wendy wrote:The groups with finite-extent symmetries, such as x3x7o, really don't have what it takes to have a horogon, because the f.e. groups represent something like integers, and you just can't keep dividing the integers as often as you want, which is what the horogon symmetry can do.
You need something with a finite-content group (ie one or more vertices on the horizon), or a kind of broken symmetry that is only 'inwards-symmetric', such as the infinite stack of horogonal cupolae support. Here, they support eg 1 -> 2, which looks identical inwards, but not outwards.
The tiling formed by 5 5 5 3 supports horocyclic symmetry, but this is because it has a horogonal-inwards symmetry at some large scale.
Marek14 wrote:... and the developer also discovered a very interesting periodic covering of hyperbolic plane composed of 40 cells (28 hexagons, 12 heptagons) and he is basing several lands on this.
Klitzing wrote:Marek14 wrote:... and the developer also discovered a very interesting periodic covering of hyperbolic plane composed of 40 cells (28 hexagons, 12 heptagons) and he is basing several lands on this.
Come on - details! - which patch?
--- rk
quickfur wrote:It looks like x3x7o.
ubersketch wrote:I like the use of horocycles in here. I wasn't expecting them actually.
Marek14 wrote:ubersketch wrote:I like the use of horocycles in here. I wasn't expecting them actually.
Well, they are something typically hyperbolical.
Marek14 wrote:Wendy, do you know of a simple way to embed an apeirogon or horocycle in the (6,6,7) tiling?
Marek14 wrote:The 6.2 version introduces some interesting challenges. One of them is the Round Table.
You see a round table. It's round, but it's also quite big. The Knights tell you to find the Holy Grail, which is in the center.
The challenge is: how to find a center of a big circle in hyperbolic plane? The circle is big enough that you don't see the edge when you're deep inside.
I have my own idea for an algorithm, but I would like to hear your ideas as well Remember that you play on (6,6,7)-grid and there is an item (Dead Orb) which you can carry in vast quantities and drop on cells to mark them.
How to find the Holy Grail?
ubersketch wrote:Marek14 wrote:The 6.2 version introduces some interesting challenges. One of them is the Round Table.
You see a round table. It's round, but it's also quite big. The Knights tell you to find the Holy Grail, which is in the center.
The challenge is: how to find a center of a big circle in hyperbolic plane? The circle is big enough that you don't see the edge when you're deep inside.
I have my own idea for an algorithm, but I would like to hear your ideas as well Remember that you play on (6,6,7)-grid and there is an item (Dead Orb) which you can carry in vast quantities and drop on cells to mark them.
How to find the Holy Grail?
Dang, I didn't know about the Dead Orbs. This is an interesting mathematical problem.
quickfur wrote:has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?
quickfur wrote:Kinda tangential, but this gives me an idea: has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?
quickfur wrote:Kinda tangential, but this gives me an idea: has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?
Marek14 wrote:Well, as for anisotropic spaces, check out this:
https://www.youtube.com/watch?v=2LotRqz ... e=youtu.be
HyperRogue now supports the Solv geometry; I'm not sure if there is a better implementation of that one around.
Marek14 wrote:Interesting! Do the remaining two Thurston geometries (Nil and universal cover of (SL,R)) also occur as subspaces of higher-dimensional product spaces?
mr_e_man wrote:Marek14 wrote:Interesting! Do the remaining two Thurston geometries (Nil and universal cover of (SL,R)) also occur as subspaces of higher-dimensional product spaces?
They do occur as subsets of higher-dimensional pseudo-Euclidean spaces. You might call it spacetime (with more than one time dimension). And the metric and the symmetry are maintained.
I could give details (e.g. Nil occurs in ℝ^{12} with signature (+)^{6}(-)^{6}), if you're interested. But I'm not even sure if I'm interested. I worked on this for a while, but it seems like the intrinsic view of the geometry is more useful than the extrinsic, especially when there's such a large difference in dimension.
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