ParametricPlot3D[{(e^(-x))Cos[θ], (e^(-x))Sin[θ],0}, {x,0,∞},{θ,0,2π}]

ParametricPlot3D[{(E^(-r)) Cos[θ],(E^(-r)) Sin[θ],0},{r,-5,5},{θ,0,2 π}]

Table[{k,u,v}, {k,-3,3}]

Table[{u,k 2π/5,-v}, {k,-3,3}]

ParametricPlot3D[
{{-3,u,v},{-2,u,v},{-1,u,v},{0,u,v},{1,u,v},{2,u,v},{3,u,v},{u,-((6 π)/5),-v},{u,-((4 π)/5),-v},{u,-((2 π)/5),-v},{u,0,-v},{u,((2 π)/5),-v},{u,((4 π)/5),-v},{u,((6 π)/5),-v}},
{u,-5,5},{v,0,5}]

{{-3,u,v},{-2,u,v},{-1,u,v},{0,u,v},{1,u,v},{2,u,v},{3,u,v},{u,-((6 π)/5),-v},{u,-((4 π)/5),-v},{u,-((2 π)/5),-v},{u,0,-v},{u,((2 π)/5),-v},{u,((4 π)/5),-v},{u,((6 π)/5),-v}} /. {x_,y_,z_}:>{(E^(-z))Cos[y], (E^(-z))Sin[y], x}

ParametricPlot3D[
{{(E^(-v)) Cos[u],(E^(-v)) Sin[u],-3},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],-2},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],-1},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],0},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],1},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],2},{(E^(-v)) Cos[u],(E^(-v)) Sin[u],3},{(1/4) (-1-(Sqrt[5])) (E^v),(Sqrt[(5/8)-((Sqrt[5])/8)]) (E^v),u},{(1/4) (-1-(Sqrt[5])) (E^v),-(Sqrt[(5/8)-((Sqrt[5])/8)]) (E^v),u},{(1/4) (-1+(Sqrt[5])) (E^v),-(Sqrt[(5/8)+((Sqrt[5])/8)]) (E^v),u},{(E^v),0,u},{(1/4) (-1+(Sqrt[5])) (E^v),(Sqrt[(5/8)+((Sqrt[5])/8)]) (E^v),u},{(1/4) (-1-(Sqrt[5])) (E^v),(Sqrt[(5/8)-((Sqrt[5])/8)]) (E^v),u},{(1/4) (-1-(Sqrt[5])) (E^v),-(Sqrt[(5/8)-((Sqrt[5])/8)]) (E^v),u}},
{u,-5,5},{v,0,2}]