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I'm trying to fit 5 pentagons around a pentagon, but it's not right. Please help.

my attempt

p = Table[{Cos[θ], Sin[θ]} // N, {θ, π/2 + 2 π/5, -3 π/2 + 2 π/5, -2 π/5}]
Graphics[{Line[p],
  Table[Rotate[
    Translate[Line[p],
     1.5 {Cos[t - 18], Sin[t - 18]}],
    t/2],
   {t, 72 Degree Range[5]}]}]
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    $\begingroup$ This needs more detail to be understandable. $\endgroup$
    – Szabolcs
    Commented Apr 30, 2020 at 19:19
  • $\begingroup$ What you want is the first step of generating the "pentaflake" fractal. That page goes in detail on how the pentagons fit together. $\endgroup$ Commented May 2, 2020 at 11:37
  • $\begingroup$ Thank you for the reference @J.M. Did you get a notice from my comment on question 84939? $\endgroup$
    – shirha
    Commented May 2, 2020 at 14:04
  • $\begingroup$ Did you not check your inbox for responses? $\endgroup$ Commented May 2, 2020 at 14:15

2 Answers 2

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I think this might be what you seek:

Graphics[{
    FaceForm[GrayLevel[0.5]], EdgeForm[{Black, Thick}],
    RegularPolygon[5],
    FaceForm[GrayLevel[0.8]],
    RegularPolygon[#, {1, 3 Pi/10}, 5] & /@ CirclePoints[{2 Cos[Pi/5], 3 Pi/10}, 5],
  }
]

a crown of polygons

All pentagons have radius 1, for convenience. The central pentagon is centered on the origin, so the outer pentagons have centers at a distance equal to twice the apothem of the central pentagon, and at positions equally spaced around a circle which can be obtained from CirclePoints using an appropriate angular shift.

Looking at the figure, I noticed that the outer pentagons all have the same angular with respect to the horizontal axis, which simplifies matters considerably, so they are constructed using RegularPolygon using the center locations obtained from CirclePoints above, and with a constant radius and angular position with respect to the $x$ axis.

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  • $\begingroup$ Just learned what an apothem is. 2 Cos[π/5]. Thanks. It looks like you are using the form of RegularPolygon[{x, y}, {r, θ}, n] but your θ is static. Where does the rotation come from? $\endgroup$
    – shirha
    Commented Apr 30, 2020 at 21:02
  • $\begingroup$ When you generate 5 of them with different {x,y}, do they all just orient themselves with respect to {0,0}? $\endgroup$
    – shirha
    Commented Apr 30, 2020 at 21:11
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    $\begingroup$ @shirha There is no need for rotation of the external pentagons. They are all oriented in the same way. That is what I was referring to in the answer that simplified matters considerably. Given their five-fold symmetry, the five-fold rotation around the origin would make each original and rotate one indistinguishable, so in practice no need to rotate. $\endgroup$
    – MarcoB
    Commented Apr 30, 2020 at 21:11
  • $\begingroup$ Wow. I see it now! It's like an optical illusion. Thank you @MarcoB for such a great solution. It's greatly appreciated. $\endgroup$
    – shirha
    Commented Apr 30, 2020 at 21:18
  • $\begingroup$ Of course, 2 Cos[π/5] is just the same as GoldenRatio. $\endgroup$ Commented May 2, 2020 at 11:39
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PolyhedronData["Dodecahedron", "Net"] /. Polygon[x_] :> Polygon[Take[x, 6]]

enter image description here

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  • $\begingroup$ Thanks! Do Nets have Faces and/or Vertices? $\endgroup$
    – shirha
    Commented Apr 30, 2020 at 21:41

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