esian ▁coordinates ▁( x , y , z ). ▁ ▁The ▁R au zy ▁f ract al ▁is ▁constructed ▁this ▁way : ▁▁ 1 ) ▁Inter pret ▁the ▁sequence ▁of ▁letters ▁of ▁the ▁infinite ▁Trib on acci ▁word ▁as ▁a ▁sequence ▁of ▁unit ary ▁vectors ▁of ▁the ▁space , ▁with ▁the ▁following ▁rules ▁( 1 ▁= ▁direction ▁x , ▁ 2 ▁= ▁direction ▁y , ▁ 3 ▁= ▁direction ▁z ). ▁▁ 2 ) ▁Then , ▁build ▁a ▁" st air " ▁by ▁trac ing ▁the ▁points ▁reached ▁by ▁this ▁sequence ▁of ▁vectors ▁( see ▁figure ). ▁For ▁example , ▁the ▁first ▁points ▁are : ▁▁▁▁▁▁▁▁▁▁ ▁etc ... Every ▁point ▁can ▁be ▁colored ▁according ▁to ▁the ▁corresponding ▁letter , ▁to ▁stress ▁the ▁self - similar ity ▁property . ▁▁ 3 ) ▁Then , ▁project ▁those ▁points ▁on ▁the ▁contract ing ▁plane ▁( plane ▁orthogonal ▁to ▁the ▁main ▁direction ▁of ▁propag ation ▁of ▁the ▁points , ▁none ▁of ▁those ▁project ed ▁points ▁escape ▁to ▁infinity ). ▁ ▁Properties ▁▁ ▁Can ▁be ▁t iled ▁by ▁three ▁copies ▁of ▁itself , ▁with ▁area ▁reduced ▁by ▁factors ▁, ▁ ▁and ▁ ▁with ▁ ▁solution ▁of ▁: ▁. ▁ ▁St able ▁under ▁ex ch anging ▁pieces . ▁We ▁can ▁obtain ▁the ▁same ▁set ▁by ▁ex ch anging ▁the ▁place ▁of ▁the ▁pieces . ▁ ▁Con ne cted ▁and ▁simply ▁connected . ▁Has ▁no ▁hole . ▁ ▁T iles ▁the ▁plane ▁period ically , ▁by ▁translation . ▁ ▁The ▁matrix ▁of ▁the ▁Trib on acci ▁map ▁has ▁ ▁as