![]() Try coloring our new "Angry Birds" tessellation. To see Alain's new tessellations and to see this one tessellate visit Ĭheck out the new tessellations from Hawthorne Elementary School NEW STUFF: Alain Nicolas, the great French tessellation artist, has posted a gallery of new original tessellations that are quite amazing. Escher to celebrate Escher's birthday: June 17 NEW STUFF: Chris Watson has posted a tessellated portrait of M. NEW STUFF: Gabriel Sotillo is eager to show his first tessellation: Quails How to Make an Asian Chop (stone stamp).It also explains the equivalence between the upper half space and the disc model, which may be useful for you cause I'd expect calculations in the upper half space model to be a little easier (this is just a guess, I do have no experience with this kind of calculation on the computer). (Of course the geodesics are then only those parts of the circles which are lying in the interior of the unit disc).īy googling I found this link which I've chosen randomly from the result list, which explains this and other facts about hyperbolic geometry. In the disc model of hyperpolic two-space these are precisely (ordinary euclidean) circles (including straight lines) which meet the boundary of the unit disc orthogonally (in the euclidean sense), so from a euclidean point of view it is a rather simple geometrical task to draw such lines and to calculate their intersection points. I'm not sure whether this is of any help, but you should note that the lines in these pictures bounding the quadriliterals your are referring to are just (hyperbolic) geodesics. Datar's can build tessellations either with a polygon centered at the origin, or with a vertex at the origin and can also tessellate "motifs" (polygons or polylines such as Escher's fish). Datar's program offers more flexibility than Joyce's applet: e.g. There is also code that goes with it, which I have not yet looked at. I found the explanations in Ajit Datar's master's thesis the most helpful for learning how the process of generating the tessellations works. ![]() It's much less efficient than Hatch's code, but efficiency is not one of my requirements at this point. This applet draws regular and quasiregular tessellations organized by polygons, so it's easy to put print statements into the update() method and output the locations of vertices of each polygon as the polygon is drawn. The most helpful solution I've found for my need - cranking out the locations of vertices of each polygon - is David Joyce's Hyperbolic Tessellations applet and its source code. But they're grouped by line segment rather than by cell, and converting from the former to the latter does not seem easy. I can extract these vertex coordinates by putting in print statements. I just found some source code for drawing hyperbolic tilings (see bottom of the page), which includes generating locations of vertices. I'd also be interested in hyperbolic tilings besides Escher's, e.g. If an algorithm could help me generate $k$ rings of cells around the origin, that would be most convenient. I want to generate a finite area of the plane (of course?). But I don't know how to do it.ĭo you know of software to do this? Or can someone help me with an algorithm? ![]() I don't need to generate fish, just quadrilaterals. the formula to convert those coordinates to the Cartesian plane, using the Poincare Disk model.įor example, in Circle Limit I, each fish seems equivalent to a quadrilateral (four other fish touch its edges), and each vertex is surrounded by either 4 or 6 fish.a way to generate the coordinates in the hyperbolic plane, for each vertex of several cells (polygons) in such a tiling and.What I want to do is generate the coordinates (in the Cartesian plane, for a graphics display) of vertices in such a tiling. Escher's "Circle Limit" drawings, which use a Poincare disk model to illustrate tilings of the hyperbolic plane. ![]() I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. ![]()
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