| Related sites for http://aleph0.clarku.edu/~djoyce/poincare/poincare.html |
| Penrose_Tiles Links in the Geometry Junkyard. | | The_Penrose_Tiling_at_Miami_University A summary of the history behind the Penrose tiling in the math department at Miami university. | | Pentagons_that_Tile_the_Plane Known solutions and a new recipe. | | Quasi-Periodic_Tilings Artist Eleni Mylonas uses tilings. | | References_about_Tessellations Compiled by Doris Schattschneider, Moravian College. | | Samantha\'s_Wallpaper_Page Wallpaper patterns and symmetry groups. | | The_Seventeen_Wallpaper_Groups Examples in traditional Japanese patterns. | | Squared_Squares_and_Squared_Rectangles Tilings of Squared Squares and Squared Rectangles, constructed using javascript. | | Symmetries,_Patterns_and_Tessellations Constructed with Geometer's Sketchpad by Allan Bergmann Jensen. | | Symmetry_and_Pattern__The_Art_of_Oriental_Carpets Resources compiled by the Math Forum. | | Symmetry_and_the_Shape_of_Space With interactive demos using Geometer's Sketchpad (requires Macintosh). | | Symmetry_and_the_Shape_of_Space Chaim Goodman-Strauss, Arkansas. With interactive demos using Geometer's Sketchpad (requires Macintosh). | | TESS Tessellation software. 11 rosette, all 7 frieze and all 17 wallpaper groups included. | | Tessellated,_Interlocking_Concrete_Pavers_and_Molds Pictures of some concrete tilings. | | Tessellating_Wallpaper_Patterns Original tessellating desktop wallpaper tiles. | | Tessellation_and_Photography A study for the endless possibilities of tessellating the plane based on the work of Dutch graphic artist M.C. Escher. | | Tessellation_Links Compiled by Suzanne Alejandre for the Math Forum. | | Tessellation_Tutorials_by_Suzanne_Alejandre Tutorials and templates for making tessellations using ClarisWorks, the Geometer's Sketchpad, HyperCard, HyperStudio, and straightedge and compass, including step-by-step instructions for classroom | | Tessellations A company selling foam puzzles, many based on planar tessellations (tilings) and fractals. | | Tessellations__Technology_and_Culture Examples and Java applets. | | Tessellations_Using_Geometer\'s_Sketchpad Description of a class project. | | Tessellations_with_Java Examples and source code. | | Tiling_Plain_and_Fancy Tiling from the mathematical and historical viewpoint by Steve Edwards. Provides a number of examples of various types of tiles that tile the plane and includes periodic tiling plus Penrose's aperiod | | Tilings_and_Geometric_Ornament Applying principles of computer graphics to the creation of geometric ornament, as a continuation of the tradition of ornamental design using modern tools and algorithms. | | Totally_Tessellated An introduction to tessellations explaining the basic underlying mathematics, and with many examples from real life. | | 17_Wall_Paper_Symmetry_Groups_to_Create_a_Regular_Division_of_the_Plane Graphics. | | Wallpaper_Groups Images and descriptions of the 17 plane symmetry groups. | | William\'s_Geometric_Construction_Page Activities to create patterns using simple resources, such as making equilateral triangle from a circle. Suitable for upper primary and lower secondary ages. | | Aanderaa_Instruments Instruments and Sensors for Land, Sea and Air | | Adcon_Telemetry_GmbH Provide meteorological sensors, but specialty is wireless telemetry. | | Advanced_Designs_Corp_ Doppler Weather Radar and Weather Data Displays, Doppler Radar, Weather Radar, Doppler, Radar | | Advanced_Sensing_Products ASP is a manufacturer standard PRTs, precision RTDs, and different types of thermocouples. | | Alltec_Corp_ A manufacturer of lightning protection and grounding systems. | | AllWeatherInc A source of weather sensors, environmental monitoring systems, data collection products, and display and recording devices. | | Applied_Technologies,_Inc_ Manufacturer and supplier of scientific instrumentation, software, and systems for meteorological and atmospheric research. | | Atmospheric_Radar_Systems_(ATRAD) Atmospheric Radar systems for scientific and commercial applications. | | Axinum_Innovations_&_Technologies Manufacture of Siphoning Rain Gauges and Data Loggers | | B__K__Consimpex_Meteorological_Systems Satellite remote sensing & metereological system integrators in India. | | Belfort_Instruments Offer a variety of weather instruments for the measurement of Wind, Precipitation, Temperature, Humidity, Pressure, and Visibility. | | BOLTEK A leader in affordable thunderstorm tracking technology and Lightning Detection Systems. |
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Hyperbolic TessellationsHyperbolic TessellationsIntroduction (You can now create your own hyperbolic tessellations. See the Java applet page.)A regular tessellation, or tiling, is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex. No doubt, the tessellations of the Euclidean plane are well-known to you. They are: {3,6} in which equilateral triangles meet six at each vertex; {4,4} in which squares meet four at each vertex; and {6,3} in which hexagons meet three at each vertex. A notation like {3,6} is called a Schläfli symbol.There are infinitely many regular tessellations of the hyperbolic plane. You candetermine whether {n,k} will be a tessellation of the Euclidean plane, the hyperbolic plane, or the elliptic plane by looking at the sum1/n + 1/k. If the sum equals 1/2, as it does for the three tessellations mentioned above, then {n,k} is a Euclidean tessellation. If the sum is less than 1/2, then the tessellation is hyperbolic; but if greater than 1/2, then elliptic.You might ask why. For a tessellation {n,k}, there are k regular polygons at each vertex. So the angle at each vertex is 360°/k. Since a regular n-gon has n equal angles, each being 360°/k, therefore the angle sum is n360°/k.Now, in the Euclidean plane a triangle has an angle sum of exactly 180°; in the hyperbolic plane less; and in the elliptic plane more. By breaking a polygon into triangles you can determine that the angle sum of an n-gon is exactly (n - 2)180° in the Euclidean plane; less in hyperbolic; more in elliptic. Therefore, if n360°/k equals (n - 2)180°, then {n,k} can only be Euclidean; if less, hyperbolic; and if more, elliptic. A little algebra (divide by n360° and add 1/n), and you see that statement is becomes this: if 1/n + 1/k equals 1/2, then {n,k} can only be Euclidean; if less, hyperbolic; and if more, elliptic.The Poincaré DiskThe hyperbolic plane can not be metrically represented in the Euclidean plane, but Poincaré described ways that it can be conformally represented in the Euclidean plane. One of those is to represent the hyperbolic plane as the points inside a disk. For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles. What this means will be clear in the examples displayed below.The regular tessellation {5,4} of the hyperbolic planeA regular tessellation is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex. For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation. It may look like the sides of the pentagons are curved, but that's just because of the representation we're using. In the actual hyperbolic plane they would be straight. Also, the pentagon in the middle looks larger, but, again, that's due to the representation. You just can't put an infinite plane in a finite region without a lot of distortion. Variations of this{5,4}-tessellation are also available.The dual tessellation {4,5} of the hyperbolic planeFor a dual tessellation you reverse the roles of the faces and the vertices.The dual of a {5,4} tessellation is a {4,5} tessellation, that is, a tiling bysquares, five squares meeting at each vertex. (Here, "square" means regular quadrilateral, a four-sided figure with the same angle at each vertex. It doesn't mean the corners are right-angled, so maybe "square" isn't the best term.) For this picture, the diagonals are drawn so you can see the straight lines that go off to infinity. Other regular tessellations are available including {8,4}, {4,8}. {6,6}, {3,12}, and {12,3}.Some quasiregular tessellations of the hyperbolic planeA quasiregular tessellation is built from two kinds of regularpolygons so that two of each meet at each vertex, alternately. We'll use the notation quasi-{n,k} to denote a quasiregular tessellationby n-gons and k-gons.Every regular tessellation {n,k} gives rise to aquasiregular tessellation quasi-{n,k} by connecting themidpoints of the edges of the regular tessellation. In the Euclidean planethere are just two quasiregular tessellations: quasi-{3,6}arises from both {3,6} and {6,3}, whilequasi-{4,4} comes from {4,4}. (Of course,quasi-{n,n} is the same as {n,4}.)Since there are more regular tessellations of the hyperbolic plane than ofthe Euclidean plane, there are more quasiregular tessellations, too. Here are some of them. First a quasi-{5,4} tessellation. The pentagons are in red or yellow while the squares are in orange. It looks like a plaid disk. |
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