I think it depends on the size of the stones you’re using compared to the size of the grid. Also except when one is in the counting stage of Japanese/Korean rules, there won’t be a situation where all the board is full, and even then there would have to be lots of captures probably from kos to achieve that.
We still do this when distance and precise coordinates are not as important as the actual destination:
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There was a joke for a while about the Irish transport system in Dublin.
There are busses like in other places of course and trains but the trains are more for getting to suburbs or across the country rather than getting around the city.
It is somewhat better now (dublinpublictransport dot ie)
or maybe more officially at transport for Ireland website.
I’m probably way off topic at this point.
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Taipei metro system is relatively simple and grid-like
I think my point is ancient people would have a similar mental image of a web of road, spread out and starting from some key cities, very much like star points, etc, if we don’t think of Go as a territory game, but a route connection game. Each fixed stones are like starting points to grow a connect route. It might not start off as a 2-person competition game, but like connect the dots.
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time to play go on the map of the tokyo metro 
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I think people physically could play ancient turf wars in the Tang Dynasty capital Chang’an, took control of different palaces, markets, shops, etc.
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@mekriff
I learnt a new word from the Chang’an article:
According to Wikipedia, a 闕 / 阙 què is
a freestanding, ceremonial gate tower in traditional Chinese architecture first developed in the Zhou Dynasty (1046-256 BC), […] used to form ceremonial gateways to tombs, palaces and temples throughout pre-modern China down to the Qing Dynasty (1644-1912), [and reached their] peak during the Han Dynasty (202 BC - 220 AD), [today often] seen as a component of an architectural ensemble (a spirit way, [神道 shéndào]) at the graves of high officials during China’s Han Dynasty.
Some related words in English:
| Word | Root |
|---|---|
| tower | lt. turris (gr. túrrhis) |
| column | lt. columna |
| obelisk | lt. obeliscus, (gr. obelískos) |
| monument | lt. monumentum |
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A introduction of the mural, and the modern reconstruction of its structure based on it.
The location on the map
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A dual graph does NOT necessarily mean a self-dual. 2nd half of your statement is true. DUAL(DUAL(X)) = X.
Nice drawing! You almost got it. In the dual from the graph of “19 lines by 19 lines”, you have 18x18+1 vertices, 684 edges, and 361 faces, which is exactly the dual of the original graph of 361 vertices, 684 edge, and 18x18+1 faces.
Of course the dual of the 19x19 graph (either you see it as lines, or equivalently, squares) has a different shape (not a self-dual), but there exists a dual graph for it which switches vertices and faces. The dual doesn’t mean anything with regard to the game. I just wanted to have fun.
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Sorry if I confused you with duality problems and dual graphs, or self-dual. Of course switching vertices and faces does not give you the same graph, but there exists a dual graph; and the dual of the dual is the original 19x19.
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A good maths student reads carefully what is stated before correcting someone: I said duality is its own inverse, not that a dual graph is its own inverse: the morphism that maps a plane graph to its dual is an involution. That is, your function DUAL( • ) is its own inverse.
I’m unsure what is almost about my drawing.
I see the graph of the goban as vertices being intersections (in Go terminology), with an edge between vertices whenever the associated intersections are connected in the sense of liberty (in Go terminology). Hence for a 4x4 goban (to save counting) we get this graph:
Giving me the dual graph:
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I think I see what you mean, in that if one for clarity does the same idea of pushing the point to infinity in Vsotveps post
then there’s the number of faces in one as the intersections of the other like in this one Vsotvep has
I can see a 4x4 board kind of turn into a 4x4 board, except without any borders, and massive edge faces to play on.
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Yes, so yebellz’ squares-board with the outer rim removed, and any vertices on the outer rim being collapsed into a point would give the dual graph of the graph extracted from the intersections-board.
Hence the dashed lines are seen as infinity, the blue dots are collapsed into a single point (at infinity).
The phrase “duality” is used to mean many different things across various areas of mathematics (see List of dualities - Wikipedia and Duality (mathematics) - Wikipedia). These things are only weakly related at a broad, conceptual level.
@Vsotvep is discussing the specific concept of a Dual Graph, whereas @mbot inexplicably linked to the article about the concept of Duality in Optimization Theory when they first brought it up. Of course, this concept from optimization theory is not at all related nor necessary nor helpful in seeing that different visualizations of the Goban are clearly equivalent.
However, ultimately, all of this grasping to apply formal mathematical concepts is just unnecessary, pretentious, condescending pedantry, since it is quite obvious, without all of this, that the two visualizations are equivalent.
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Not necessarily. Many forms of duality from different fields can be described as the same concrete concept using category theory, the difference only being a matter of which category you’re interpreting your functions in.
Also, optimisation theory uses many concepts from geometry or algebra in more or less the same way as they are used in geometry or algebra. I personally know nothing substantial about optimisation theory, but I wouldn’t be comfortable to hastily conclude that the things we’re discussing here couldn’t be formulated in a meaningful way using optimisation theory.
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This got me thinking for a bit, since “a dual graph being its own inverse” is an interesting sentence: it begs the question what is meant with the inverse of a graph. Usually the inverse graph is the graph with edges xy for x and y distinct vertices if ond only if xy is not an edge of the original graph. This notion of course only makes sense for simple graphs, not for multigraphs (graphs are simple if every two vertices are connected by at most one edge).
Hence, the interesting sentence would ask for a plane graph for which the dual graph is isomorphic to the inverse of the dual graph (assumed that the dual graph is a simple graph).
For example this graph has a dual that is its own inverse:
The dual graph in red: the inverse graph would have the blue edges, which gives a graph isomorphic to the red dual graph.
Or another example:
There is also a graph that is its own inverse of which its dual is its own inverse as well: the graph with a single vertex and no edges. In fact, this graph is even self-dual 
The typical traditional goban is made up of 19x19 cells (where one can place stones). Most of these cells looks like this:
Except for the corners and edges, which look different, like this:
, 
, and their rotations.
The alternative visualization (as discussed in this thread) replaces these cells with identical cells that all look like this:
Note that there is a black border on that cell, which might be difficult to see with dark mode on these forums, so here is a version cropped slightly larger to show a single cell with parts of its neighboring cells:
Of course, some cells are also decorated with hoshi points.
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I wouldn’t impose the own-inverse constraint. Could you quote a source?
Your drawing is perfect, but you seemed reluctant to accept it because it’s not a self-dual 
Just out of curiosity, is any of this duality math and theories, help with visualizing or counting and make them easier? Any tricks or could be taken away from morph them in different graphs? Like they can be stretched out and able to line them up somehow for each side?
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