Hyperbolic coordinates?

Higher-dimensional geometry (previously "Polyshapes").

Hyperbolic coordinates?

Postby Marek14 » Mon Aug 19, 2013 7:53 pm

Imagine a hyperbolic plane with {5,4} tiling. Is there a simple way how to assign coordinates to the vertices (I suppose you might need three) so you could find, from coordinates of one point, coordinates of the neighbouring points and the directions which way to go to reach a specific another coordinate (for example origin)?

Or consider a hyperbolic city with this arrangement of blocks: how should they name their streets so people wouldn't get lost in it?
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Re: Hyperbolic coordinates?

Postby wendy » Tue Aug 20, 2013 7:17 am

I really never had a coordinate system here. I mainly used triangulation to set up relations.
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Re: Hyperbolic coordinates?

Postby Marek14 » Tue Aug 20, 2013 7:46 am

wendy wrote:I really never had a coordinate system here. I mainly used triangulation to set up relations.


Yes, but would that be practical if you had to live there? How would inhabitants of hyperbolic world give directions to travelers?

If we want to retain some relation to distance, two numbers for coordinates are clearly not enough (since there's more integral points in distance x than possible small two-number combinations). Would three numbers be enough? In particular, could a system similar to Gaussian integers exist in hyperbolic plane?
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Re: Hyperbolic coordinates?

Postby wendy » Tue Aug 20, 2013 7:53 am

Not really. If you were living at the scale where a house-block is the size of a cell of 4,5, then eighty blocks away is enough for one to get lost in a big way (like there are 100,000,000 blocks give or take some, in that range). It would be much worse than getting lost in a foreign city, for example.

For example, if you were to put the earth in the size of one of these cells, at 4000 miles diam, the whole universe fits inside the size of the sun of 864000 kms., and that's only something like 432 blocks away.
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Re: Hyperbolic coordinates?

Postby Marek14 » Tue Aug 20, 2013 8:00 am

wendy wrote:Not really. If you were living at the scale where a house-block is the size of a cell of 4,5, then eighty blocks away is enough for one to get lost in a big way (like there are 100,000,000 blocks give or take some, in that range). It would be much worse than getting lost in a foreign city, for example.

For example, if you were to put the earth in the size of one of these cells, at 4000 miles diam, the whole universe fits inside the size of the sun of 864000 kms., and that's only something like 432 blocks away.


That makes me want to see a videogame implemented in such a city...

Hmm, so let's see. If the streets had consistency of, say, {oo,3} there WOULD be a simple way to mark the crossroads: you could mark the origin as 0, three branches as A,B,C and then give labels to further crossroads based on whether you go left or right to get there (like ALLR for "go through A branch, then go left twice and right once"). Since there are infinitygons, any combination would be unique. And computing distance (at least taxicab distance) between any two points is then very simple: if they start with same string, cut it off maximally and add the number of remaining symbols, if they don't start with same string, just add the symbols.

Which leads me to believe that the finitegonal tesselations might be doable by expressing them as a suitable group. A (6,8,10) tesselation, for example, has three kinds of edges, so directions can be given as "traverse a-edge (6-8), b-edge (6-10) or c-edge (8-10)", with simplifying rules stating that a^2, b^2, c^2, (ab)^3, (ac)^4 and (ad)^5 are all identities (more rules would be probably needed). Ideally, you'd end with 1-1 mapping of points and valid strings forr a given tesselation. Then, you could simply post waystones pointing to neighbouring vertices (with their labels) and you wouldn't get lost...
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Re: Hyperbolic coordinates?

Postby wendy » Tue Aug 20, 2013 8:19 am

I remember seeing a game that did this, a link-map in H3 or something.
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