example non periodic generated set named mathematician physicist investigated sets 1970s implies shifted copy match original constructed exhibit reflection symmetry symmetry right remarkable properties notably non periodic means lacks symmetry self similar patterns occur larger larger scales obtained inflation finite patch occurs infinitely times implemented physical structure produce reveals symmetry underlying long range order methods construct discovered including matching rules subdivision rules cut project schemes background history periodic simple examples plane covering plane tiles overlaps gaps tiles normally finite number shapes called set said admit tile plane plane tiles familiar squares periodic perfect copy obtained translating tiles fixed distance given direction translation called period informally means finite region repeats periodic intervals periods said non periodic set said tiles plane non periodic sets called earliest subject received new interest 1960s noted connections decision problems particular introduced square plates colored edges known tiles posed problem determine given set tile plane matching colors adjacent edges observed problem exist set time set exist student proved incorrect thesis obtained set described reduction appear published monograph detailed modification set requiring color matching required easily achieved modifying edges tiles like puzzle pieces fit prescribed edge paper simplified techniques proof technique obtain set development set introduced paper based squares attempt tile plane regular necessarily leaves gaps showed work gaps filled star related shapes traces ideas found work acknowledging inspiration found matching rules imposed decorations edges shapes obtaining set viewed completion finite pattern subsequently reduced number discovering independently discovered investigated properties discovered property explained nature findings publicized january mathematical games column explained method construct families parallel lines cut project method obtained dimensional projections dimensional cubic structure approach viewed set points tiles shapes obtained connecting edges types described individually common features case tiles constructed shapes related pentagon golden ratio basic tile shapes need supplemented matching rules order tile rules described labeled edges patterns tile faces alternatively edge profile modified obtain set original uses shapes pointed star boat roughly star diamond thin ensure non periodic matching rules specify tiles meet different types matching rule tiles common indicate different types tiles different colors figure right second uses called combined matching rules prohibit combination composed called robinson notes robinson interior angles degrees axis symmetry form pair acute robinson angles degrees non convex interior angles degrees axis symmetry form pair robinson angles degrees smaller acute matching rules described ways approach color colors black white require adjacent tiles matching use pattern circular arcs shown left green red placement tiles tiles share edge patterns match edges rules force placement certain tiles example concave necessarily filled corresponding figure center row lower image left called ace conway looks like enlarged tile way similarly concave formed meet short edge necessarily filled right fact seven possible ways tiles meet figures star left sun right fold symmetry reflections remainder single axis reflection vertical image figures apart ace sun force placement additional tiles uses pair referred context equal sides different angles ordinary shaped tiles tile plane periodically restrictions tiles assembled tiles form allow periodic sufficient force figure shows kinds tile robinson thin corners angles degrees short diagonal form pair acute robinson thick angles degrees long diagonal form pair robinson contrast larger acute matching rules distinguish sides tiles tiles certain particular ways ways describe matching rules shown image right form tiles assembled curves faces match color position edge tiles assembled edges fit ordered combinations angles add degrees rules allow seven combinations appear arises ways combinations angles facial allow construction complex tiles chickens features constructions golden ratio local symmetry properties common features involve golden ratio approximately ratio chord lengths lengths regular pentagon consequently ratio lengths long sides short sides robinson follows ratio long lengths short tiles length sides short diagonal thin long diagonal sides thick ratio area larger robinson triangle smaller areas thick thin larger smaller robinson found pentagon left larger halves thick linear dimensions scaled compared small shaded triangle base ratio areas local symmetry sense points surrounded configuration tiles configurations symmetry center point mirror lines reflection symmetry passing point symmetry group symmetry generally preserve patch tiles center point patch large conway proved colored curves close loop region loop symmetry furthermore curves color close center point global symmetry rotating yield closer centers symmetry leads mathematical contradiction type global symmetry center point sun star inflation common features follow structure given rules referred inflation composition collections tiles rules tile smaller tiles shape allow larger tiles composed smaller ones shows scaling self similarity thought originally discovered way pentagon smaller half net half diamonds observed repeated process gaps filled stars diamonds boats process indefinitely obtained symmetry robinson triangle method described robinson different sizes robinson arising called tiles arising called tiles smaller tile denoted robinson triangle larger tile acute contrast smaller tile denoted acute robinson triangle larger tile lengths lengths tiles related tiles ways size enlarged version lengths tile tile joined common length instead identified reduced version lengths joining tile tile common length yields tile appears ambiguity robinson ways mirror images axis symmetry triangle choice fixed matching rules furthermore matching rules determine smaller compose larger ones follows mutually locally set tiles generate example subdivided tiles composed canonical way form tiles mutually locally figure tiles tiles written assuming larger size convention tiles summarized matrix equation formula combining enlarged tiles tiles yields formula enlarged tile tiles tiles matching rules force particular tiles tile form half half enlarged tiles tiles similar way tiles composition example formula number iteration construction determined power matrix formula number ratio numbers sufficiently large pattern golden ratio similar result holds ratio number thick thin starting collection tiles given single tile plane collection proceeds sequence steps called generations generation tile replaced new tiles scaled versions tiles original rules guarantee new tiles arranged accordance matching rules repeated generations produce original shape smaller smaller tiles rule dividing tiles subdivision rule consequences applications inflation yield method constructing known generation non periodic symmetry pattern shifted match entire plane bounded region matter large repeated infinite number times finite patch differentiate determine position shown shows particular number distinct type infinite generation yields method methods use bars cut project schemes related topics german mathematician demonstrated covering called distinguish non overlapping equivalent constructed single tile kinds overlapping regions allowed tile decorated colored patches covering rule allows overlaps compatible coloring suitable tile transforms covering similarly obtained thick remaining space filled thin considered realistic model growth overlapping quasi unit cells analogous unit cells crystals constructed matching rules maximize density certain atomic clusters related variants mutually locally selecting allows produce non periodic corners pentagon labeled succession established order clockwise points label define robinson points numbers define related boat star instance matching rules reduced specific restriction angles permitted obtained underlying symmetry obtained original smaller ones applying rules cut project method art aesthetic value long appreciated remains source interest visual appearance formal defining properties attracted attention similarity decorative patterns middle_east noted presented evidence examples medieval islamic geometric patterns shrine artist artwork derived projecting shadow plane observing embedded fat tile produce non periodic art historian observed similar motifs san_francisco new billion transit center planned exterior white metal skin pattern floor molecular tiles location university_of_oxford october includes section paving entrance pedestrian street paved form work finished
Penrose tiling