Chess Piece Sizing - Proposing and Testing Some Benchmarks

Hi All,

So, I’m designing a chess set, and have been researching what makes for good proportions among various chess pieces, and between all the pieces and the board. This site has been really helpful in providing insights into this area - I particularly enjoyed @Loubalch's post, ‘A System for Sizing Chess Pieces and Boards’. I’ve also looked at various reproductions from Chess Bazaar to get an idea of the chess piece sizes of various historic chess sets (I assume they are at least somewhat accurate, if at least proportionately, and if so then the sizing details of each piece has been most informative). And of course I’ve looked at some of the USCF and Fide guidelines on sizing, but given Fide has ignored its own guidelines for its official set (something that still boggles me), I’m not taking these as absolute rules.

It seems that most rules of thumb focus on:

• the size of the King (base diameter as a % of the chess square, and then the base as a % of height), and
• the base of the Pawn (the four-pawns-to-a-square rule, which seems to me a little old fashioned but included in USCF guidelines, and the two pawns-to-a-square rule which seems more contemporary and to my eye somewhat more aesthetically pleasing)

There is also the rule about the bishop being able to pass through the King and Queen when placed on diagonally opposing squares (demonstrated so well by @ifekali - I'd post the link but apparently new members can't post links or pics...!) but this doesn’t really help set the size of the Queen relative to the Bishop per se (the result can be achieved with various sizes of Queen and Bishop, with one being disproportionately larger or smaller than the other).

Leaving aside height for now and focusing on base sizes, there is effectively good guidance for the King and Pawn, some vague guidance for the Queen and Bishop, and no guidance about the Rook and Knight, though I guess Knights (and most Rooks) seem to have the same base size as a bishop.

This has been a bit problematic in designing the right sized chess pieces. This is particularly so because in many chess sets (especially, I think, in the older designs, but I’m happy to be corrected), the King can be substantially bigger than the Queen and subsequent pieces. Similarly, the Pawn can be substantially smaller than the other pieces. This means that even knowing the right size for the King and Pawn, it doesn’t follow that the other pieces should simply be sized in the middle, and / or evenly distributed between King and Pawn.

In the absence of other guidelines, and based on observations of some classic chess sets, plus a little bit of my own subjective aesthetics, I’m come up with some benchmarks for sizing the various chess pieces (with the exception of the King, which seems very well covered). I’d love to get feedback on these; I wouldn’t go so far as to call these benchmarks ‘ideal’, as so much is still down to personal, subjective taste and indeed the seating arrangement and other environmental factors. Rather, it would be interesting if these benchmarks could delineate sets which are a little more on the ‘crowded’ side (preferred, perhaps, in Eastern Europe), vs sets that are less crowded (preferred, perhaps, in Western Europe, from what I can tell); or sets that have, say, pawns or minor pieces that are on the small or large side relative to the rest of the set.

Anyway, the benchmarks I’ve come up with (unless someone else has already formulated something very similar), are:

1. Base of the Rook + Bishop + Knight in single file = width of 2 squares
2. Base of 2 Queens + Pawn in single file = width of 2 squares
3. Base of the Rook + Bishop + Knight + Pawn in a diagonal fit within a 2x2 square

These benchmarks would seem somewhat arbitrary, but here are some statistics on various sets, based on CB reproduction proportions and stats on the FIDE official set. I included a couple of older style sets like the 1851 Morphy and 1900 Marshall sets, as well as more popular sets like the Lardy, Cavalier, Dubrovnik (of course!) and FIDE, to get a broad representation of styles. 'Target Pawn Size' is what I calculate based on the 2x2 square rule, assuming R+B+N was the width of 2 squares. All measurements in mm, for the base diameter of each piece.

Notes:

• Target Pawn size is rounded to the nearest mm, and based on R+B+N length being equivalent to 2 squares. The ‘target square size’ is just estimated to illustrate how the sizing of the set would look relative to the King, and is effectively about half the size of R+B+N. I’m not entirely familiar with the size of boards available - I thought 55mm was the norm, with 50 and 60mm available. Interestingly, CB list the French Lardy Staunton set with a recommended board size of 52mm - a size which I didn’t think was common, and isn’t even offered in CB’s own board selection. So I’ve put down somewhat uncommon / theoretical board sizes for the ‘target’ board size estimates (which again, are more for illustrative purposes).

• The target Pawn size takes a bit of calculation, but if you’re curious, it’s below following (not sure if it’s the simplest form of the equation, but it gets the job done). It assumes the rook / biggest piece is placed in the corner of the 2x2 square, followed by the Bishop and Knight (which I think matters, if the rook is of a different size, as is the case in the FIDE set).

P = (2√2) / (√2 + 1) x (R+B+N) - R - 2/(√2 + 1) x (B + N)

• The resulting Pawn size seems to generally be slightly smaller than that recommended by the two-pawns-to-a-square rule / Loubalch’s 0.765 (or 0.586) ratio - by about 1mm for a 55mm square board - but somewhat bigger than the four-pawns-to-a-square rule.

What do people think? Are these benchmarks reasonable and useful, or am I missing something? (apart from determining suitable heights, which I’ve put off for now).

And what I’m particularly curious about: how do people’s favourite (or least favourite!) sets line up with the proposed size benchmarks? Do they confirm what people like/dislike about various set dimensions?

Kohpablanca,

Glad to see I'm not the only one interested in the dimensional aspect of sets and boards. I like your work-up. I'll need to take some time to absorb it before commenting.

It was my feeling that since the pawns make up half of all the pieces on the chessboard, I felt the dimensions of the pawns should be considered independent from all the other pieces, except the king. Once the king and pawns are scaled accordingly (the largest and smallest pieces on the board), the remaining pieces can be aesthetically proportioned to 'fill in the gaps.'

You might be interested in the following Linear Slope Design I came up with that uses a constant height differential between adjacent pieces, providing the set with an aesthetically pleasing linear slope.

I've worked up the design for three different size sets:

Kohpablanca,

Here's the dimensional table I worked-up for the Staunton Castle 1950 Dubrovnik set on a 2-1/4" chessboard (all dimensions were measured first-hand). Notice how close the ratios of the king and pawn are to those I recommended (Kd=76.5%, Pd=58.6%). A very well-balanced combination indeed!

Kohpablanca,

Okay, I get the first two equations, but I'm not sure how to visualize the third, fitting the diameters of the rook, bishop, knight, and pawn into a 2x2 square?

Simplifying the first two formulas, I come up with the following where: Queen diameter = (Qd), Rook diameter = (Rd), Bishop diameter = (Bd), Knight diameter = (Nd), Pawn diameter = (Pd), and Square size = (S).

1) Rd + Bd + Nd = 2S

2) 2Qd + Pd = 2S

Selecting four of my sets that spec out very close to ideal suggested by my system, and applying your first two formulas, I came up with the following results:

3.75" 1950 Dubrovnik set from Staunton Castle on a 2.25" board - [2S = 4.5"]

1) 1.5" + 1.5" + 1.5" = 4.5"

2) 3.2" + 1.27" = 4.57"

-------------------------------------

4.1" Soviet Era set from Chess Bazaar on a 2.25" board - [2S = 4.5"]

1) 1.5" + 1.5" + 1.5" = 4.5"

2) 3.26" + 1.32" = 4.58"

-------------------------------------

4.125" New American set from The Chess Piece on a 2.375" board - [2S=4.75"]

1) 1.6" + 1.6" + 1.6" = 4.8"

2) 3.5" + 1.4" = 4.9"

-------------------------------------

4.375" Piatigorsky (Steiner) set from Offical Staunton on a 2.375" board - [2S=4.75"]

1) 1.55" + 1.55" + 1.55" = 4.65"

2) 3.6" + 1.3" = 4.9"

-------------------------------------

Kudos, I think your formulas are very creative and represent a close approximation when applied to well-balanced chess sets.

Good. You two have met. Let me grab the popcorn. This will be fun to watch...

Welcome aboard! Should be an interesting flight.

Here's the set that started it all (first pic below). A nice set at a nice price, but I kept feeling that the pawns were just too small. So much so, they looked like they belonged to a smaller chess set. Yet, when I measured the pawns they were right in line with the '4 pawns to the square' guideline recommended by FIDE, which is based on the pawn diameter of the original 1849 Staunton chess set by Jaques of London (and has yet to be updated for the 20th, let alone, the 21st century).

I had a number of other chess sets lying about and started replacing pawns until I found ones that looked a lot more balanced. When I did the 'pawns in the square' test (pigs in a blanket), I found that two of the larger pawns fit almost perfectly (when placed diagonally) inside the square. From there, I did the calculations and discovered that a perfect fit occurred when the pawn diameter was exactly 58.6% the size of the square. It was later I discover that the square root of .586 equaled .765 (76.5%), which turned out to be smack dab in the middle of the recommended range for the diameter of the king. Finding this more than coincidental, I took this as a sign from the Deus ex Numerus that these were the ideal dimensions for a well-balanced chess set. Amen! And who am I to argue with the Gods?

Here's the same set with the larger (2 to the square) sized pawns. Compare the two sets in the following post. Which set looks more balanced to you? Feel free to chime in!

Here are the two sets side-by-side, so to speak, to make it easier to compare.

Four to the Square (small) Pawns (they were too short as well)

Two to the square (larger) pawns

Proportionally, I felt the larger pawns looked much better, but YMMV.

Chess Set Sizing - Pieces vs Board...

I tend to agree with loublach's perspective on the size of pawns expressed above. 

However...

For the overwhelming majority of chess sets (pieces) manufactured, and offered for sale by chess equipment retailers, the primary sizing criteria for purchasers of sets vs boards is that of square size vs King base diameter...

In those cases a good rule of thumb is...
King base diameter = 0.75 * square size
or equivalently....
Square size = 1.33 * King base diameter

Any variation from these ratios, or the size of individual pieces and pawns, becomes a matter of personal preference...

http://blog.chesshouse.com/how-to-select-the-right-size-chessboard-for-pieces/

@Loubalch!

Thanks so much for your replies, the statistics on the different chess sets, and the pics. That’s been really helpful, not to mention encouraging.

Firstly, to clarify one of my sizing principles: in looking at the base diameters of the R+B+N+P, it's similar to the 2-pawns-in-a-square (diagonally) principle. So for a 55mm board, place the 4 pieces in a diagonal, to fit in a 110mm x 110mm square (ie "2x2 squares"). I'd post a diagram, but unfortunately being a new member I can't post links or pics, which is incredibly frustrating. Anyhow, I think this check is easy enough to do / visualise, even if the maths is a little messy. To simplify the latter, if R=B=N, then the target size of the Pawn under this principle is 0.858 x R (or B or N, of course).

Next, regarding piece height - thanks, I did see your suggestion on a uniform, linear progression of piece heights, from King downwards. I've decided against that idea, for a few reasons:

1. The asymmetric line-up of chess pieces means that for one side, the progression is indeed K-Q-B-N-R, but obviously for the other side it would be K-B-N-R, so that the top of the pieces can never descend uniformly in the desired manner.

2. As far as I can tell, classic and other acclaimed chess sets have not sought to achieve this outcome in piece heights

3. Researching the development philosophy of the FIDE chess set, I noted that the designer Daniel Weil was trying to reference the parthenon's facade (as I suppose an architect is wont to do!), with the pieces representing columns and their heights alluding to the triangular roofline.  Alas, I can’t post a link, but it’s at designweek .co.uk/issues/march-2013/daniel-weil-redesigns-the-chess-set/ (just copy and paste, and delete the space between designweek and .co). In any case, even with this goal in mind, he didn’t use a uniform height decline from King to Rook, which he presumably would have if the aesthetics allowed for it.

4. Lastly (and perhaps, least), FIDE’s recommended heights for pieces do not follow a straight line.

With regard to the sets you provided measurements for - they all have equal bases for Rook, Bishop and Knight, so my calculated target Pawn size is pretty straightforward - just 0.858x the base of the minor pieces. This gives:

I think this is a pretty good result, in that my estimated target Pawn size is sometimes a little larger than actual (as in the Dubrovnik and Piatigorsky sets), and sometimes a little smaller (Soviet Era and New American). But in each case, only by a small margin. Actually, I couldn’t ask for a better outcome!

FWIW, here's some background to my thinking on the size of the Pawn. You mentioned that, given they make up half of all pieces on the chessboard, they should be considered independently from all other pieces except the King (that and, presumably, square size - hence your usage of the 2-pawns-to-a-square rule). I think we agree that pawn sizing is particularly important given their preponderance in a chess set. Where we differ, though, is that I think pawn sizing should take into account the other major and minor chess pieces, rather than just the King, since:

1. The major and minor pieces outnumber the King 7 to 1, and so their relationship to the pawns will be more prominent
2. Some (older) chess sets have somewhat outsized Kings, with much smaller Queens and other pieces. Sizing the Pawn to the outsized King would lead to a disproportionately large Pawn.

My Pawn sizing outcome leads to pawns that are slightly smaller than what is suggested under the 2-pawns-to-a-square rule; the difference is marginal - as mentioned, about 1mm difference on a 55mm square board.

On this point, Loubalch, I noted in one of your earlier threads, ‘Mis-scaled Chess Sets’, that @bananamoon posted a picture of their favourite chess set which they found to have suitable proportions (Edit — link here:  https://www.chess.com/forum/view/chess-equipment/mis-scaled-chess-sets#comment-29947300 ). Anyway, I thought the proportions looked fine, even if the King was a little small relative to the chessboard’s 55mm squares, as Bananamoon noted. And you mentioned that you thought the set looked good too, even with the issue of the King size, mostly because the pawns filled the squares sufficiently, and the other pieces also fit. So… I looked into that set, which was from Chess Bazaar - link: chessbazaar .com/tournament-series-staunton-chess-pieces-with-german-knight-in-sheesham-box-wood-3-75-king.html (again, copy ‘link’ and delete the space between chessbazaar and .com). The King is 96mm high, but only has a 36mm base, which is small vs a 55mm square board (0.655 ratio). BUT, here’s the thing: the Pawn is only 27mm, which on a 55mm square means just under a 0.5x ratio, conforming to the 4-pawns-to-a-square rule. And yet, they didn’t look too small. This suggested to me that there is a reasonably broad range for acceptable Pawn size, relative to the square size, and that other factors might influence the outcome. I noted that for the set in question, the Pawn has what I think is quite a big head, which may offset the otherwise ‘small’ base.

This example, as well as the general 4-pawns-to-a-square rule, prompted me to look for a rule that would provide a pawn size somewhere between what you’ve suggested, Loubalch, and the 4-pawns rule. The one I’ve come up with does that (albeit towards the larger end of this spectrum, which I think suits my aesthetic). The results from the 4 sets you pointed out, as well as the 6 sets I detailed in the original post, are also supportive of the sizing rule, since my calculated pawn size was close to the actual pawn sizes of the chess sets, with no material bias of being consistently smaller or larger.

Well, that’s another long post…!

PS - Loubalch, your tables look great! How do you format the colors here? (or do you format them in another program, and then copy/paste into the post?)

@RussBell - Thanks. Agree that personal preference is always a big factor.

But the same could be said of say art and beauty, and yet there is art theory, and various principles that lead to pleasing aesthetics, even if such principles are generalities and not always adhered to.

Further, the square size to king base diameter criterion is primarily a rule of thumb for people choosing a chess set and board combination, and relies on the designer of the chess set to properly scale all other pieces relative to the King; it does not provide useful guidance for someone actually designing a chess set. So in the extreme, a King diameter that is 0.75 the square width may be fine, but if I design the Queen to be 0.95x the square width, or the Pawn to be 0.1x the square width, the set would probably be regarded by the vast majority of people to be incorrectly scaled.

Sorry for this thread disappearing temporarily! Was muted due to trumped up charges of spam, but I’ve been exonerated.

Also, looks like I can now include links, so I might edit my earlier post and put some working links. Oh, sweet, sweet freedom!

Kohpablanca,

Glad we're back on track. The reason I don't stress the relationship between the pawns and other major pieces is based on the assumption that in a well-balanced chess set the all the back-row pieces would be proportionally scaled to the dimensions of the king. This being so, by scaling the pawns to the king, they would also be proportionately scaled to the other back-row pieces as well. To express this in mathematical terms: if A=B and B=C, then A=C. If the other pieces (A) are proportional to the king (B) and the king (B) is proportional to the pawns (C), then the other pieces (A) will be proportional to the pawns (C), which eliminates the need to scale the pawns and pieces separately. For if the king isn't properly scaled with the other back-row pieces then, by definition, it isn't a well-balanced chess set, regardless of the dimensions of the pawns.

Plus, I like the idea of initially balancing the largest and smallest members of a set, for if the extremes aren't balanced, it matters little what falls in between, 

Loubalch,

Yes, great to be back on again!

I think my issue is that we shouldn’t assume the back row pieces are scaled properly (and indeed, if you were designing a chess set, you would need to somehow formulate what an appropriate sizing should be).

Just as we can’t assume that the pawns are appropriately scaled to the king (as you noticed in one of your sets, which kicked off your research into proper sizing), so in the same way we can’t assume the other back row pieces are scaled appropriately. Your assumption works, as you say, ‘in a well balanced chess set’; I’m trying to deduce some principles that indicate if a set is indeed well balanced.

All this is perhaps more important when one is designing every piece from first principles, and not just choosing a fixed set where the balance has already been struck (or not, as the case may be).

Thanks, Endgame. I’ll see if I can post a few pics tomorrow, otherwise maybe on Saturday. It’s almost 1am here in Sydney so I should get some sleep!

Kohpablanca,

We can compare chess piece dimensions in the horizontal plane (using their diameters) or in the vertical plane (using their heights), but if these comparisons are made separately we're still operating in only one dimension at a time. Why not compare chess pieces the way our brain perceives them, as three-dimensional objects, by measuring and comparing their apparent volumes as well. Since I have neither the time nor the inclination of measuring the actual physical volumes of all six pieces (king, queen, bishop, knight, rook, and pawn), I decided on adopting a short-hand method by visualizing each piece as a cylinder using each piece's diameter and height as the diameter and height of an imaginary cylinder. Visualize, if you will, each piece snugly enclosed in a glass cylinder, and what we're comparing are the volumes inside those cylinders. I started fiddling around with such a design a few years back and figured I'd get back to it when I coughed up enough 'ca'ching' to acquire the necessary software and the time to learn how to use it. Theoretically, the Engineer's Design (below) was my initial sketch.

I was curious to see how a set I designed using the 1D method, in this case, my 4" Linear Slope Design, would spec out when measured volumetrically. I was surprised to see the ratios of the 1D system (.765 and .586) showing up here as the volume rations of the queen and bishop to that of the king, as these ratios were not massaged into the original design.

I took it a step further and designed a set from scratch that factored in the volume ratios as a basic criterion of the design. Here I was able to bring the queen and bishop volume ratios closer to the 1D ratios (.765 and .586), without resorting to the trick of reducing the measurements to a hundredth of an inch to make it work out.

CAD and 3D printers. Just my 2 cents.

If you want to work up a CAD rendering based on either of these dimension sets, be my guest, brother! As computer graphics are not my ken (I'm Lou!).

Love to, but can’t - I don’t dabble in that. You know does? Izmet. 

I was talking about rendering as 3D drawings not the manufacturing of the pieces.