Chess Piece Sizing - Proposing and Testing Some Benchmarks

I'm actually working on a set in CAD. Thanks for the dimension posts! It's helped!

Great! Please post pictures of the set when you when it's finished.

Actually, those are what started me down this rabbit hole. I recently found a CAD app for the iPad (Shapr3D), and figured in this day and age it’s not unreasonable to design something to be 3D-printed. And I’ve had an idea for a chess set for forever, which I thought may now be possible to realise.

Following are some drawings for my idea. Keen to hear what people think about the proportions (though I understand it’s probably a little hard to tell just from 3D drawings). I think it might be best to not detail the measurements in this post, so as not to influence people’s opinions — but happy to disclose them in a later post. For now, suffice to say I’m planning for a 95mm high king for a 55mm square board. I want the set to be as practical and up to tournament standards as possible... with one main exception.

And yes, Loubalch, I’ve considered the volume of each piece — quite a lot, in some instances (but I’ll get to that later).

The general look and feel I’m going for in the set is towards a more modern / minimalist aesthetic, but with easily recognisable Staunton features which make the set practical rather than just ornamental. Most modern chess sets (like the Man Ray, Yves Tanguy or the Bauhaus sets), are too abstract, with little or no reference to the Staunton design, which make them impractical. The only modern ones I really like are the Berliner and the Noj Stage 1 Blitz. The Herman Ohme set is also really nice, but the pieces are impractically tall and their ease of recognition is borderline (mostly due to the slightly ambiguous king and queen distinction).

Anyway, my design for the white pieces:

As for the black pieces, my idea has been to make them square and angular, almost harsh, in contrast to white’s softer, circular and curved form. The same design, essentially, but diametrically opposed. A mirroring of the opposition of the pieces in the game, and indeed the opening board setup with the opposing king and queen — same, yet opposite. This was somewhat tricky, as I obviously wanted to ensure the black and white pieces were of the same size (volume), even with a different form. I’ve managed to keep the differences down to < 0.5%, which I think will be unnoticeable visually and by feel / weight. My OCD tendencies had to be checked at some point!

So, with the black pieces:

I’m no expert but are there tools you can throw your wireframes into another software to add textures like Maya? Then you can place them over backgrounds or landscapes even like a 2D chessboard perhaps?

i figured that’s how you can easily illustrate the piece : square ratio that has been Mathed out in a few postings here.

Oh, that is way beyond me for now...! But I’m a about to pull the trigger on a pro subscription for the app, or at least the 7-day free trial. That will allow me to draw up a chessboard and place the pieces on the squares. I just need to transcribe all my drawings onto paper, just in case I lose my sketches in the changeover to the pro version. Unlikely, but I don’t want to risk losing all my work getting to this stage — it’s been many, many hours!

I just want to say this is a really interesting thread. I need to pore over the figures for a while but I really appreciate the work that's gone into this project.

Make a back of your computer before the upgrade. Any upgrade. Restore from backup as needed. Always Good practice to do this anyway. 

Ah, thanks Skeleton! Hope you keep following, and let us know your observations / findings when you go through the numbers!

[Be forewarned, This is a long, detailed, and quite possibly, a boring exercise in "mathematical noodling." And may only be of interest to those seriously interested in the mechanics of chess set design (at least the way I seem to be approaching it). If it comes off a bit abstruse, I apologize and blame it on my OCT.]

Kohpablanca,

In the Engineer's Chess Set design, mentioned in post #17 above, I went beyond the basic system of matching chess sets and boards to develop a system, similar to your own, for establishing the dimensions for all the pieces, not just the king and pawns. Starting with the constants developed for my Basic System (.586 and .765), I came up with other constants, derived from these two numbers, that inform the dimensions of all the remaining pieces.

In designing the Engineer's Chess Set (the working title), I began with some basic design principles.

Axiom #1: A good chess set should be designed for a specific size chessboard.

Axiom #2: The King’s diameter should occupy 76.5% the side of the square.

Axiom #3: A pawn’s diameter should be such that two pawns will fit exactly into a square when placed diagonally. This occurs when a Pawn’s diameter is 76.5% the diameter of the King.

Axiom #4: The Queen should be scaled, height and width, to that of the King.

Axiom #5: The remaining pieces should be scaled to the King, Queen, and, to each other using the following formulas. As an example, ideal dimensions are calculated for a 2.5” chessboard.

The following calculations are derived from six constants - .586, .765, .875, .935, .905, .450, all of which are mathematically interrelated.

CALCULATING PIECE DIAMETERS FOR A 2.5" CHESSBOARD (S=2.5")

The diameter of the King is derived from the size of the chess square (S):

• K_{D =} (.765 x S) = (.765 x 2.5”) = 1.91” ≈ 1-15/16” (1.9375”)

The diameter of the Queen is derived from the diameter of the King:

• Q_D = (.935 x K_D) = (.935 x 1.9375) = 1.81” ≈ 1-13/16” (1.8125”)

The diameter of the Bishop is derived from the diameter of the Queen:

• B_D = (.905 x Q_D) = (.905 x 1.8125”) = 1.63” ≈ 1⅝” (1.625”)

The diameter of the Knight is derived from the diameter of the Queen:

• N_D = (.905 x Q_D) = (.905 x 1.8125”) = 1.63” ≈ 1⅝” (1.625”)

The diameter of the Rook is derived from the diameter of the King:

• R_D = (.905 x K_D) = (.905 x 1.9375”) = 1.74” ≈ 1¾” (1.75”)

The diameter of the Pawn is based on the diameter of the King:

• P_D = (.765 x K_D) = (.765 x 1.9375”) = 1.48” ≈ 1½” (1.5”)

CALCULATING PIECE HEIGHTS FOR A 2.5" CHESSBOARD (S=2.5")

The height of the King is derived from the size of the square:

• KH = (S/.586) = (2.5”/.586) = 4.266” ≈ 4¼” (4.25”)

The height of the Queen is derived from the height of the King:

• QH = (.875 x KH) = (.875 x 4.25”) = (.875 x 4.25) = 3.72” ≈ 3¾” (3.75”)

The height of the Bishop is derived from the height of the King :

• BH = (.765 x KH) = (.765 x 4.25”) = 3.25” ≈ 3¼” (3.25”)

The height of the Knight is derived from the height of the Bishop:

• NH = (.935 x BH) = (.935 x 3.25”) = 3.04” ≈ (3.0”)

The height of the Rook is derived from the height of the Queen:

• RH = (.765 x QH) = (.765 x 3.75”) = 2.87” ≈ 2⅞” (2.875”)

The height of the Pawn is derived from the height of the King:

• PH = (.586 x KH) = (.586 x 4.25”) = 2.49” ≈ 2½” (2.5”)

You may have noticed that throughout our calculations we are using five constants – .586, .765, .875, .935, and .905. As it turns out, these are not random numbers. There exists an interesting relationship between numbers. Starting with the number - .586, the next three constants (.765, .875, and .935) are derived through a square root progression of first constant, .586, as follows:

• Second Constant = (√.586) = .765
• Third Constant = (√.765) = .875
• Fourth Constant = (√.875) = .935

 The final constant, .905, is the median between the third and fourth constants .875 and .935.

• Fifth Constant = (.875 + .935) / 2 = .905
  

 If all this isn’t strange enough, the diameters of the queen, bishop, knight, rook, and pawn can all be derived by ADDING pairs of constants.

CALCULATING PIECE DIAMETERS USING THE FIVE CONSTANTS (For 2.5" boards ONLY)

• Q_D = (C₃ + C₄) = (.875 + .935) = 1.81”
• B_D = (C₂ + C₃) = (.765 + .875) = 1.64”
• N_D = (C₂ + C₃) = (.765 + .875) = 1.64”
• R_D = (C₃ + C₅) = (.875 + .905) = 1.78”
• P_D = (C₁ + C₃) = (.586 + .875) = 1.46”

Comparing these figures with those derived above, the results are nearly spot-on.

EXAMPLE

Using these formulas, here are the dimensions of a chess set designed specifically for use with a 2.5” chessboard.

Interestingly enough, the original constants (.586 and .765) popped up again, this time as the volume ratios of the queen and the bishop, with respect to the king's volume (highlighted in yellow above).

Now this is how you can start talk about scaling. Hope the you can see the slight changes. 

Oh, lots to think about; thanks loubalch!

I’ll have to consider the various numbers and dimensions, but one thing I would observe is that the ADDING of the ratios to get similar base dimension results to that calculated earlier is actually just coincidence, and would not work for any other square size. This is because the ratios are precisely that — ratios, without units. So if you redid your calculations based on say a 2” square board, you would obviously come up with smaller base estimates for pieces, while the ratios (being constant), would be unchanged when added up.

Also, one thing I didn’t understand: I thought you calculated a larger diameter for the Rook, but in the table you made it the same as the Bishop and Knight. What was the reason for this?

Which table? I see 1.78in for the rook.

This was written up several years ago. Some of the formatting got screwed up during the cutting and pasting. I had worked up a few slightly varied Dimension Tables for this design, and have since replaced with the correct one. Also added some clarifications. Sorry, once I found this write up, in my eagerness to post it I didn't review it carefully enough before posting. Hopefully, it makes better sense now (hopefully). Thanks all for your patience.

Sorry, updated to the working dimension table.

Agreed, the shorthand method of arriving at the dimensions by adding the various constants ONLY works for a 2.5" chessboard. I included it because I found it to be a, "wow, this is an interesting coincidence!" The earlier calculations, based on the size of the chessboard would work for any size board. Thanks for pointing that out.

Posted the wrong table (since updated). I drew up a few with slightly different measurements just for comparison.

Loubalch,

Looking at the numbers, there’s obviously a reliance on the 0.765 and 0.586 ratios (particularly the latter, which arguably gives rise to the former and subsequent ratios, though you might say that you derived the 0.765 ratio independently based on recommended King diameter ranges from various sources). The 0.586 in turn is driven by the 2 pawns to a square idea.

I appreciate the recurrence / pattern of the ratios used to then drive other piece dimensions. It is, perhaps, like the occurrence of the golden ratio / phi / the fibonacci series in nature, which pops up in all sorts of places, and there is a numerical beauty in that. Even more so, it is generally accepted that the golden ratio is aesthetically pleasing, which would potentially make it especially useful in designing chess pieces. I had some initial thoughts of employing phi for just such a purpose, but quickly found that I couldn’t really derive any dimensions that would fit commonly accepted ranges for king and pawn sizing, and promptly gave up.

So I guess I’m keen to determine (at least for myself) whether, for whatever reason, the ratios you’ve proposed are indeed suitable to make an aesthetically pleasing chess set (beyond, say, numerical cohesion). Obviously aesthetics are going to be subjective, and while the golden ratio could claim to be aesthetic by common opinion, it is hard for other ratios to make the same claim. Still, it is not an entirely futile task, I think. As you can probably tell from my earlier posts, the way I’ve approached the task of determining ideal chess piece proportions has been to consider the proportions used by well known and loved chess set designs (as well as official guidelines for tournament play). If I can come up with principles that lead to proportions similar to such sets, with differences being small and/or evenly distributed around my ‘target’ dimensions, then I’d be quite confident of their aesthetics. There would be a kind of empirical evidence behind my chosen dimensions.

My uncertainty around the 2 pawns in a square rule is that, while I think it leads to perfectly acceptable pawn sizing, it is right at the upper end of pawn sizes. I may be wrong, but I suspect that if you compare your target pawn sizes to the pawn sizes of well known and loved chess sets, say relative to the king (since the exact square size the designer had in mind is not always certain), most sets would have pawn bases similar to yours or significantly smaller, and very rarely larger. (The Mechanics Institute set may well be an exception! But it looks like their Queen base is bigger than the King’s, based on the HoS pictue, so I’m not sure that is a good data point).

This empirical approach (if you can call it that) was why I was encouraged to see some of the set dimensions you posted early on, where the pawn and other piece sizes quite closely matched my target sizes calculated based on a few simple formulas (formulas which I came up with from observation of various classic chess sets, but are not as mathematically harmonious / integrated as your system).

Do you think such a proposed ‘empirical’ approach to chess piece sizing is reasonable? Not to say that a more mathematical / engineered approach is in any way invalid. The empirical approach would not necessarily work if tastes and preferences have significantly changed over time — say, towards bigger pawns, or a more similar sizing of the Queen relative to the King (reflecting greater gender equality ?!) But so far, I’ve been surprised that my estimates held up reasonably closely even to the more contemporary FIDE set.

One last thing: regarding piece volumes, I think the actual piece design affects the final volume of a piece so much that it’s too hard to determine target volumes based solely on base and height dimensions. In particular knights, though shorter than bishops and with generally the same base, are often bigger (by volume) than bishops, given their more stout proportions. I also feel that rooks, given their greater value in the game, should be noticeably bigger than either bishops or knights (the FIDE set designer reflected this in the larger rook base; I’m not sure about volumes).

Kohpablanca,

As we say here in the States, "there's more than one way to get to Cleveland" (or in your case, Sydney). I'm reminded of the incredible coincidence of Newton and LaPlace both discovering the calculus, at the same time, by independent paths! In short, YES! I believe there is more than one way to design an aesthetically pleasing chess set that adheres to accepted standards (my way ain't the only way). My mathematical approach is just that, one approach. I'm also reminded that Mount Everest can be scaled from Nepal in the south, or from China in the north, both paths leading to the same summit. So, your empirical approach is as valid as mine. It doesn't matter how we get there so long the end product is a beautiful, functional, well-balanced chess set. So, keep-on keeping on!

THE GOLDILOCKS DILEMMA

You've just bought a beautiful reproduction of the set used in some obscure chess tournament in the foothills of the Himalayas before you were born. The set arrives in perfect condition, you love the yaks as knights and all the pawns are sherpas, yet, no matter what size chessboard you mate it with, things just don't look right! Either the king is too crowded or the sherpas are swimming in their squares, even with full packs! Your new set is a beauty alright, and your chessboards are magnificent, but harmony is not to be had. You begin to feel like Goldilocks, one board too big, the other too small, and NONE are just right.

There are only three ways to avoid the fate of Miss Scaling. First, you can design your own chess set to match the board of your choice. Second, you can design a board to fit your set; or third, you can purchase sets that are already balanced to play on your favorite chessboards. Either way, you must have a set of guidelines that defines what a balanced chess set looks like!

If anyone is interested, I can post the Dimension Tables for the sets I own that rate high based on my sizing scheme.