flat surface plane geometric shapes called tiles overlaps gaps mathematics higher dimensions variety periodic repeating pattern special kinds include regular regular tiles shape regular tiles shape corner arranged patterns formed periodic categorized groups lacks repeating pattern called non periodic uses small set tile shapes form repeating pattern geometry higher dimensions space filling called space real physical materials cemented squares decorative patterns functions providing water resistant pavement floor wall historically islamic art decorative palace twentieth century work use ordinary geometry geometry artistic effect employed decorative effect form class patterns nature example cells found history building wall decorations formed patterns clay tiles decorative mosaic small squared blocks called widely employed classical antiquity displaying geometric patterns early documented study wrote regular possibly explore explain structures years later russian proved periodic plane features seventeen different groups work marked unofficial beginning mathematical study prominent contributors include etymology latin small piece clay stone glass word means small square square turn greek word corresponds everyday term refers applications clay overview dimensions topic geometry studies shapes known tiles arranged fill plane gaps according given set rules rules varied common ones gaps tiles corner tile lie edge created obey rule regular identical regular tiles identical regular corners having angle adjacent edges tile shapes form regular triangle square regular shapes infinitely fill plane gaps types possible different constraints example types semi regular kind regular having arrangement corner irregular shapes fact kind geometric shape artist famous making irregular tiles shaped like animals natural objects suitable contrasting colours chosen tiles differing shape striking patterns formed physical surfaces church floors formally cover plane number closed sets called tiles tiles intersect boundaries tiles shapes formed finite number tiles given geometric shape create shape said tile plane conway criterion sufficient necessary set rules deciding given shape tiles plane periodically reflections tiles fail criterion tile plane general rule found determining given shape tile plane means problems concerning example types convex pentagon tile plane remains problem extended spaces plane swiss pioneered defining spaces dimensions defined symbol easy describe example symbol triangle square makes possible describe example regular sided symbol methods exist describing regular common configuration simply list number sides square configuration regular noted mathematics introduction use technical terms discussing edge intersection bordering tiles straight line point intersection bordering tiles terms point identical arrangement fundamental region shape repeated form example regular plane squares meeting squares sides necessarily identical edges tiles edge edge adjacent tiles share tile shares partial tile edge edge sides edges tiles familiar brick wall edge edge long rectangular brick shared bordering bricks normal tile equivalent disk intersection tiles single connected set set tiles uniformly bounded means single radius single radius tiles condition tiles long thin tiles particularly interesting type spiral spiral discovered unit tile published pentagon irregular regular tile plane internal angle regular pentagon special variation tiles belong class tiles transforms symmetry group admits called forms regular highly edge edge regular shape regular squares regular semi regular uses type regular arrangement semi regular mirror image pair counts described configuration example semi regular squares regular configuration square non edge edge plane possible including family use sizes square square touching squares size groups symmetry independent directions categorized groups exist claimed seventeen groups represented palace spain disputed variety surprised modern researchers regular group symmetry direction categorized seven groups describing possible patterns describe groups plane use different best known example tiles forcibly create non periodic patterns belong general class use tiles periodically process method generating class generated way tiles surprising self properties non periodic tile construction tiles appear infinitely thought non periodic pattern entirely symmetry lacking symmetry types infinite repetition bounded patch certain finite groups reflections patches rule generate patterns assemblies tiles called illustrates scaling symmetry word build study structures order tiles squares coloured edge placed edges adjacent tiles colour called suitable set tile plane known machine represented set tile plane machine halt halting problem problem deciding set tile plane tiles square tiles decorated patterns symmetry square tile split contrasting colours tile plane periodically randomly colour colour tile understood times arbitrary colours applied later discussing displayed colours avoid ambiguity needs specify colours illustration affects tiles shape different colours considered identical turn affects questions symmetry colour states normal plane set available colours tile coloured colour tiles equal colour meet curve positive length guaranteed colour generally respect produce necessary treat colours seven colours needed picture right regular studied triangle non convex form way copies arbitrary form symmetry fold symmetry centres sides belongs group fundamental domain construct minimal set translation starting centre divide diagonal half triangle fundamental domain triangle area constructed cutting shape tile allowed exists convex equal results plane uses tile defined set points closest points set defining points think geographical regions region defined points closest given city post office cell defining point convex dual useful numerical simulation possible defining points maximize minimum angles formed edges randomly placed points construct random plane higher dimensions extended dimensions certain stacked regular crystal pattern fill tile dimensional space including cube regular truncated naturally occurring found crystals kind triangle spherical triangle tile sphere dimensions called dimensions regular similarly dimensions possible dimensions uniform constructed construction convex property space non possible non geometry uniform plane regular edge edge filling plane regular faces mapping uniform space uniform uniform cells dimensional space group families compact convex uniform generated constructions represented rings family art architecture create decorative motifs ancient times mosaic geometric patterns later larger tiles plain individually decorated decorative wall islamic architecture tiles buildings frequently appeared graphic art inspired use symmetry places visited spain circle limit drawings use geometry circle limit prepared pencil ink study showing required geometry explained single component series infinitely far away rise like rockets limit lost reaches boundary line designs appear woven printed patterns design motifs patch shapes main genre paper folding connect molecules twist repeating fashion manufacturing manufacturing industry reduce material yield losses sheet metal cutting shapes objects like car doors drinks cans nature provides known example nature cells term describes pattern example flower tree bark fruit flowers including species patterns nature formed cracks sheets materials patterns described gilbert known random crack networks gilbert mathematical model formation needle like crystals similar structures model named allows cracks form starting randomly scattered plane crack opposite directions line initiation point slope chosen random creating irregular convex lava flows display result forces causing cracks lava extensive crack networks develop produce columns lava example array columns giant causeway northern_ireland pavement characteristic example found neck rare rock formation rock fractured rectangular blocks natural patterns occur packed according plateau laws require minimal surfaces present problem pack cells tightly possible lord proposed packing solid cubic slightly curved faces proposed structure uses surface area separate cells equal volume foam puzzles recreational mathematics given rise types puzzle traditional puzzles irregular pieces wood cardboard modern puzzles mathematical basis example figures regular squares puzzles authors uses recreational mathematics example invented wrote tile shape smaller copies shape inspired articles amateur mathematician found new square problem integral square sides length integral squares extension plane squares sizes natural numbers james proved possible examples
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