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I have here an example of how a rotation matrix operates a parametric representation of a square.

R15 = {{Cos[π/12], -Sin[π/12]}, {Sin[π/12], Cos[π/12]}};
R30 = {{Cos[π/6], -Sin[π/6]}, {Sin[π/6], Cos[π/6]}};
R45 = {{Cos[π/4], -Sin[π/4]}, {Sin[π/4], Cos[π/4]}};
R60 = {{Cos[π/3], -Sin[π/3]}, {Sin[π/3], Cos[π/3]}};
R75 = {{Cos[(5 π)/12], -Sin[(5 π)/12]}, {Sin[(5 π)/12], Cos[(5 π)/12]}};

r1 = {t, -1};
r2 = {1, t};
r3 = {t, 1};
r4 = {-1, t};
rsq15 = {R15.r1, R15.r2, R15.r3, R15.r4};
rsq30 = {R30.r1, R30.r2, R30.r3, R30.r4};
rsq45 = {R45.r1, R45.r2, R45.r3, R45.r4};
rsq60 = {R60.r1, R60.r2, R60.r3, R60.r4};
rsq75 = {R75.r1, R75.r2, R75.r3, R75.r4};
ParametricPlot[{sq, rsq15, rsq30, rsq45, rsq60, rsq75}, {t, -1, 1}]

How would you minimize the amount of code necessary?

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If you need a more literal refactoring of your code, rather than bill's Line drawing, consider:

R15 = RotationMatrix[π/12];  (* thanks to J. M. ! *)

rr = {{t, -1}, {1, t}, {t, 1}, {-1, t}};

sqrs = NestList[#.R15 &, rr, 5];

ParametricPlot[sqrs, {t, -1, 1}]

enter image description here

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    $\begingroup$ R15 = RotationMatrix[π/12]; is shorter. FWIW, here's how to sidestep the use of MatrixPower[] altogether: rr = {{t, -1}, {1, t}, {t, 1}, {-1, t}}; sqrs = Array[rr.RotationMatrix[(1 - #) π/12] &, 5]; ParametricPlot[sqrs, {t, -1, 1}] $\endgroup$
    – J. M.'s torpor
    May 26 '17 at 15:31
  • $\begingroup$ Very nice! Thanks for the lesson. :-) $\endgroup$
    – Mr.Wizard
    May 26 '17 at 15:32
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You could use graphics objects:

Graphics[Rotate[Line[{{1,1}, {1,-1}, {-1,-1}, {-1,1}, {1,1}}], #]
    &/@ Range[Pi/16, Pi/2, Pi/16]]

enter image description here

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Personally, I would recommend that such figures be generated from primitives, as in bill's answer. Nevertheless, allow me to present a slight curiosity based on this math.SE answer:

PolarPlot[Table[Sec[Mod[θ - k π/12, π/2] - π/4], {k, -3, 2}] // Evaluate, {θ, 0, 2 π}]

squares

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rr = {{t, -1}, {1, t}, {t, 1}, {-1, t}};
rotations = Map[RotationTransform[#][rr] &, Range[0, 2 Pi, 2 Pi/5]];
ParametricPlot[rotations, {t, -1, 1}]

enter image description here

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    $\begingroup$ You can also use Graphics@Table[Line@CirclePoints[{1,θ},4], {θ, 0, π, π/8}] $\endgroup$ May 26 '17 at 19:24
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You've got several choices for the basic square:

Line[{{1, 1}, {1, -1}, {-1, -1}, {-1, 1}, {1, 1}}]

or, with FaceForm[], EdgeForm[Black], either of

Polygon@CirclePoints[Sqrt[2], 4]
Rectangle[{-1, -1}, {1, 1}]

Then I'd probably use GeometricTransform:

Graphics[{
  FaceForm[], EdgeForm[Black],
  GeometricTransformation[
   Rectangle[{-1, -1}, {1, 1}],
   RotationMatrix /@ (Pi Range[8]/16)
   ]
  }]

But this explicitly constructs all the coordinates:

Graphics[{
  FaceForm[], EdgeForm[Black],
  Polygon@ Transpose[
    CirclePoints[Sqrt[2], 4] . Transpose[RotationMatrix /@ (Pi Range[8]/16), {2, 3, 1}]] 
   ]
  }]
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