http://de.evo-art.org/index.php?action=history&feed=atom&title=Bending_Circle_LimitsBending Circle Limits - Versionsgeschichte2026-05-27T05:29:13ZVersionsgeschichte dieser Seite in de_evolutionary_art_orgMediaWiki 1.27.4http://de.evo-art.org/index.php?title=Bending_Circle_Limits&diff=3905&oldid=prevGbachelier: Die Seite wurde neu angelegt: „ == Reference == Vladimir Bulatov: Bending Circle Limits. In: Bridges 2013. Pages 167–174 == DOI == == Abstract == M.C.Escher’s hyperbolic tess…“2015-01-28T11:20:12Z<p>Die Seite wurde neu angelegt: „ == Reference == Vladimir Bulatov: <a href="/index.php?title=Bending_Circle_Limits" title="Bending Circle Limits">Bending Circle Limits</a>. In: <a href="/index.php?title=Bridges_2013" title="Bridges 2013">Bridges 2013</a>. Pages 167–174 == DOI == == Abstract == M.C.Escher’s hyperbolic tess…“</p>
<p><b>Neue Seite</b></p><div><br />
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== Reference ==<br />
Vladimir Bulatov: [[Bending Circle Limits]]. In: [[Bridges 2013]]. Pages 167–174 <br />
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== DOI ==<br />
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== Abstract ==<br />
M.C.Escher’s hyperbolic tessellations “Circle Limit I-IV” are based on a tiling of hyperbolic plane by identical<br />
triangles. These tilings are rigid because hyperbolic triangles are unambiguously defined by their vertex angles.<br />
However, if one reduce the symmetry of the tiling by joining several triangles into a single polygonal tile, such tiling<br />
can be deformed. Hyperbolic tilings allow a deformation which is called bending. One can extend tiling of the<br />
hyperbolic plane by identical polygons into tiling of the hyperbolic space by identical infinite prisms (chimneys).<br />
The original polygon being the chimney’s cross section. The shape of these 3D prisms can be carefully changed by<br />
rotating some of its sides in space and preserving all dihedral angles.<br />
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The resulting tiling of 3D hyperbolic space creates 2D tiling at the infinity of hyperbolic space, which can be thought<br />
of as the sphere at infinity. This sphere can be projected back into the plane using stereographic projection. After<br />
small bending the original circle at infinity of the 2D tiling becomes fractal curve. Further bending results in thinning<br />
the fractal features which eventually form a fractal set of circular holes which in the end disappear.<br />
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== Extended Abstract ==<br />
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== Bibtex == <br />
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== Used References ==<br />
[1] M.C. Escher. M.C. Escher, His Life and Complete Graphic Work. 1992.<br />
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[2] M. von Gagern and J. Richter-Gebert. Hyperbolization of euclidean ornaments. The Electronic Journal<br />
of Combinatorics, 16R2, 2009.<br />
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== Links ==<br />
=== Full Text === <br />
http://archive.bridgesmathart.org/2013/bridges2013-167.pdf<br />
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[[intern file]]<br />
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=== Sonstige Links ===<br />
http://archive.bridgesmathart.org/2013/bridges2013-167.html</div>Gbachelier