 # The shape of the stones

Let me share a question that is tormenting me from many years.
Which is the best math curve (or curves) that design the perfect shape of a Go stone?
Yes, there are a lot of stones in the market, but the shape I have in mind is the classical old-style competition 12mm thick minimum, not the flattened one.

It is often said that the Go stone is a lens shape. If this can be true for a very thin stone, when it becomes thicker the shape is quite different.

My best guess until now is that you can get the perfect shape using 2 catenary curves.

Let me remember the catenary equation: y(x) = a cosh( x/a )
Even if it is similar to a parabola under certain parameters, the catenary is another thing.

I would like to check if there are two catenaries (the first for the bell of the stone with a>1, the second one for the side with ‘a’ around 0.5) which should have coincident tangents in their intersection in order to ensure a smooth junction of the twos in the stone shoulder.

Has anyone the same ‘trip’ like me?

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Here is one way to do it

But coinciding tangents isn’t enough in my opinion. Ideally would like to have a continuous 2nd order derivative as well to get nice shading for 3d renders

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The first idea is to intersect two spheres to obtain the Go stone shape. However I’m not convinced that this is the real shape.

The following are the kind of shapes I have in mind. the following - instead - are the thinner Go stone than can be (maybe) obtained intersecting two spheres: The article (yes, I’m not the only one so crazy to think this kind of things) suggests to add the intersection of a torus in order to obtain the final shape. So, playing with r0, r1 and r2 you can obtain any kind of stone you like. Since the stone is forged consuming the shell material starting from a initial shape of a cylinder, and the movement of the stone-maker is not properly a pendulum movement, I don’t think the real shape of the hand-made stones coincides with the intersection of two spheres and a torus.

See this video for details:

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I think the traditional shape of the go stone is …

1. not precisely defined by any specific mathematical formula or geometric shape,
2. a subtle aesthetic matter defined only by the hand of the master and the shape of their tools,
3. varying across different cultures.
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Obviously… but the question is: which is the best mathematical function that can fit the traditional shape.
Often the beauty in nature is produced by proteins that repeat the same schema highlighting at a macroscopic level what is hidden at a molecular or atomic level.

I like often to mention in this regard the Romanesco broccoli which is an incredible example of fractal structure. Even a repetitive movement of a human being can produce nice shapes that can be well described by a mathematical function. The game is, what is that function for the go stone?

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Best is somewhat subjective, but there are many techniques for approximating a given shape with parametric functions to arbitrary accuracy. To name a few (and see related articles listed therein):

These techniques are widely used in computer graphics and modeling. Typically, deciding between these and their many variations comes down to weighing their various algorithmic advantages and disadvantages (e.g., computational complexity vs accuracy for what needs to be modeled).

However, it seems that you are interested in the most beautiful approach, which is a highly subjective question. Perhaps, you are looking for a reasonably accurate approximation that is also most elegant in a mathematical form?

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You may have to adjust and play with the parameters a bit and carefully decide exactly where to end each arch, but I believe that you could smoothly transition between two catenary arches (one rotated 90 degrees relative to the other).

I don’t think the lack of a pendulum movement is relevant. I believe the stone maker is just grinding the stone along the curved groove, periodically rotating the stone as it goes, such that the curve of the stone winds up similar to the curve of the groove.

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So, any math question needs a well-defined answer.

First thing we should therefore investigate: does there exist a uniform “perfect” traditional shape? For this it’s probably best to investigate how go stones are made.

There are of course two answers, the way the Japanese make their stones, and the way the Chinese do. The first is a shape with a convex curve on both sides, the latter has a flat side (technically still convex) and a convex side.

Chinese is probably easiest to answer physically, but very troublesome mathematically. The stones are made by putting liquid stone on a plate and letting it cool down.

Thus effectively, this becomes a problem of surface tension, the materials involved, gravity, and viscosity dependent on outside temperature. The first two are pretty easy to measure, the latter brings in trouble.

Mathematically, this kind of problems are in the realm of solving (partial) differential equations. This is very rarely doable exactly, and thus we will almost surely have to use numerical methods to find the perfect curve. The shape is probably comparable to those to shapes of water droplets on hydrophobic surfaces, of which I’m positive has been studied (but too lazy to search for).

The Japanese stones are made by hand, by grinding the stones on a grinder, so presumably the shape is decided by the shape of the grinder, of which I can’t find much reference…

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Exactly, beauty is simple.
A Bezier curve is somehow complex in terms of mathematical description (I suppose since I never wrote an equation of a Bezier curve of arbitrary complexity).

If (and only if) I can use a function for the bell profile of the stone and the function with different parameters for the edge of the stone, then I think to have obtained the result I’m looking for.
The surface of the stone should be obtained for revolution around the axis.

I don’t know if the catenary is the right curve. As soon as I have enough time, I’ll try to build such a model as you suggested.

Yes, I searched a lot but the high-quality stones I’m referring to are still hand made (including the tools like the grinder used in the manufacturing process), so - apart from some short sequence in few videos - I wasn’t able to find other information.

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This is an interesting suggestion. Second derivative should add gracefulness to the junction. Otherwise you risk to have a repentine change in the curvature before and after the junction (even if the junction is smooth).

Beauty is highly subjective, but if we are judging by simplicity, then I think an ellipse is perhaps the simplest approximation that might match reasonably well with the side-view of the traditional Japanese bi-convex stone shape, especially for the larger stone sizes that you were particularly interested in.

Of course, I’m sure the masters will say that the ellipse is not quite right and the shape has much more subtlety and nuance, but it looks like an almost exact match when I overlay an ellipse on the size 42 (11.9 mm) stone:

With an ellipse defining the side-view, we’ve reduced the shape of the stone to essentially just one free parameter: the ratio of its height to its width (i.e., length of semi-major vs semi-minor axis). Of course, the rest of the 3D shape is defined by rotating the ellipse around it’s semi-minor axis in 3D space.

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In the picture you posted the 40, 42, and 50 probably fit quite perfectly with ellipses of given parameters. The 44 seems instead to have a sharper edge on the geodesic ring. It is more similar to the set of three I posted earlier. When the thickness of the stone becomes too large, it seems to be a little graceless. As you told, it is a matter of personal preferences, and I like a lot thicker stones… but not so thick like 48 and 50. I have 10 mm shell stones in my go set, but it is only my budget that stopped me from buying a set with little ticker stones (12-13 mm).

So as a first approximation I must say that ellipse is the best curve… but is missing something on the edge. Like this in the place they call ear (耳). The name itself suggest to me something that protrudes (at least slightly).

From one of the most appreciated manufacturer in the world (http://www.kurokigoishi.co.jp/)

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By the way, my photos are from Kuroki Goishiten as well

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First attempt to try a catenary as a target shape.
(let me use this post like a notebook, so including errors and wrong directions)  Don’t know… maybe.
This shape is thick 1.2 and large 2.0. Almost there as a real stone in cm.

Let me try to work on ears to increase the slope at around |x|=0.75 with a square root function.  Taking apart proportions that can be distorted by axis scales not even, it seems to me that square root function can work. Now the game is to find the right x for the junction.

Let me check the derivative of both curves to check for which x value they are equal. There are two points that I should investigate.

The first one is too far from the edge, so I will focus on the second one.
To better see where the derivatives are equals it is more useful to plot the difference between the two and equal it to zero:  To find the exact value we must solve computationally a system of two equations: Solution over the reals:
x1 = 0.450972
x2 = 0.766747

x2 value confirms that 0.75 can be a good candidate for the junction of the two curves but doesn’t means that x2 itself is the exact point of the junction. In fact, for x2 the derivatives are the same but y2 for this value of x is not the same for the two functions:

y2(x2) for catenary = 0.253150…
y2(x2) for square root = 0.255527… so there is still a gap. It’s small, yes (2 thousandths of a millimeter) but it is a gap.

Let zoom-in the original picture in the x2 point with the two functions running close one to the other (same derivative) but with a gap, just to make this visible. I want remove the gap.

Someone can think that it would be enough scale a little bit the square root function to adjust this curve to the point where the two functions have the same derivative. But stretching one curve with a scale factor would change also the derivative in the point we found experimentally. It doesn’t work.

I should arrange a system of two equations and two variables to find the exact values of the variables (if they exist) that fix this gap.

First of all, which variables I must choose?

One is the x2 point (the x value for which the difference of the two derivatives is equal to zero) since the value I found is a mere experimental value found looking to a plot that seemed to be working for my purposes.

The other variable is the parameter of the square root function that allows me to adjust its shape with the objective to create the junction in the right place without any gap. Since the 0.9 constant determines the intersection with the x-axis, I don’t want to change it because I want a stone 1.8 cm wide. It isn’t the candidate. So, the only other parameter I can play with is 0.7 which multiply the square root. This is the y scale factor of such function so it is what I’m looking for…

Secondarily, which equations should I put in the system?

The first equation is simple, the difference between the two derivatives equaled to zero.
The second is the difference between the two equations equaled to zero because this is equivalent to say that the gap doesn’t exist anymore in correspondence with x2 (the solution of x in the first equation of the system).

Let me start with the second equation because it can be written straightforward. We need simply to equal it to zero. = 0
For the first equation, let me write the plain form for the derivative and equal it to zero: = 0
Now let see if WolframAlpha can find a solution for this system considering x and a variables.

Unfortunately not.

Computational time exceeds (I don’t have a pro account) so I must invent something to reduce the computational effort.

Let me simplify the cosh and sinh part of our equations, since I think this is the more complex function to manage for the engine.

We are interested in a good representation of such functions around the point x=0.75, so let expand those functions in two Taylor series neglecting orders superior to 5:    we will check later if the approximation is good enough in x2 (solution of our system).

Now I can write the system using this simplified versions of the hyperbolic functions and let the engine work for me: Now it works and the numerical solution found by WolframAlpha is: So, the parameter a has changed in order to cancel the gap between the catenary and the square root in the junction point as expected. But the point for which the derivatives are the same now has moved forward respect to x2=0.75 envisaged initially. Now we have a solution with x2=0.770978.

Let see if the junction is smooth enough.

y2(x2) for catenary is 0.24907(9)02584617236
y2(x2) for square root is 0.24907(6)43015856845

we got a pretty good approximation. The error is of the order of 3 millionths of a centimeter. To be precise 26 nanometers. Or if you prefer the gap in the junction is meters.
Ok, I like it.

Let see again the two curves in a closeup just in the junction point as we made earlier:  So we checked even visually that the curves fit very well at x=0.77.

Now we can take a look to our stone’s shape: I cannot find the way to represent the plot with even axis units using WolframAlpha, so the shape appears distorted in the picture above. Using Google instead, I can represent better the real shape that I have obtained with the given parameters.

If we assume the axis units like real centimeters, we have to do here with a stone 12mm thick (quite thick for a Go stone, even if there are thicker stones out there) and 18mm in diameter. We know that the standard diameter for a stone is around 22mm, so we can play with parameters of the catenary to make the sone wider.
We can also play (more easily) with the square root function moving the intersect of x-axis from 0.9 to 1.1 in order to obtain a 2.2cm stone in diameter. Obviously we must adjust also the scale factor and repeat all the calculations with the above derivatives to find the new juncture point. I would expect that - in this case - a smooth junction will fit the curves around 0.5 or 0.6.

I leave this job to someone more patient than me but crazy enough like me.

I would like to see a 3D rendering of such stone, but this is much beyond my capacity. I don’t have 3D software and I don’t know how to transport math functions in it. Let me leave this game to someone so crazy to try this challenge.

In principle, we can repeat this approach looking for other candidate functions. I like the way in which square root terminate the edge (or ear, 耳) of the stone (it is a parable at the end of the day).
I leave also the game to try other functions to someone else.

What did I say somewhere before?
"Beauty is simple" 