Suppose we consider two 'fundamental' kinds of patterns that occur in go; shapes and formations.

A shape is defined as a number of stones (2 or more, all of the same color) that are all connected
(directly or indirectly).
A formation is defined as a number of stones (all of the same color), none of which are connected.

Both shapes and formations are considered modulo translation, reflection, rotation (in steps of 90 degrees) and
coloration (interchanging the colors of the stones). So a shape or formation that lacks symmetries can occur
in 16 possible variations.

To illustrate these concepts, have a look at this image:

http://i.imgur.com/I6gxLrf.jpg

Here we see 3 white shapes, 7 black shapes, 1 formation of 7 stones (in both black and white color).

So all white shapes touching the bottom and left side are all the same shape (as they can
all be obtained via reflections, rotations and translations).

So, while there can be multiple shapes on the goban for each color, there can only be at most one formation on
the goban for each color.

The idea is that such formations can be understood to stand for the way a number of shapes can be distributed on
the goban. So if you have two shapes on the goban, there is a formation of two stones corresponding to the way
those two shapes are distributed on the goban.

Given this definition of the concepts 'shape' and 'formation', are there more possible shapes or more possible
formations on a 19x19 goban?

For instance, there are 188 possible formations of two stones, while there are only 5 possible shapes of four
stones. The biggest possible shape on a 19x19 goban consists of 360 stones (as it needs to have at least
one liberty), while the biggest possible formation has 181 stones.

What do you mean by:
- (a) connected directly -- do you mean the two stones are immediately adjacent to each other, vertically or horizontally ?

Click Here To Show Diagram Code

[go]$$B (a) ?
$$ . . . . . . . . .
$$ . . . . . . . . .
$$ . . X X . . O . .
$$ . . . . . . O . .
$$ . . . . . . . . .[/go]

- (b) connected indirectly -- do you mean a diagonal ?

Click Here To Show Diagram Code

[go]$$B (b) ?
$$ . . . . . . . . .
$$ . . . . . . . . .
$$ . . . X . . O . .
$$ . . X . . . . O .
$$ . . . . . . . . .[/go]

BTW, the term 'shape' already has certain existing and well-established meanings in Go that are quite different from your usage here.
(Similarly, for the term 'connected'.)

If I understood your definition of (a), I think an existing term already used by some people is a 'string'.

May I ask how you got the numbers 16 and 188?

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O . O . . . . . . . . . . . . . . . . |
$$ -----------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O . . O . . . . . . . . . . . . . . . |
$$ -----------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O . . . O . . . . . . . . . . . . . . |
$$ -----------------------------------------[/go]

And so on, for 17 total "formations" for the 1-Row:

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O . . . . . . . . . . . . . . . . . O |
$$ -----------------------------------------[/go]

Another 17 "formations" for the 2-row.
Another 17 "formations" for the 3-row.
And so on, until:
Another 17 "formations" for the 19-row:

Click Here To Show Diagram Code

[go]$$Wc
$$ -----------------------------------------
$$ | . . . . . . . . . . . . . . . . . . O |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O . . . . . . . . . . . . . . . . . . |
$$ -----------------------------------------[/go]

No, by connected directly, I mean two stones that are right next to each other (either horizontally or vertically) and hence touching each other. By connected indirectly, I mean two stones that are connected via other stones. For instance, if you have three stones in a row (e.g. at coordinates A1, A2 and A3), the outer stones are connected indirectly to each other via the middle stone.

The 16 is easy. In the image I have shown, you can see 8 variations of a white shape that lacks
symmetries (as opposed to the other two white shapes that have a symmetry).
Those are the white shapes touching the bottom and the left side.
You can have the same 8 variations in black, so in total there are 16 possible variations
of that shape (16 variations in case one only considers such shapes modulo translations and not
modulo rotations, reflections and interchanged colors).


The 188 possible formations of two stones is understood as follows. If we have two stones on the goban that
are not touching each other, we can take the translation (moving the formation around on the goban without
changing the spatial relationship between the stones) where one of the stones is in the bottom left corner and
the other stone is located on or below the diagonal line from bottom left to upper right (reflecting
it in the diagonal to obtain the variation where the 2nd stone is on or below that diagonal).
The 19th triangular number is 190 (=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19), and that would
correspond to all possible locations for the 2nd stone. But we have to subtract two of those locations;
one where both stones are in the bottom left position (as they would be mapped on top of each other and hence
not constitute a formation of two stones) and one where both stones are right next to each other (as they would
be touching each other and constitute a shape rather than a formation).

To avoid confusion it would help if you used the term "chain" instead of "shape". Shape in go refers to something completely different, while chains are exactly what you are describing.

Well, not completely different, except that shape traditionally refers to clusters of stones which are located
in close proximity, regardless of whether they are touching each other.

So a shape like a 'farmers hat' or '(empty/filled) triangle' fits my concept:

http://senseis.xmp.net/?HatShape
http://senseis.xmp.net/?EmptyTriangle

While a shape like a 'tiger mouth' would be classified as a formation rather than a shape from the way
I'm differentiating between the concepts of shape and formation:
http://senseis.xmp.net/?TigersMouth


I guess we could also call it a chain, but somehow the concept of a chain doesn't seem to fit very well to
dumpling shapes, as a chain invokes associations of something in one dimension rather than two dimensions.
A chain is more like a string than a sheet, so to speak, while a shape seems more general, referring to both
one-dimensional structures, like a line, as well as two-dimensional structures like a plane.

My reasoning to differentiate between formations and shapes (in the sense I've tried to define them in this
thread) is that the nature of go is such that they behave in different ways. For instance, a shape can be captured
in one move (when it is in atari), regardless of how many stones are comprised in that shape. While a formation of
more than 4 stones can never all be captured in one move (even if each of the stones in the formation is in atari).

So the property of being connected is kind of fundamental in go, as once stones are connected, they live or die
together. If two stones are connected, you can not capture one of these stones without capturing the other.

I'm speaking of connections here as being in contact, similar to how electricity flows between two points of a
metal structure as long as there is a way for electrons to hop from one atom to the next and traverse between
the two points in that fashion. If you have three pieces of metal A, B and C and A touches B and B touches C,
electricity can flow from A to C, even though A doesn't necessarily touch C directly (a kind of transitivity
of connections).

Again, not in the traditional sense of a connection, like a bamboo joint, which I would consider more of a
potential rather than an actual connection (as opposed to the direct vs indirect connection).

I think the point he was trying to make, is that you trying to redefine the term "shape" as what we now call "chain", will only breed confusion, as the term "shape" is used for a very very different concept from what you use, and has been in use for a long time, and in a lot of literature.

To make that change, you'd have to convince the whole western go community why the term "shape" is better than chain, with a good enough reason to be worth changing all that literature(or making it obsolete).

So, if you want people to actually discuss your idea, instead of your terminology, I'd change it.

_________________
a1h1 [1d]: You just need to curse the gods and defend.
Good Go = Shape.
Associação Portuguesa de Go

I think these concepts are all somewhat fuzzy. Like what's the difference between a group and a shape?
http://senseis.xmp.net/?Group
http://senseis.xmp.net/?Shape

Perhaps 'form' would be an alternative for 'shape', as 'form' doesn't seem to have an established meaning.

http://senseis.xmp.net/?Dango

As you can see, they speak of both a 'dumpling shape' as well as a 'dumpling chain'.
So a string/chain doesn't really seem a universally accepted concept to apply to a collection of stones that
are all strictly connected.

But whether one prefers to speak of a shape, a string/chain or a form doesn't really matter, as long
as it's clear what's meant by the word in the context of this discussion.

Let's return to the original question. Are there more possible forms or more possible formations
on a 19x19 goban?

Hmm, because the distinction between a thing and the shape of a thing is critical to the meaning of the word "shape" in English (and, I assume, in most European languages), it's hard to explain to you the difference between groups and shapes without using the word "shape". (Would it help if I told you that the shape of a thing is it's formal aspect, or it's morphology?)

I could show you a painting, and ask you to tell me what you saw, and you would say "I see angels, prophets, sibylls, gymnasts, clouds..." Or instead I could ask you to tell me what shapes you saw, and you would say "I see circles, ovals, ellipses, triangles, rectangles..." But, of course, you can't say that it is a cherub's cheek but not a circle, or vice versa!

Likewise, if I show you a board position and ask you to show me the groups, you would say "I see one dead group in this corner, and a living group in that corner, and a weak group under attack on the top side..." If I ask you to show me the shapes, you say "I see a hane here, and a counter-hane there, and a hanging connection there, and a bamboo joint next to it..."

I don't really understand your diagram, but I wonder if you wouldn't enjoy reading about spightonians.

"I'm really proud of myself: usually, on a Saturday, I'd rent a DVD and order an extra large, four-cheese pizza. But yesterday I only ordered a large. I'm trying to get in shape, you see."
"Oh? What shape? Round?"

Groups and chains have shape (e.g. flexible shape, good shape for attacking; clumsy shape, heavy shape). This is not the same as having a shape (empty triangle, bamboo joint, a square). Go players will talk about having "bad shape", or making "a bad shape". Neither is what you mean. It doesn't bother me, but why overload an already heavily burdened word when there's a perfectly good one that fits?

I think what you're asking is "how many chains (totally connected collections) of stones are there?" and "how many totally disconnected collections of stones (i.e. collections in which no two are adjacent) are there?", where two stones are called "connected" if you can walk from one to the other along (horizontally or vertically) adjacent pairs of stones, and you'd like to know the answer modulo reflection, rotation and translation. Is this right? In any case, I'm not a combinatorialist, so I can't answer you; but I'd bet that the former was significantly easier than the latter.

The diagram just shows a somewhat arbitrary collection of patterns or structures to illustrate the distinction
I'm trying to make between two very general categories of patterns.

My motivation is to see if there is a way to identify certain fundamental categories of patterns as a means
to comprehend more complicated patterns in terms of these more fundamental patterns.

At the most fundamental level of patterns in go, the most elementary distinction seems to be whether stones are
connected or not.
In case one likes to program something like a tsumego generator or a bot that can learn autonomously by
pattern recognition, one would like to employ a kind of systematic approach towards classifying patterns
one can encounter in the game of go.

In that respect it often makes sense to consider things modulo reflections, rotations, translations and colorations.
Just like at gochild most tsumego (the ones that lack symmetries) occur in 16 possible variations, because this
allows one to focus on the underlying patterns as opposed to any of those 16 specific instances of that abstract
pattern.
Similar to how children learn counting from merging two or more collections of concrete objects (having a batch
of 3 apples and another batch of 5 apples and merging them to obtain a batch of 8 apples) before abstracting
from concrete objects to recognize that 3+5=8 at a higher level of abstraction.
Abstraction is a wonderful thing and it seems very suitable as a way to analyze patterns in go. For instance
group theory which deals with things like symmetries.

Spightonians do seem interesting as a kind of systematic exploration of possible moves, based on their immediate
context (a 3x3 section of the goban surrounding the move), though they don't seem to take into account moves
on an edge or in the corner of a goban.
But I reckon these are patterns that are less fundamental as the ones I'm considering, since they involve
stones of multiple colors (whereas the patterns I'm after only concern stones of a single color).

I guess one approach towards answering the question is to start with a small goban, like 2x2 and gradually work
your way up towards larger gobans (3x3, 4x4, 5x5, ..) and observing how the proportion of the number of possible forms
to the number of possible formations tends to change.

See, I guess I don't understand whether you see this question you're asking purely as a fun math problem, or whether there is some question related to how we play Go that you are trying to answer...

When I look at the shapes in your diagram, I mostly see dumplings and empty triangles. This is more-or-less a consequence of the limits you've put on the shapes you're looking for - you want chains or strings of stones, but in general it's inefficient to connect stones into chains until nearby enemy stones force some sort of firm connection. (This is one reason I thought you might be interested in looking at the spightonians.) When we try to understand good shapes, we're normally looking at sets of interrelations between stones. Out of multiple local interactions, we build a recognizable local pattern - for example, one orientation for two kosumi is the tiger's mouth. If you were going to try to analyze shape this way, you would start by looking at likely candidates on a 2x2 grid, then look at 3x3 and 4x4 and so on. (Another reason I thought the spightonians might interest you.)

If you're just interested in this as a math problem, I apologize for derailing the thread - it seemed that maybe you didn't understand what we normally mean when we talk about "shape", and I was trying to help you out. If you're interested in it as a question about Go strategy, I think a back-and-forth would be helpful. You can explain to us what puzzles you, we can try to convey to you how other Go players think about good shape.

Well, not just as a pure math problem. More as a kind of exploration of certain concepts that delineate
certain classes of patterns. This is not intended as a way to figure out how to play go in any optimal
fashion from the perspective of a human player, but more of a way to figure out all possible ways one
can play go from the perspective of a bot.
Try to imagine yourself as a bot that plays go, initially knowing only the rules of go and making
random moves.
After each game, you analyze what happened during the game and you try to memorize certain patterns
that occurred during the game. Bad patterns (you might want to recognize those so you know what
patterns to avoid) as well as good patterns.
The question is, how do you store these patterns in a kind of database efficiently, so you can quickly
look up a pattern in that database. When you focus on the essential patterns (that is, modulo
translations, rotations, reflections, colorations), there are less patterns you need to memorize.
Also, as a bot you probably want to have a method to generalize patterns, so you might first learn
the pattern of a ladder and later you might generalize this pattern by taking into consideration
related concepts like a ladder breaker.

So I'm not interested in aspects like whether a particular chain would be good or bad in a typical
game where it occurs, but more fundamental properties of such chains, such as symmetries it might
have, which has implications for the way such chain patterns are stored in the database of patterns
a bot might accumulate as it plays games and updates its database of patterns after each game.

An analogy might be drawn with an algorithm that seeks to analyze substances and store them in a
database. One approach might be to store each and every kind of molecule in that database (based
on chemical properties), but then the database might get too big. A more sophisticated approach
might be to analyze substances in terms of their more fundamental constituents (atoms) and to
figure out certain patterns, so one can characterize whole ranges or potential molecules in a
kind of grammar of their atomic elements and the possible ways in which those atomic elements
can be combined into molecules. So a carbon atom typically has 4 possible connections it
can make to other atoms, while oxygen atoms have 2 possible connections and hydrogen atoms
have one possible connection. Knowing this, one can formulate a kind of grammar that characterizes
all possible molecules that can be created with carbon, oxygen and hydrogen atoms as constituents
and in that fashion you might have a compact description of a finite number of ways these can
be combined which specifies an unlimited number of molecules that can be created from these
constituents.

Likewise the number of patterns that can occur in go is huge and if you just memorize them all
in the least intelligent fashion, you're likely to run out of memory as a bot. Hence you're
forced to figure out patterns at a higher level of abstraction, which allows you to formulate
compact descriptions that characterize huge number of patterns that fit that description.

Right, and that's exactly the problem with your approach - no chain of stones is good or bad per se without understanding where the enemy stones are. A chain of 8 stones in a 4x4 sub-grid with no neighboring stones is always a mistake. So the problem with giving heuristics is precisely to get the machine (or the man) to identify the interactions between individual stones, not the gross silhouette of the stones. It's not worthwhile for an AI to identify, oh idk, chains with prime numbers of stones if factoring the chain isn't part of the game.

For example we remember the "broken shape" sub-pattern and look for it as an element of larger groups. It's not always bad for the broken group, but the prima facie evaluation that it is often bad makes it worth recognizing and studying.

You could compare it to looking for patterns in the configuration of oxygen atoms while ignoring the presence or absence of any other element. That's pretty silly! There are configurations of oxygen that are impossible without other elements to bind it into place, and the pattern to recognize is the molecular structure as a whole.

Perhaps I've misunderstood: you want to store tens of thousands of patterns, rather than 361 stones? Where's the computational benefit? How does storing patterns help you, say, read faster, or read fewer moves? (I agree that hard-wiring bulky fives, L-groups and snapback shapes into a bot's mind would be nice, if that's what you mean, but I don't think that has anything to do with your original question.)

How would storing 361 positions for stones in a database help a bot learn go from practice?
Seems more sensible to store a database of patterns somewhat similar to a collection of tens of thousands of patterns
as you can find them on the gochild tsumego website for instance. My original question was about patterns and seeing if
we can understand these patterns in terms of fundamental concepts (like chains and formations) that might allow
one to store patterns in a more efficient fashion, rather than just storing patterns as they occur without employing
any underlying patterns at a higher level of abstraction that characterize whole ranges of patterns.

No it's not, because the point is that more complicated patterns (like patterns where stones of multiple
colors are involved, like a ladder pattern) can be understood in terms of more fundamental patterns (like
chains or formations).
Your argument is like saying to people who study language that there is no point in classifying words into categories
of verbs or prepositions, since it doesn't make sense because words occur in sentences and nobody just utters a
preposition without the context of a sentence.

To understand things, you often look for the most fundamental patterns, even if purely for theoretical purposes.
Take quarks for instance, you can't even have a single isolated quark. It's impossible. But that doesn't mean it's
pointless to think of matter in terms of quarks and leptons (as more fundamental constituents of atoms).

But, the stronger you get, the less those patterns crop up. In the same way, you can't learn tesuji from studying professional games, because no professional will allow themselves to get into a situation where a tesuji can be effectively used against them.

My real objection is: what do you propose this bot will do when its opponent plays something that's not in any predetermined pattern? Presumably it needs to do things like weigh up influence vs. territory, look for weaknesses, read, or something like that - something that it can't do using patterns alone if there are no familiar patterns to exploit.