Tuesday, 16 February 2016

Comparing hexagons with squares

Some time ago I injured my lower back, and for the last week or so I have been suffering from lumbago ... something that I thought that only old people got! As a result I have been finding it difficult to sleep, and last night I tried to occupy my sleepless night by thinking about hexes and squares.

Now for gridded wargames hexes have lots of advantages over squares, but they are not easy to draw and sometimes terrain and/or units don't fit into them very easily. As a result I have swung backwards and forwards between using hexes and squares in my wargame designs. In the end I have to used hexes for wargames set in historical periods when tactics tended to be non-linear, and squares for periods when they were linear.

Last night I began thinking about the area of squares of a given face-to-face distance when compared with the area of hexes with the same face-to-face distance, and this morning I set up a simple speadsheet in order to look at the results. (Please note that:
  • I have used the approximate ratio of 4:3 when comparing the sizes of squares and hexes and
  • the distances chosen for the second chart are approximately the imperial equivalent of the metric distances shown in the first chart.)
The first chart compares squares and hexes where the face-to-face distance is in centimetres.


The second chart compares squares and hexes where the face-to-face distance is in inches.


This was quite an interesting exercise, and besides helping me get some much-needed sleep, it has given me something to think about regarding the uses of squares rather than hexes.

14 comments:

  1. Bob
    Interesting enough but I am not sure where it gets you? The area of any hexagon is three quarters the area of a square with sides the same length as the width between the faces of the hexagon. On any given table however only one of these shapes will be in use so what aspect of a set of rules does it impact on?

    I get the relevance of the 'length' of various parts of the two shapes (so firing/moving diagonally with a square grid is anomalous) but not sure there's anything more to be learned from the difference in two dimensions? Please feel free to enlighten me.

    Oh and as a by the by, have you considered offset squares (occasionally called squexes) as the hybrid panacea to your gridding problems? Functionally identically to hexagons, the area of squares and easy enough to construct.

    Cheers

    Andrew

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  2. Interesting. I know some people simply do not like grids, finding them difficult to become second nature, which I thought strange, until I tried using squares.

    I have boardgames for years, so hexes are very natural to me, my mind just accepts them and manages them very easily, then I tried squares and shuddered to a halt .. That in itself is interesting, after all, a grid is just a grid regardless of format .... But apparently not.

    Squares feel very regimented, chess like and do not flow easily - hexes much less so.

    Is it just me .... I don't think so as the boardgame industry has had 50 years to evolve to the most useful system and it still stays with hexes.

    For those of us that game in small spaces, hexes do offer the additional advantage that in any given space, you do get more 'cells' or locations than you would with similar sized squares in relation to the depth of the battlefield.

    Having said all of that, I continue to experiment with squares. I like WWII and so facings, turrets, fire arcs etc seem to be my biggest stumbling block.

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  3. What about the way Crossfire does things, which is basically like an area movement boardgame, only with figures.

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  4. Rumblestrip (Andrew),

    As I tried to explain in my preamble, this was a mental exercise done late at night whilst I was unable to sleep. I did it in order to help me to sort out in my own mind whether or not to persist with hexes rather than squares for my future mini-campaigns. My mini-campaign board is 3'/90cm x 2'/60cm, and I currently manage to get a 8 x 6 grid of Hexon II 10cm hexes on it. If I moved over to 10cm squares, the grid would be 9 x 6. In other words, my squared grid would have 6 more grid areas (i.e. It would have 11% more grid areas on the same size of board). This is a not insignificant increase in the number of grid areas without an increase in the total size of my tabletop and each grid area would be one-third bigger.

    I have thought about using offset square grids, but as I already have a load of Hexon II and Heroscape hexes, I would prefer to use them or a normal squared grid.

    All the best,

    Bob

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  5. Norm,

    You have pointed out very succinctly why hexes are more popular than squares ... and why people who will just about accept hexes seem to baulk when it comes to squares. Interestingly Joseph Morschauser advocated squares because he found that the people who used his rules had fewer problems adapting their thinking to moving their units on a grid of squares. It is also interesting to note that the British 1956 Army War Game used maps with a grid of squares.

    Good luck with your own experiments.

    All the best,

    Bob

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  6. Phf,

    I've never used CROSSFIRE, but I have used board games that employed area movement. Personally I prefer a more regimented grid system ... but I am always open to persuasion.

    All the best,

    Bob

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  7. I was going to suggest "staggered squares" (which I think is what Rumblestrip was suggesting). They are a reasonable alternative between hexes and a grid . . . but they are much much easier to apply to a table surface than hexes.


    -- Jeff

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  8. Bluebear Jeff (Jeff),

    My only objection to the use of staggered squares is that the ranks of squares only run in one direction ... and that objection is purely an aesthetic one.

    All the best,

    Bob

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  9. Bob

    "My mini-campaign board is 3'/90cm x 2'/60cm, and I currently manage to get a 8 x 6 grid of Hexon II 10cm hexes on it. If I moved over to 10cm squares, the grid would be 9 x 6."

    Ah I see - it's a practicality thing to do with space rather than a functionality aspect of rules and gaming. Sorry I didn't get that the first time round.

    Trust your sleeping is better. My own insomniac musings would be the probabilities associated with multiple different dice combinations. Try that - it's like counting sheep but on steroids!

    Cheers

    Andrew

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  10. Rumblestrip (Andrew),

    As you have now gathered, what I am trying to do is to fit the maximum number of grid areas into the space that I have available for my min-campaigns. I currently use hexes, but wondered if a move over to squares might be better. I may yet give it a try just to see if the extra grid areas that would fit might improve the wargames I fight.

    As to my sleeping ... well last night I fell asleep very quickly, but at about 3.00am I must have turned over and the pain woke me up. I finally managed to get back to sleep at about 4.30am, only for the cat to wake my wife and I at 6.30am!

    All the best,

    Bob

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  11. Bob,

    I like both squares and hexes, so I have nothing to add there.

    I do want to sympathize with your having to cope with back pain. I had full lumber vertebral fusion 3 years ago, and pain in one part of my back or another since then, and the pain can be unbearable. I originally thought I'd at least get some painting done while on the mend, but I tend to hunch over a bit when I paint, so that was out (and that's not even taking the distraction from the pain into account). In addition, I hated how I felt when taking prescription pain medication. I found relief by taking ibuprofen and acetametophin at the same time (as my doctor advised).

    In any case, take care, and I hope you're back in action soon!

    Best regards,

    Chris

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  12. Chris,

    Thanks very much for your expression of sympathy ... and the advice about the right sorts of medication to take. Like many medical problems, if you haven't suffered from a bad back it is almost impossible to understand how difficult it can make normal living.

    My back problems started some time ago, and began with a bit of muscle strain after I overstretched trying to take a suitcase down from on top of a wardrobe. I thought that it was empty ... and it wasn't! It had just begun to get better when I the muscle was damaged yet again during a hurricane at sea. A short but heavy woman came through the doors of a lift just as the ship lurched. She shot across the open space between the lift doors and the wall with her arms flailing, and collided with me so hard that I then fell back against the handle of a door, hitting the exact spot where I had damaged my back muscle. Having used me to stop her forward movement, she fell over, and trying to help her up I strained the muscle even more.

    Since then the muscle has caused me a great amount of pain, and has made swiveling my body at the hips or turning over in bed almost impossible without considerable effort and agonizing pain. It is only during the last few days that the pain has begun to abate thanks to the combined use of painkillers, heat pads, and a rub-in ointment that the local pharmacists recommended. Luckily I can sit and type on my computer keyboard, but I have yet to try to do any painting as - like you - I tend to hunch over when I paint.

    I am hoping that another week or so of treatment will see the problem solved. If not, then I will have to pay another visit to my doctor.

    I hope that your back problem gets better eventually.

    All the best,

    Bob

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  13. Bob,

    Is your formula correct?

    When keeping the face-to-face distance identical (let's say x), a square has area x*x, but a hex has area x*x*3/(2*sqrt(3)). That's 0.87*x*x. Thus, a hex is not 75% of the area of a square, but 87% of that area.

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  14. Phil Dutré,

    I used a formula that I found in an old maths text book that I was given when I first started teaching in 1973, and that was in the era of what was called in the UK 'new maths'. In other words, it was not the style of mathematics that I was taught where one worked from the brackets outwards and not in a linear left-to-right manner used in 'new maths'. Looking at it now, I can see that I may not have transferred the formula correctly when I set up my initial spreadsheet. Your formula looks the same but is set out in a way that I understand ... and when the numbers are added, it is 87% and not the approximate 75% that I used on the charts I produced.

    All the best,

    Bob

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