touching up these guys.

Here was my problem.  In an hour, a human being might walk, pretty briskly, 4 miles - 72 inches (6ft) at the 1:3600 scale.  That is the length of my war games table.  As my moves represent 1 hour, that's the length of the table in just one move. 'We can't be having with that,' quoth Granny Weatherwax. 

Of course, the time scale I'm using is that 1 minute represents 1 hour.  But in one minute, at 4mph, one will travel maybe 120 yards, which at my chosen ground scale is 1.2".  Hardly what you'd call 'stepping out', eh?  The answer had to lie somewhere in between.  It was sheer intuition and no more that led me to the movement value m that satisfied the condition 3600/m = m/60, or, more specifically in inches: 
72/m = m/1.2. 

They sure look rough, don't they?  I admit this is a very
experimental technique, but I'm hoping the final product
will pass muster.
 The last equation gave us the  ballpark figure for a standard move (numbers in inches):
72/m = m/1.2
72x1.2 = m^2
m 86.4,
m ~ 9.3

Russian howitzer and crew.  I'm not sure how 'Russian' the
piece is, but as the two guns that came with it were
definitely French, this is how they will go.  Minifigs.
Of course, 9.3, approximate anyway, isn't a particularly tractable number to work with, but it is in the right ballpark.  So, as a 'first pass' I rounded it to 10 inches.  Now, some quick research indicated that a standard 'quick march' would take a foot soldier 100 yards (120x30-inch paces), which approximates to an 8-inch (7.75") move under my proposed scaling scheme.  'Double time' takes a soldier about 180 yards (180x36" paces). This would be as near to 14 inches as makes no never mind.

French 8pr cannon and crew.  
At this point I am inclined to bring this 14-inch 'double time' down to a 12-inch (30mm) maximum, for march column along a road. Cross country, let us reduce it to 10-inch (25cm - our 'standard), to take into account the casual irregularities of the ground.  Our 'column of manoeuvre', 'assault column', column of companies', or however it is to be styled, has an 8-inch move; line (single rank) is down to 6-inch.  Back in July I formulated my movement rules as follow:

"Just to make things simple, and as 6km an hour is a pretty fast rate of travel, I'm inclined therefore to round things down, thus:
Infantry in march column: 25cm (10 inch) + 5cm (2 inch) on a roadway.  The assumption here is that nearing the battlefield, the troops are probably moving 'at the double'.  All other movement is geared around this benchmark.
Infantry in skirmish order: 25cm (10 inch) (I have some doubts about this provision)
Infantry in battlefield (or assault) column: 20cm (8 inch).
Infantry in line: 15cm (6 inch).
Infantry in square: 5cm (2 inch).
Light Cavalry in march column: 50cm (20 inch) + 10cm (4 inch) on a roadway.
Light Cavalry in battlefield column: 40cm (16 inch).
Light Cavalry in line: 30cm (12 inch)
Heavy Cavalry (includes 'heavy' Dragoons) in March Column: 40cm (16 inch)
Heavy Cavalry in battlefield column: 35cm (14 inch)"
Horse Artillery (3-4pr, 'light' 6pr) limbered: 35cm (14 inch)
Horse Artillery manhandled: 10cm (4 inch)
Foot Artillery ('heavy' 6pr, 8-9 pr, 5.5"-7" howitzers) limbered: 30cm (12 inch)
Foot Artillery manhandled: 5cm (2 inch)
Heavy Foot Artillery (12pr; 8" howitzer): 25cm (10 inch).
Heavy Foot Artillery may not be manhandled except to change front.
At the scale we are looking at, this might well be too much detail.  In that case, the default rules for all artillery will be those for the 'Foot Artillery'. 
Foot Routing: 30cm (12 inch) These guys aren't hanging around!
Light Horse Routing: 60cm (24 inch)
Heavy Horse Routing: 50cm (20 inch)
Heavy cavalry in line: 30cm (12 inch)  
French 12pr (I think) and crew.  Two of  these guys seem to have  been  painted up as
 Russians, and the 'Westphalian (?)' dude with the rammer  had been adapted
 from a Grenadier figure.  I replaced the missing slow match with modelling wire,
 and added plumes to the two crewmen without them by jamming  partially  depilated
 cotton bud  ends over the existing standard artillery headgear ornamentation.
Next time I'll dilate upon the combat rules and game mechanics.
Another gun crew sans ordnance.  The nearest figure is actually
a Wurttemburger artilleryman from the Minifigs range.  He
 will become repatriated as a Frenchman. A lot of these arrived
unpainted,  They have received a black undercoat, with white
dry brushed over the top.  The white details have been added.


  1. Is it April 1st already? You've made my brain hurt with all these formulas!

    I remember doing A level biology where we had to compare the volume to surface area ratio of a mouse to an elephant. No wonder a mouse has to race around just to keep warm!

    These old mini figs actually look quite nice now - but I have had a drink...

    1. I did have a bit of a think about volume to surface ratios as well. Apparently it used to be thought that as mass was proportional to volume, and surface area to Volume^2/3, then maybe the rate of metabolism would also be proportional to Mass^2/3. At that, Kleiber's 'Law' has its sceptics, though I find it so convenient to accept!

  2. Replies
    1. I've certainly persuaded myself I must be on the money - or at least close by.

  3. I'm afraid I have to agree with Stryker - this made my brain hurt and now I need a drink.

    All I know, and for comparison I repeat it here, is that for 1" equals 100 yards, and one turn equals 30 minutes, my French line infantry brigade of 1815 can cover 10" in column, 3" in line and 18" in march column, not that they would be doing the latter anywhere near the enemy. They can do 11/2" in square. Their cavalry companions get around at 14" or 15" and have a march column movement of 22". Heavy cavalry is a couple of inches slower. Limbered heavy artillery is 9", manhandled 2" and march column 14". Horse artillery is 12", 4" and 21" respectively. If they rout they can go up to 18" if they are foot or 27" if they are mounted.

    These distances are surprisingly close to what you have calculated.

    Elite troops move quicker and poor troops slower and there is a slight variation between nationalities and over time.

    If cavalry get into a winning streak they can go double or possibly even more. Good luck getting them back.

    There is also a case for formed movement versus unformed movement, the latter not being routs but when troops are withdrawing disordered.

    Fascinating stuff. I'm looking forward to what you have to say about combat.

    1. I think once you get your ground and time scales worked out then it comes down to what seems reasonable in terms of actual movement. 1inch:100yard is the same as my 1:3600 scale, but your 1 turn represents the half-hour; although not in the ball park I have suggested, it seems quite reasonable to me.

      For the rest your column moves are more generous, the linear a little less. Actually, I would have guessed from them and your ground scale you were using the 1-hour bound!

      There is bound to be a certain amount of 'informed arbitrariness' in the decisions we make about our game mechanics.

  4. Oh dear. I fear I am going to be a negative Ned again. I don't reaĺly speak or read mathematics so I can't comment on the formula but where did you get your march rates from??

    120 ppm may be a typical parade quickmarch now but not for long distances and troops are fit and well fed. I have route marched at 3 miles IN an hour (ie 50 minutes marching, 10 resting) when I was young and fit but even then I'd not have been fit to fight after even 3 hours.

    Having double checked 2 period manuals (availabe on line let me know if you want links ) to confirm memory, the British marched at an ordinary pace of 75ppm while the French were zippier at 90 ppm. (Check video of British guards trooping the colour and slow marching) 120 ppm was the equivalent of double time. Anything faster and all order is lost within 50 yards and it was all about order not speed.

    Presumably you could jigger the time factor in the formula to get the same on table moves using a slower rate?

    1. You will observe that my 12 and 10-inch moves are carried out in some sort of route column, which you might use if you needed to move a unit in a hurry somewhere. Somewhat arbitrarily I reduced the 'double quick march' to something that seemed more reasonable on the battlefield.

      The 100 yards per minute equates to 6000 yards per hour - roughly 3.3 miles per hour. A bit 'hasty', especially as the 8-inch move is slightly faster than 100yards anyhow. A 6" columnar move would be very close to your 75ppm (how long the 'pace: 30-inch or 36? A 36-inch pace, by the way, even I find unlikely, unless you're actually running. Apparently this was a sort of jog. I daresay you would not use this in any formation other than a route march.

      But see, all this formula-crunching was really designed to produce numbers that seemed reasonable. I don't think one can do much better than that. My move distances might seem too zippy - and would probably better 'fit' my original ground scale idea of 1:3000. What this does is substitute 'pace' (30-inch) for the yard (36-inch).

      There is a good deal to be said for this, but it will require me to do some more number crunching. Just by the way, the 1:60 time scale is already a 'fudge' in that case, rounded from the 'stricter' square root of ground scale, which is 1:55. But who wants a 55-minute bound?

      I might revisit this as a blog post - I think it might be worth it.

    2. 29 inch pace actually but i seem to recall the French was shorter. We were supposed to take a 30 inch pace but my shortlegs always strughled with it.

      As a point of interest people didn't normally march faster in column, except for emergencies, they are just easier to manouvre.

      Regardless of 75, 90 or 120 paces your observation on how easily troops could cross the table going at a normal march rate. Its obvious that most troops spent most of a battle standing around (hurry up and wait). That's a conundrum I struggle to reproduce in a game.

    3. In my view, the solution to the 'hurry-up-and-wait' conundrum is built into the sort of rules mechanism I have in mind. Rule writers in the past have, whether they knew it or not, done much the same thing.

      I think we have to remind ourselves what we are trying to do as rule set writers. The 'm' in the G/m=m/T is simply the compromise made between the ground scale and the time scale - and the figure scale - oh, and the fact that our 'bound' system is not dynamic but an attempt to reproduce as best we can the dynamics of battle by a series of static 'snapshots.'. The 'down time' when our soldiery isn't actually doing anything is subsumed within this compromise.

      On balance, then, I'm not terrifically fussed if my 4mph (being the absolute fastest my foot can go, and that only on a good road) is a rather on the optimistic side. Look at it this way. My table is 6ft long, which represents a whisker over 4 miles in my ground scale. My time scale is 1 move the hour. At a route march of 12 inches per turn (which is 'my' 4 miles per hour), on a road, mark you, it would take half a day (6 hours), to travel these four miles. Suddenly my 4 mph transforms itself into 2/3 of 1 mile per hour.

      Seen in that light, it is clear that the boys didn't spend that whole time marching, but must have spent a good deal of it standing or sitting or lying around. Even if their rate of march, whilst marching, was a steady 2 miles per hour, two-thirds of the morning will have been spent halted for any number of reasons.

      Incidentally, 75 paces per minute at 29" the pace, is a shade over 2 miles an hour.

      I am not totally wedded to the 1:3600 ground scale, though, the 1:3000 has almost as strong a claim to be given preference. At that point my 4mph move (12") becomes 3.33mph (again, my fastest foot movement rate), without changing anything else.

      My table length becomes 3.33 miles. It would take our road route march half a day to travel 3.33. miles...

      It seems to me whichever scale I use, the compromise allows for a great deal of slack either way. The whole exercise was indeed to determine what our ballpark numbers should be. Because some of the resulting numbers are intractable for gaming purposes, we must allow ourselves an interval that will permit us handier numbers to work with, yet will still give us, to paraphrase 'The Mikado', the verisimilitude for what would otherwise be a bald and unconvincing narrative.

      Ross, I have it in mind to reproduce this conversation - or much of it - as a substantive posting. would that be OK with you?

    4. looking back at the substantive posting, I think that 10" was supposed to be the 4mph speed, not the 12". That rather gives your argument more force, so a rethink might after all be in order. First thought: reduce all foot movement by 2", and the square to 1".

      This of course is a bummer, for, without a calculator, I'm having to do all my sums by hand. Fortunately, I know how to find square roots with pen and paper... :-)

  5. Ion, it seems that biological science is another interest that we share in common; ag scientist me, so I enjoyed the reference to metabolic live weight!

    My thoughts were along the lines of Ross Mac's, viz. the difference between a recreational march through the countryside, compared with progress across the business end of a battlefield.

    We came across a version of this in our re-fight of Eylau. The combination of the approximated time scale that we were using and the normal movement rates in the rules that we use meant that it was impossible for L'Estocq to perform his 'match winning' march around the back of the battlefield so as to be able to blunt Davout's attack. We debated with allowing march moves in favour, in this case, of manipulating the scenario and having him arrive near Anklappen, rather than his actual 'arrival point' in the area ,northern' area of the battlefield. The standard movement rates worked well for the other formations in the game, you see.

    On another occasion, in a retreat-pursuit game from the 'Campaign of Nations' that we are part of, we allowed a march move (double-rate) and it worked well.

    Nothing particularly useful in this comment, I realise, but the experience may be of interest? :)

    1. Not wishing to hi-jack the Archduke's post, but marching to battle is important in many battles and a challenge to get right. I was happy using SPI's Napoleon's Last Battles boardgame to try out some Waterloo what-ifs. Of course getting the timing of the Prussian's march (and subsequent on table moves) right is essential for Waterloo.

      Napoleon's Battles has a forced march rule. I rarely use it (there is a serious down side of straggler casualties), but in terms of Waterloo it could be interesting, but then I got to thinking of the state of the roads and that maybe they are force marching just to move at normal movement rate.

      Fascinating stuff.

    2. I've only just seen these postings, hence the delay... I have to admit I have occasionally been surprised at how nippy the lads could be when pressed, but it is not always clear that the times given in battle accounts are accurate.

      Some rule sets, notably the DB* sets, allow multiple 'march' moves in a single bound (turn). This has its points, and translates to as an action continuous for most of the period of the bound, unlike other actions, that might take place for small fractions of that time.

      But I'm not enthusiastic about it, and prefer to leave it to a route march to bring people as fast as may be from one part of the field to another. Come to think of it, an Eylau action might be a fine method of testing these relative times.

      At that, we might have to 'reverse-engineer' the timings to set up the scenario. Lestocq's Prussians (together with Russian stragglers, apparently) struck Davout's leading elements at roughly 4:30pm, which indicates he must have arrived at Althof at least an hour earlier, more likely two, it being 3 miles at least to Kitschitten from there, in a snow storm, and having to drive Davout's men from thence and Auklappen.

      Three miles according to my ground scale is a little under 54 inches. Even at a 12" forced march (which I have since decided to reduce to 10" and that on a good road), it would take more than 4 moves to cover the distance.

      The question arises; which to tweak: march speed, or arrival time?

  6. I'd go for tweaking the scenario. Special scenario rules are have less widespread impact, of course.
    Another answer, perhaps, is to deal with the illusion of time. Allow it to flex as required, rather than treat it as a linear variable. From my reading of the history of wargames rules, Empire began the 'revolution', albeit crudely, with the 'telescoping time concept'. Bob Jones took it further with piquet's card-based determination of activity. Variations on this approach seem to have been adopted by several sets since. Grist for the mill?

    1. I'm never quite sure about these sorts of systems, James. I don't mind a degree of uncertainty and chance, but I want it to stop well short of caprice - even for a solo rule set. I have tried one card based system (WW2, Italieri's Operation Overlord set), but this required the application of tactical alertness and skill. Quite how the Picquet and similar systems work, I wouldn't know.

      One method I like of bringing off table reserves onto the table is a simple system of determining an ETA ('estimated tome of arrival) and rolling a D6 two moves before that ETA. With a score of 1, the reserve comes on 2 moves early (i.e. at once), 2, and it comes on 1 move early. Three and four means 'on time', 5 a move late, and 6 two moves late.

      Many years ago in a Napoleonic campaign, set in late winter or early spring in northern France, a battle between an Anglo-Russian force and the French required the arrival of three distinct bodies of troops from off table. As it happened, only one of then arrived 'on time'; the other two were late. In the subsequent report I inferred a fall of snow that caused the delays...

      That at least has a degree of predictability, and a degree of uncertainty as well. That the opponent sees the roll for reinforcements (and possibly the result) is not implausible, someone on the field observing something afoot in the enemy rear.

      A locally grown rule set (Vive l'Empereur - I think I still have listed the blog for it though it isn't very active) uses an initiative points system, with IPs being expendable in your own or your opponent's turn (in response to something he does). In my view this is potentially a very good system, but one tended to find that the poorer troops were seriously disadvantaged, and the way the system was worked, very much favoured the attacker. Actually the bias in favour of the attacker might have had more to do with the easy 'passage of lines' as the rules stood last time I looked. Age of Eagles allows PoLs, but at a cost in movement. I preferred to allow it only at a cost in IPs for both units involved.