Hausdorff box dimension divided by dimension of the convex hull
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I mean I just made up the ratio but the 2 quantities are well-known things… a fractal would end up at like 0.7 or something for eg. Legible things would be 1
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Why would you expect them to be if the intent is to model legibility
Another metric: straight line distance/graph shortest path
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Graph shortest path is very unlikely to honor the triangle inequality. Trying to apply spatial metaphors to graphs seems ill-fated, at best you can try to have edges represent orientational metaphors a la Lakoff and Johnson.
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I don’t understand what you’re talking about
An illegible environment will have wiggly paths that will on average be longer than Euclidean shortest path, You’re not applying the triangle inequality on the graph. You’re applying it in the embedding space where it will work fine
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A lot of graphs aren't planar! If your graph *is* planar, then each vertex has a direct relationship to the nearby vertices on the plane, but if it isn't, then it may only have an indirect relationship to the neighbor, per whatever layout you've chosen.
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All graphs are embeddable in R^3 though, by Nash theorem, so the argument generalizes
Fair enough, but I'm not sure that allows our spatial intuitions (which again, per Lakoff, are less spatial and more orientational) to be applied to non-trivial graphs.
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Not really sure what you’re arguing about tbh
We were talking about modeling legibility in the sense of James Scott’s seeing like a state book in topological terms
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You may have misunderstood my point. The question of a spatial legibility (in the sense of James Scott) metric really only makes sense when you have an embedding anyway. Other kinds of illegibility, like textual, will require other measures.


