Konstruktionsschritte:

\copyright \;Roman \;Chijner,\;Heidelberg\;2013

GS:  f

Fünfsterne

Fünfeck

\text{Parkettierung regulärer Fünfecke und Fünfsterne mit Seitenverhältnissen im Goldenen Schnitt (ähnlich der Penrose–Parkettierung)}

n_{ext} = 5 (slider from 1 to 10, step 1)
f = 0.6180339887498949 (slider from 0 to 5, step 0.1)
Zoom = 3.1 (slider from 0 to 15, step 0.1)
An = Intersect(yAxis, xAxis)
l1 = Sequence(Rotate[(Zoom; 0), 2 * π / 10 * i, An], i, 0, 10)
r2: Zoom * cos(π / 10) + 2 * Zoom * sin(π / 10) / (2 * tan(π / 5))
An1: (r2; 18°)
lPV5 = Sequence(Rotate[Element[l1, 1], 2 * π / 5 * i, An1], i, 0, 5)
VZ1, l, m, n, q, s, t = Polygon(Element[lPV5, 1], Element[lPV5, 3], Element[lPV5, 5], Element[lPV5, 2], Element[lPV5, 4], Element[lPV5, 1])
l3 = Sequence(Polygon[Element[l1, i + 1], Element[l1, i], 5], i, 1, 10)
V3, a, o, b = Polygon(An, Element[l1, 1], Element[l1, 2])
Str01 = Ray(An, (Element[l1, 1] + Element[l1, 2]) / 2)
V51, d, e, g, h, i, j = Polygon(lPV5)
lrP1 = Sequence(Dilate[An1, 1 / f^(i), An], i, 0, n_{ext})
lZ1: {Segment[Element[lPV5, 1], Element[lPV5, 3]], Segment[Element[lPV5, 1], Element[lPV5, 4]], Segment[Element[lPV5, 2], Element[lPV5, 4]], Segment[Element[lPV5, 2], Element[lPV5, 5]], Segment[Element[lPV5, 3], Element[lPV5, 5]]}
lrPV = Sequence(Dilate[lPV5, 1 / f^(i), An], i, 0, n_{ext})
lrV = Sequence(Dilate[V51, 1 / f^(i), An], i, 0, n_{ext})
lrV2 = Sequence(Translate[Rotate[V51, (i + 1) * π, An1], Element[lrP1, 2 * i - 1] - An1], i, 1, n_{ext} - 2)
lrZ = Sequence(Dilate[VZ1, 1 / f^(i), An], i, 0, n_{ext})
ZV2 = Element(lrZ, 2)
lrP2 = Sequence(Dilate[Element[lrP1, i], 1 + f^(3), An], i, 3, n_{ext} + 1)
Liste1 = Sequence(Rotate[Translate[V51, Element[lrP2, 1] - An1], 2 * π / 5 * i, Element[lrP1, 3]], i, 0, 4)
An2 = Element(lrP1, 2)
Liste2 = Sequence(Rotate[Translate[ZV2, Element[lrP2, 2] - An2], 2 * π / 5 * i, Element[lrP1, 4]], i, 0, 4)
lrZ2 = Sequence(Translate[Rotate[ZV2, (i + 1) * π, An2], Element[lrP1, 2 * i] - An2], i, 1, n_{ext} - 2)
Liste3 = Sequence(Rotate[Translate[Liste1, Element[lrP2, 3] - Element[lrP1, 3]], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4)
Liste4 = Translate(Rotate[Liste1, 3.141592653589793, Element[lrP1, 3]], Element[lrP1, 5] - Element[lrP1, 3])
Liste5 = Sequence(Rotate[Element[Liste1, 1], 2 * π / 5 * i, Element[lrP1, 4]], i, 0, 4)
Liste6 = Sequence(Rotate[Element[Liste2, 1], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4)
Liste7 = Sequence(Rotate[Translate[Element[lrV2, 2], Element[lrP2, 3] - Element[lrP1, 3]], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4)
Liste41 = Translate(Rotate[Liste5, 3.141592653589793, Element[lrP1, 4]], Element[lrP1, 6] - Element[lrP1, 4])
Liste42 = Translate(Rotate[Liste2, 3.141592653589793, Element[lrP1, 4]], Element[lrP1, 6] - Element[lrP1, 4])
Liste43 = Sequence(Rotate[Translate[Liste2, Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4)
Liste44 = Sequence(Rotate[Translate[Liste5, Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4)
Liste45 = Sequence(Rotate[Translate[Element[lrZ2, 2], Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4)
Liste46 = Sequence(Rotate[Element[Liste7, 1], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4)
Liste47 = Sequence(Rotate[Element[Liste3, 1], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4)
c = false
k = false
V52 = Dilate(V51, 1 + f, (0, 0))
ZV1: {Segment[Element[lPV5, 1], Element[lPV5, 3]], Segment[Element[lPV5, 1], Element[lPV5, 4]], Segment[Element[lPV5, 2], Element[lPV5, 4]], Segment[Element[lPV5, 2], Element[lPV5, 5]], Segment[Element[lPV5, 3], Element[lPV5, 5]]}
p = Circle(An, Element[Element[lrPV, 6], 3])
lrZz = Sequence(Dilate[VZ1, 1 / f^(i), An], i, 0, n_{ext})