Talk:Binary tiling

View in the Poincaré disk model?[edit]

Does anyone have a view of this tiling in the Poincaré disk model? I think this would illustrate better the fact that the tiles are not polygons. — Preceding unsigned comment added by TheKing44 (talk • contribs) 21:53, 28 January 2018 (UTC)[reply]

There's one in the Penrose article, but we can't just copy it directly and I don't think it does what you want it to do. (The horocycle edges are drawn as circular arcs, just as straight lines in the disk model would be, but that's not different from what we have now where the horocycles are horizontal lines. Anyone who knows enough about the hyperbolic plane to know that the circular arcs of the disk model are the wrong direction to be straight lines would also know that the horizontal lines in the halfplane model are the wrong direction to be straight lines.) —David Eppstein (talk) 22:32, 28 January 2018 (UTC)[reply]

@David Eppstein: Well, maybe a view in which the horocycles are replaced by straight edges would be useful (to demonstrate what the difference looks like). (In particular, the edges would form a bunch of regular aperigons.)

Ok, done. —David Eppstein (talk) 01:55, 29 January 2018 (UTC)[reply]

Nice! It would still be informative to see the shape of a single tile drawn "centered" on a Poincaré disk model. Then it would be more clear that the bottom pair edges are the same length as the top one. Tom Ruen (talk) 01:58, 29 January 2018 (UTC)[reply]

It's not a regular polygon, so I'm not sure what "centered" means here. —David Eppstein (talk) 04:26, 29 January 2018 (UTC)[reply]

Ha, that's why I put in quotes! Call it a centroid, or somethig close, as you like. Tom Ruen (talk) 05:59, 29 January 2018 (UTC)[reply]

@Tomruen: Or perhaps displayed in the band model, centered on a line going through a bunch of tiles. The band model preserves area near a line, so that would show the fact that the "parent" tiles are the same size as the "child" tiles. (For reference here is what the {5,4} looks like in the band model: http://bulatov.org/math/1001/band/tiling_425_band_00.png although I was thinking our band model would go through the centers of the tiles.) TheKing44 (talk) 17:52, 29 January 2018 (UTC)[reply]

Relation to this Aperiodic Tiling[edit]

The binary tiling is related to the one in figure 3 of this paper about a strongly aperiodic tiling of the hyperbolic plane (in particular, it also uses rectangles in the half-plane model, and they have a picture of the Binary tiling in Figure 1). Should we talk about in the article somehow?

(Maybe we could move this article to Böröczky tilings, and describe all such tilings (including higher dimensional versions).) TheKing44 (talk) 17:50, 29 January 2018 (UTC)[reply]

Symmetry group[edit]

Theorem 3.12 of https://www.math.uni-bielefeld.de/baake/frettloe/papers/hyp-art-final.pdf describes the symmetry group of this tiling. (Just though y'all would like to know.) TheKing44 (talk) 20:50, 29 January 2018 (UTC)[reply]