# Edit Edit an Image Page: Hyperbolic Tilings

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 Image Title*: Upload a Math Image This image is a hyperbolic tiling made from alternating two shapes: heptagons and triangles. In addition to tessellations found in the traditional Euclidean geometry, there are also tilings that can be created in non-tradition or non-Euclidean geometry. Hyperbolic tiling is a type of tessellation that is visualized in a non-Euclidean geometry called [[Hyperbolic Geometry|hyperbolic geometry]]. This particular tessellation is viewed in the [[Hyperbolic Geometry#Poincaré Disk Model|Poincaré disk model]], where straight lines are arcs that form right angles at the boundary of the disk. [[Image:HyperbolicTiling2.jpg|200px|thumb|center|A hyperbolic tiling]] The polygons in a hyperbolic tiling grow smaller as they reach the edge of the disk. This is due to the representation of the tiling in the Poincaré disk model that we use to visualize hyperbolic tilings. The Poincaré disk model does not show traditional distances, so that as we get closer and closer to the edge of the disk, the distances increase and the boundary of the disk is in fact infinite. ==Schläfli symbol== The Schläfli symbol is a notation that can be used to describe the properties of polygons in a tessellation. The symbol is in the form ''{n , k}'', where ''n'' refers to the number of edges of a regular polygon and ''k'' refers to the number of regular polygons that meet at each vertex. [[Image:Tiling_3_8mod.png|625px|right|Hyperbolic Tiling {3,8}]] Thus, if the summation of all the angles at a vertex is 360 degrees: :the angle at each vertex is $\frac{360}{k}$ :and the polygon has a summation of angles $\frac{n360}{k}$. Regarding the image shown to the right, the tessellation is made up solely of triangles, so that ''n'' = 3. At each vertex in the tessellation, there is always a meeting of six triangles, giving us ''k'' = 8. Thus, the Schläfli symbol for this tessellation is {3,8}. Also, the angle at each vertex is 35 degrees, and the summation of angles of the polygon is 135 degrees (which is less than the traditional 180 degrees, a characteristic of hyperbolic tilings). ==Types of Tilings== Most tessellations are made from regular polygons, which are polygons with edges of equal side length and vertices of equal angle. This allows the tessellation to have the same number of polygons meeting at each vertex. ===Quasi-regular Tilings=== You will notice that some hyperbolic tilings utilize two polygons instead of one polygon. These types of tilings are called quasi-regular tilings. The two polygons will alternate meeting at each vertex and are usually colored differently for emphasis. To describe a quasi-regular tiling, the Schläfli symbol is still used with an prefix ''quasi'' and with ''n'' being the number of edges of one polygon and ''k'' being the number of edges of one polygon: quasi-{n,k}. It's interesting to note that every regular tessellation can be made into a quasi-regular tessellation. You simply connect the midpoints of all the edges of the polygons in the tessellation. In the image gallery below, the first tessellation shown is a regular tessellation {5,4}. The second image shows how to connect the midpoints of the edges to create a quasi-{5,4} tessellation like the one shown in the third image of the gallery. The ''n'' in the quasi-regular tessellation now refers to the the four-sided squares, while the ''k'' now represents the five-sided pentagons alternating within the tessellation. ===Dual Tessellations=== There is another type of tessellation that occurs when the Schläfli symbols {n,k} are switched to {k,n}. This is called a dual tessellation, and an example can be found in the final image in the gallery: {4,5}. (Note, although the Schläfli symbols for the quasi- and duel tessellations are the same for this particular set of examples, they are very different tessellations). Image:Tessl_5_4.gif Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other Yes, it is.