Post by aher on Feb 4, 2013 6:20:12 GMT -6
Earlier, in a different thread, I talked about topology, and why a gamer might want to know something about it, in order to spice up his games. I argued that knowing a little about topology could be a big help if you wanted to create a dungeon in a non-Euclidean space--R'lyeh, for example.
From "The Call of Cthulhu" (written 1926, published 1928), we know R'lyeh is covered with green slimy stone vaults, colossal alien statues, bas-reliefs, and hieroglyphs. We also know that the angles of these surfaces are unearthly: "the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."
As the adventurers approach Cthulhu's tomb--the one with the squid-dragon bas-relief--and they gaze upon the door to this vault, and it seems "like a great barn-door; and they all felt that it was a door because of the ornate lintel, threshold, and jambs around it, though they could not decide whether it lay flat like a trap-door or slantwise like an outside cellar-door. As Wilcox would have said, the geometry of the place was all wrong."
As noted earlier, there's an excellent pic of this imagery here:
While there are many ways to model this, what sprung to mind was a (linked) double torus. A genus-2 torus is an orientable surface with an Euler characteristic χ(2) = -2, giving it a hyperbolic geometry. (See my earlier post if you're not sure what this means.)
So here is a pic I drew of R'lyeh as a linked double torus, showing why the adventurers might be confused whether the door was lying flat or slantwise. I apologize in advance for being a sucky artist.
And here is a pic I drew showing why a linked double torus is topologically equivalent (homeomorphic) to an unlinked double torus:
Here is another pic showing how a double torus may be modeled by a planar graph shaped like an octogon:
So, if I wanted to map out R'lyeh on a flat sheet of graph paper, I could do so using this octogonal planar model.
Here is more detail on this octogonal planar model:
Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn
From "The Call of Cthulhu" (written 1926, published 1928), we know R'lyeh is covered with green slimy stone vaults, colossal alien statues, bas-reliefs, and hieroglyphs. We also know that the angles of these surfaces are unearthly: "the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."
As the adventurers approach Cthulhu's tomb--the one with the squid-dragon bas-relief--and they gaze upon the door to this vault, and it seems "like a great barn-door; and they all felt that it was a door because of the ornate lintel, threshold, and jambs around it, though they could not decide whether it lay flat like a trap-door or slantwise like an outside cellar-door. As Wilcox would have said, the geometry of the place was all wrong."
As noted earlier, there's an excellent pic of this imagery here:
While there are many ways to model this, what sprung to mind was a (linked) double torus. A genus-2 torus is an orientable surface with an Euler characteristic χ(2) = -2, giving it a hyperbolic geometry. (See my earlier post if you're not sure what this means.)
So here is a pic I drew of R'lyeh as a linked double torus, showing why the adventurers might be confused whether the door was lying flat or slantwise. I apologize in advance for being a sucky artist.
And here is a pic I drew showing why a linked double torus is topologically equivalent (homeomorphic) to an unlinked double torus:
Here is another pic showing how a double torus may be modeled by a planar graph shaped like an octogon:
So, if I wanted to map out R'lyeh on a flat sheet of graph paper, I could do so using this octogonal planar model.
Here is more detail on this octogonal planar model:
Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn
