Reduce Code Length

since the result of the new computation depends on the previous one we can use NestList

NestList[N@({0, 0, 0.5} + 
  RotationMatrix[45 Degree, {0, 0, 1}].Transpose[#])\[Transpose] &, n01, 18]

Another one using NestList:

tf = TranslationTransform[{0, 0, 0.5}].RotationTransform[45 Degree, {0, 0, 1}];

allPts = NestList[tf, n01, 18]

This is concise and I think it will perform a little better than the methods using Transpose.

n01 = 
  N[{{5, 0, 0}, {6, 0, 0}, {6, 0, 1}, {6, 0, 2}, {5, 0, 2}, {4, 0, 2}, {4, 0, 1}, {4, 0, 0}, {5, 0, 0}}];

xform =
  AffineTransform[{RotationMatrix[45. °, {0, 0, 1}], {0, 0, 0.5}}];

allPts = NestList[xform, n01, 18];

Short[allPts, 12]

{{{5., 0., 0.}, {6., 0., 0.}, {6., 0., 1.}, {6., 0., 2.}, {5., 0., 2.}, 
  {4., 0., 2.}, {4., 0., 1.}, {4., 0., 0.}, {5., 0., 0.}},
 <<17>>,
 {{0., 5., 9.}, {0., 6., 9.}, {0., 6., 10.}, {0., 6., 11.}, {0., 5., 11.}, 
  {0., 4., 11.}, {0., 4., 10.}, {0., 4., 9.}, {0., 5., 9.}}}