Hyperbolic tilings of Mona-LisaIn Euclidean geometry you can only use triangles, squares or hexagons
Hyperbolic tilings of Mona-LisaIn Euclidean geometry you can only use triangles, squares or hexagons if you want to tile the plane using regular polygons. The total angle when rotating around a vertex of a polygon must be 360 degrees, and since the angles of all regular polygons depend on the angle sum of a triangle, these are the only options in Euclidean geometry. In non-Euclidean geometry, the angle sum of a triangle is either larger or less than 180 degrees. In hyperbolic geometry (one of the two kinds of non-Euclidean geometry), the angle sum of a triangle is always less than 180 degrees, and there are infinitely many ways to make a tiling using regular hyperbolic polygons. One way to model hyperbolic geometry, is to let the plane be represented by points inside a circle. The circle itself lies at infinity. In order to make a hyperbolic tiling, start with a hyperbolic polygon (which looks distorted for everyone living in Euclidean space), then reflect the polygon in all its sides, continue this process (in theory) infinitely many times.Make hyperbolic tilings of any image at: http://www.malinc.se/m/ImageTiling.php -- source link
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