or ic ), ▁from ▁the ▁Greek ▁roots ▁poly ▁(" many ") ▁and ▁ch or os ▁(" room " ▁or ▁" space ") ▁and ▁is ▁advoc ated ▁by ▁Norman ▁Johnson ▁and ▁George ▁O ls he v sky ▁in ▁the ▁context ▁of ▁uniform ▁pol ych ora ▁( 4 - poly top es ), ▁and ▁their ▁related ▁ 4 - dimensional ▁symmetry ▁groups . ▁ ▁Rank ▁ 4 ▁C ox eter ▁groups ▁allow ▁a ▁set ▁of ▁ 4 ▁mirror s ▁to ▁span ▁ 4 - space , ▁and ▁div ides ▁the ▁ 3 - s phere ▁into ▁t etra hed ral ▁fundamental ▁domains . ▁Lower ▁rank ▁C ox eter ▁groups ▁can ▁only ▁bound ▁hos oh ed ron ▁or ▁hos ot ope ▁fundamental ▁domains ▁on ▁the ▁ 3 - s phere . ▁ ▁Like ▁the ▁ 3 D ▁poly hed ral ▁groups , ▁the ▁names ▁of ▁the ▁ 4 D ▁pol ych or ic ▁groups ▁given ▁are ▁constructed ▁by ▁the ▁Greek ▁prefix es ▁of ▁the ▁cell ▁counts ▁of ▁the ▁corresponding ▁triangle - fac ed ▁regular ▁poly top es . ▁Ext ended ▁symmet ries ▁exist ▁in ▁uniform ▁pol ych ora ▁with ▁symmetric ▁ring - pattern s ▁within ▁the ▁C ox eter ▁diagram ▁construct . ▁Ch iral ▁symmet ries ▁exist ▁in ▁altern ated ▁uniform ▁pol ych ora . ▁▁ ▁Only ▁ir reducible ▁groups ▁have ▁C ox eter ▁numbers , ▁but ▁du op r ism atic ▁groups ▁[ p , 2 , p ] ▁can ▁be ▁double d ▁to ▁[[ p , 2 , p ]] ▁by ▁adding ▁a ▁