Identifying tiles in hyperbolic space?

In summary, the author is looking for a way to uniquely identify vertices in a hyperbolic space. He is hoping that the tilings corresponding to these graphs have some structure that a random graph doesn't.
  • #1
Coin
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"Identifying" tiles in hyperbolic space?

So I have a software project I've been working on on and off that involves the hyperbolic plane. There is something I am stuck on:

So let's say I have filled the hyperbolic plane with one of the tilings of regular n-gons. I now would like to "label" each individual tile with some sort of number or string which uniquely identifies it, in some pattern such that if I know a particular tile's "label" I can calculate the labels of the tiles which adjoin it.

So by analogy let's say this were the euclidean plane, and I had tiled it with squares:

euc_ad_hoc.png


In this case labeling is easy, my "label" is each square's coordinates (x,y), such that the square labeled (x,y) is bordered by (x-1,y),(x,y-1),(x+1,y) and (x,y+1).

When I am on the hyperbolic plane, it's less obvious to me how this would work and there doesn't seem to be a natural notion of integer "coordinates". I'm looking for a tiling that would look something like:

hyperbolic_ad_hoc.png


This is an ad hoc labeling (I made it in a graphics program, and I made one mistake...) for the "5,4" tiling, based around a sort of "polar" scheme where I tile one tile 1,1 and number "rings" out from that center with coordinates (r,t) with "r" being the ring number and "t" just being assigned around the circle. However this is just an example and doesn't actually help for my purposes unless I can look at the label (2,3) and somehow have a rule for deducing from the label that that tile is bordered by (1,1), (2,2), (3,9), (3,11) and (2,4).

Does anyone have any suggestions how to proceed? There seems to be a lot of information out there about these tilings in terms of what they mean in hyperbolic geometry, but less information about their discrete structure, and it's the discrete structure I'm interested in here.

Thanks.
 
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  • #2


Assuming your graphic represents your model, each element has 5 relatives. If you deem each n-gon a node then you have a 5-way (5-dimensional) tree. I don't see a lot of hope in designating them with two-dimensions. In a 5-way tree and with any uniquely indentified node, you know the position relative to neighbor nodes of your given node.

Or am I completely misunderstanding your question?
 
  • #3
jim mcnamara said:
Assuming your graphic represents your model, each element has 5 relatives. If you deem each n-gon a node then you have a 5-way (5-dimensional) tree. I don't see a lot of hope in designating them with two-dimensions. In a 5-way tree and with any uniquely indentified node, you know the position relative to neighbor nodes of your given node.

Or am I completely misunderstanding your question?

Right. So if I look at things this way, then what I am looking for is some way to "name", or label with some other kind of mathematical object, each vertex, such that the vertex is uniquely identified. The reason I want to do this is that in my particular case I am exploring the graph using something like a random walk, and I need some way to tell, when I cross an edge from one vertex to the next, whether the vertex I have just come to is one that I have visited before.

The reason I am hoping this is possible is that the graphs corresponding to these tilings have a lot of structure that an arbitrary degree-5 regular graph doesn't. (And the two-dimensional "coordinates" aren't themselves important, I just used that as an example because that's the way I'd do it in Euclidean space.) I guess I'm just trying to figure out if this is a question that's been studied before.

(And to be clear the picture above is one of the tilings I'm looking at but I'm overall looking at a class of tilings such that for some combinations of (n,k) you have a tiling of regular n-gons where k of these n-gons meet at each polygon vertex. Using the graph terminology I guess you could say each (n,k)-tiling is an infinite planar regular graph of degree n whose dual graph is a regular planar graph of degree k? There are some more examples of these tilings in the link at the top of my initial post.)
 
  • #4


Hmm. A quad-edge map in n-dimensions on a closed surface? I dunno.
Quad-edge data structures are used in three dimensional mappings. They provide what you seem to want. I do not have clue on generalizing.

The only reference I have is:
L J. Guibas & J Stolfi,
"Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams", ACM Transactions on Graphics, 4(2), 1985, 75-123

which is old. Sorry I can't provide something more substantive. Stolfi has libquad code here in C:

http://www.ic.unicamp.br/~stolfi/EXPORT/software/c/2000-05-04/libquad/
 
  • #5


I find this project very interesting and challenging. Identifying tiles in hyperbolic space poses a unique problem due to the non-Euclidean nature of this space. Unlike the Euclidean plane where coordinates can easily be assigned to each tile, the hyperbolic plane does not have a natural notion of integer coordinates.

One possible approach to this problem could be to use a coordinate system that is specific to the hyperbolic plane, such as the Poincaré disk model or the Beltrami-Klein model. These models allow for a mapping of points in the hyperbolic plane to points in the Euclidean plane, which could then be used to assign coordinates to each tile.

Another approach could be to use a labeling scheme similar to the one used in the example provided, where tiles are labeled based on a polar coordinate system. However, as you mentioned, this may not be efficient for determining the neighboring tiles.

In addition to these suggestions, I would also recommend exploring the literature on hyperbolic tilings and discrete hyperbolic geometry for potential solutions. There may be existing methods or algorithms that could be adapted for your project.

Overall, identifying tiles in hyperbolic space is a complex problem that requires a deep understanding of the underlying geometry and careful consideration of the chosen labeling scheme. I wish you the best of luck in finding a solution and look forward to seeing the results of your project.
 

1. How do you identify tiles in hyperbolic space?

To identify tiles in hyperbolic space, you first need to understand the concept of hyperbolic geometry. This type of geometry deals with non-Euclidean spaces, where the parallel postulate does not hold. To identify tiles, you can use a variety of methods such as using the Poincaré disk model or the Klein model. These models help to visualize hyperbolic space and identify tiles using geometric constructions.

2. What are some common types of tiles found in hyperbolic space?

Some common types of tiles found in hyperbolic space include regular polygons such as triangles, squares, and hexagons. Other common tiles include hyperbolic triangles, which are formed by three curved lines and have angles that add up to less than 180 degrees. Hyperbolic quadrilaterals, pentagons, and heptagons are also commonly found in hyperbolic space.

3. How do you determine the properties of tiles in hyperbolic space?

To determine the properties of tiles in hyperbolic space, you can use various mathematical formulas and equations. For example, the area of a hyperbolic polygon can be calculated using the Gauss-Bonnet formula. The angles of a hyperbolic polygon can also be calculated using the hyperbolic law of cosines. Additionally, the perimeter of a hyperbolic polygon can be determined using the hyperbolic trigonometric functions.

4. Can tiles in hyperbolic space have infinite sides?

Yes, tiles in hyperbolic space can have infinite sides. In fact, there are infinitely many different types of hyperbolic polygons with an infinite number of sides. These polygons are called apeirogons and are similar to regular polygons, but with an infinite number of sides. Apeirogons have unique properties and play an important role in the study of hyperbolic geometry.

5. How is identifying tiles in hyperbolic space useful?

Identifying tiles in hyperbolic space is useful in various fields such as mathematics, physics, and computer science. In mathematics, it allows for the exploration of non-Euclidean geometry and the study of different types of tilings. In physics, hyperbolic space is used to model various phenomena such as black holes and the expanding universe. In computer science, hyperbolic tilings are used in the creation of complex patterns and designs for graphics and digital art.

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