tran[{x_,t_},p_]={xCosh[p]-tSinh[p],tCosh[p]-xSinh[p]};
trang[{x_,t_},p_]={x-pt,t};
hist150R=FromCAState[CAEvolveList[Reversible[ElementaryRule[150]],{ConstantList[203,0],CenterList[203,{1}]},100]];
hist150=FromCAState[CAEvolveList[ElementaryRule[150],CenterList[203,{1}],100]];
RTrans[ran_,hist_,v_]:=With[{w=Length[First[hist]],h=Length[hist]},LabelWrapper[Surround[Graphics[{AbsolutePointSize[.5],(Point[ran[{Last[#]-Ceiling[w/2],1-First[#]},v]]&)/@Position[hist,1]},AspectRatioAutomatic,PlotRange->{{-w/4,w/4},{-h/2,0}}]],StringForm["v = ``",v]]]
Show[GraphicsRow[RTrans[tran,hist150R,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[trang,hist150R,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[tran,hist150,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[trang,hist150,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[tran,hist150R,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[trang,hist150R,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[tran,hist150,#]&/@Range[0,.6,.3]]];
Show[GraphicsRow[RTrans[trang,hist150,#]&/@Range[0,.6,.3]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0]]&)/@px150},AspectRatioAutomatic,PlotRange->{{0,203},{-101,0}}]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0.25]]&)/@px150},AspectRatioAutomatic,PlotRange->{{0,203},{-126,-25}}]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[trang[Reverse[{-1,1}#1],0.25]]&)/@px150},AspectRatioAutomatic]]];
px150=Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[150]],{ConstantList[300,0],RandomList[300]},200]],1];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0]]&)/@px150},AspectRatioAutomatic]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0.25]]&)/@px150},AspectRatioAutomatic]]];
px150=Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[150]],{ConstantList[100,0],RandomList[100,2,.1]},100]],1];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0]]&)/@px150},AspectRatioAutomatic]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0.25]]&)/@px150},AspectRatioAutomatic]]];