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Moved from a direct page edit of Chris's Go Pages / Thickness:
Thickness, influence, and power are all the same concepts.
Alex: I disagree with this last statement. Influence need not be thick and indeed is often somewhat thin, although thickness usually has influence. "Power" is ambiguous and, as far as I know, doesn't have a specific technical meaning.
Well, I probably should have put that they are related concepts.
However, I'm trying to keep these pages simple, and for beginners. When I refer them to Sensei's, I want them to find easy-to-understand definitions, and I'm finding more and more that this isn't the case.
It's still a draft, and it's still being written, but I'd appreciate your comments in the discussion section.
I've had a long period struggling with the differences between these concepts. I'm not saying I know the truth by now, but I do have a much more comfortable understanding now.
If you bring back the purpose of the game to: put more living stones on the board thatn the opponent then everything comes down to making your existing stones stable and create areas where you can put more stones. Hence,
LIFE means stones that will remain.
TERRITORY is empty area that can be filled with stones
THICKNESS (or strength) is stones that are likely to remain
INFLUENCE is empty area that likely will be filled with stones
A living group makes territory. A thick group exerts influence.
Eyespace is what makes these concepts interfere: you need some territory behind the lines to speak of life. But basically the above relationships apply.
I like your definitions. I think that most beginners have a relative idea (if incorrect) idea that alive stones have two eyes. I like classifying dead stones as "stones that can be captured." The corollary is that alive stones are "stones that can't be captured."
Territory, I think they also understand, but they sort of get mixed up between moyo and territory.
Thickness and influence, at least at our club, are terms that are thrown around a lot, but not really explained.
Anyway, I appreciate the comments, and I'll update the pages (and hopefully simplify them) as I hear more feedback from the beginners...
Hicham: Imagine a handicap 9 game. The handicap stones dertainly exert influence throughout the whole board, but you can't possibly call them thick?
Alex: An even better example might be kikashi. Quite often, the kikashi stones provide influence, which is what makes them useful. No one would consider a kikashi stone thick, however.
Thick stones are unlikely to die, whereas influence is simply about gaining an advantage in nearby fighting. Kikashi stones can provide an advantage in fighting, so they have influence, but that influence usually takes the form of sacrificing the kikashi stones to gain that advantage.