Bild

GS: f

Fünfsterne

Fünfeck

Konstruktion\; des\; flachen\;  Penrose\; Kachel\; mit\; Hilfe\; zwei\;Fünfecke\; im\; goldene\ Verhätnise

Reguläre\;Fünfecke \;und\;  Fünfsterne\; beim \;Parkettierung \;die\;  mit \; Seiten\; in \;den\;  goldenen \;Schnitt\;Verhältnis\; sind\; (ähnlich\; der\; Penrose–Parkettierung)

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Gitter aus fünfsternigen Figuren in Verhältnissen des Goldenen Schnitts

Gitter aus fünfsternigen Figuren im 
Verhältnissen des Goldenen Schnitts

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

\text{Konstruktion der flachen Penrose Parkettierung mit Hilfe zweier Rauten im Verhältnis des Goldenen Schnitts}

f - Verhältnisse (Animation)

Zoom = 7.35 (slider from 0 to 15, step 0.01)
r2: Zoom * cos(π / 10) + 2 * Zoom * sin(π / 10) / (2 * tan(π / 5))
An1: (r2; 18°)
An = Intersect(yAxis, xAxis)
l1 = Sequence(Rotate[(Zoom; 0), 2 * π / 10 * i, An], i, 0, 10)
u2 = Vector((Element[l1, 1] + Element[l1, 2]) / 2)
au2 = Length(u2)
u1 = Vector(Element[l1, 2])
au1 = Length(u1)
au21: au1 / au2
lPV5 = Sequence(Rotate[Element[l1, 1], 2 * π / 5 * i, An1], i, 0, 5)
f = 0.6180339887498949 (slider from 0 to 1, step 0.02)
n_{ext} = 5 (slider from 1 to 10, step 1)
lrPV = Sequence(Dilate[lPV5, 1 / f^(i), An], i, 0, n_{ext})
v01, g_1, h_1, i_1, j_1 = Polygon(Element[Element[lrPV, 1], 3], Element[Element[lrPV, 2], 4], Element[Element[lrPV, 2], 3], Element[Element[lrPV, 2], 2])
l3 = Sequence(Polygon[Element[l1, i + 1], Element[l1, i], 5], i, 1, 10)
Str01 = Ray(An, (Element[l1, 1] + Element[l1, 2]) / 2)
lrP1 = Sequence(Dilate[An1, 1 / f^(i), An], i, 0, n_{ext})
lrP2 = Sequence(Dilate[Element[lrP1, i], 1 + f^(3), An], i, 3, n_{ext} + 1)
c = false
k = false
A = (46.92985855547167, 34.06674738896126)
B = Point(Str01)
C = (63.02685625617909, -0.21985771354539851)
D = (41.446602092212494, 56.96597965283478)
Liste9 = Sequence(Rotate[lrP1, π / 5 * i], i, 0, 9)
v02, b_2, d_1, e_1, f_1 = Polygon(B, Element[Element[lrPV, 1], 5], Element[Element[lrPV, 1], 3], Element[Element[lrPV, 1], 1])
v021 = Translate(v02, Element[Element[lrPV, 2], 3] - B)
Aa01: (An + Element[Element[lrPV, 1], 3]) / 2
Aa02: (Element[Element[lrPV, 2], 3] + Element[Element[lrPV, 1], 3]) / 2
Aa03: (Element[Element[lrPV, 2], 3] + Element[Element[lrPV, 3], 3]) / 2
v01a1 = Rotate(v01, -1.25664, Element[Element[lrPV, 2], 4])
v01a2 = Rotate(v01, 1.25664, Element[Element[lrPV, 2], 4])
an: 2 * Zoom * sin(36°)
d102 = Distance(Element[Element[lrPV, 2], 3], Element[Element[lrPV, 1], 3])
d201 = Distance(B, Element[Element[lrPV, 1], 3])
d202: au1 * f
d101: au1 / f
lP201 = Translate((0, 0), Vector[u1 * (2 + 1 / f + f / 2)])
Vieleck11'' = Translate(Rotate[Rotate[v021, 0.31416], 1.5708, Rotate[Aa03, 0.31416]], lP201 - Rotate[Aa03, 0.31416])
lP101: (Element[Element[lrPV, 1], 5] + Element[Element[lrPV, 2], 4]) / 2
v01a11 = Translate(v01a1, Element[Element[lrPV, 3], 4] - Element[Element[lrPV, 1], 5])
v011 = Translate(v01, Vector[u2 * (2 + d102 / au2)])
v01a12 = Rotate(v01a11, -1.2566370614359172, Element[Element[lrPV, 3], 4])
au11: d101 * f
v022' = Mirror(Rotate[v021, -1.25664, Element[Element[lrPV, 3], 3]], Element[Element[lrPV, 3], 3])
v022'' = Mirror(v022', Line[B, Aa01])
lP202 = Translate(lP201, Vector[u1 * (f / 2 + 1 / f)])
v0211 = Translate(v021, Vector[u2 * (2 + d102 / au2)])
lP300 = Translate(Aa03, Vector[u2 * (1 + d102 / au2)])
v01a21 = Translate(v01a2, lP300 - Element[Element[lrPV, 2], 3])
u5 = Vector(Element[l1, 5])
lP301: lP300 + u5
v01a22 = Rotate(v01a21, 1.2566370614359172, lP301)
v01a23 = Rotate(v01a22, 1.25664, Element[Element[lrPV, 4], 4])
v01a13 = Translate(v01a11, Vector[u1 * (1 + 1 / f)])
v0211' = Mirror(v0211, Element[Element[lrPV, 4], 3])
v01a1221 = Translate(v01a22, -lP301 + Element[Element[lrPV, 4], 3])
lP302: Element[Element[lrPV, 4], 3] + u1
v01a131 = Translate(v01a13, lP302 - Element[Element[lrPV, 4], 4])
v022''1 = Translate(v022'', lP302 - Element[Element[lrPV, 3], 3])
u4 = Vector(Element[l1, 4])
lP303: Element[Element[lrPV, 4], 3] + u4
lP304: lP302 + u4
u3 = Vector(Element[l1, 3])
lP305: lP304 + u3
v022''1' = Mirror(v022''1, Line[lP305, lP304])
lP306: lP303 + u5
v01a13' = Rotate(v01a13, 4.39822971502571, lP306)
lP307: lP303 + u3
v01a211 = Translate(v01a21, lP307 - lP300)
lP308: lP302 + u3
v01a23a = Translate(v01a23, lP305 - Element[Element[lrPV, 4], 4])
v02111 = Sequence(Rotate[v0211, -0.62832 * i, Element[Element[lrPV, 4], 3]], i, 1, 2)
v02111a = Element(v02111, 1)
v02111b = Element(v02111, 2)
lP309 = Element(Element[lrPV, 4], 3)
lP310 = Element(Element[lrPV, 5], 3)
v0212 = Translate(v0211, lP310 - lP309)
v0212a = Translate(v02111a, lP310 - lP309)
v011a = Mirror(v011, lP309)
lP311 = Element(Element[lrPV, 4], 4)
Vieleck11''a = Mirror(Vieleck11'', lP311)
lP312 = Element(Element[lrPV, 5], 4)
v01a24 = Translate(v01a23, lP312 - lP311)
v01a22a = Translate(v01a22, lP312 - lP311)
lP313: lP302 + u1 / f
v01a212 = Translate(v01a21, lP313 - lP301)
v0211a = Translate(v0211, lP305 - lP300)
v01a22b = Translate(v01a22, lP313 - lP301)
lP314: lP313 + u1
v01a22c = Rotate(v01a22b, 4.39822971502571, lP314)
lVV1: {v01a1, v01a11, v01a13, v01a2, v01a21, v01a211, v01a212, v01a22, v01a22a, v01a22b, v01a22c, v01a23, v01a23a, v01a24, v01a131, v01a13', v01a1221, v01a12, v011a, v011, v01}
lVV2: {Vieleck11'', Vieleck11''a, v02111a, v02111b, v0211a, v0212a, v022'', v022''1, v022''1'}
lVV1' = Mirror(lVV1, Line[B, Aa02])
lVV2' = Mirror(lVV2, Line[B, Aa02])
Liste1 = Sequence(Rotate[{lVV1, lVV1'}, π / 5 * i], i, 0, 9)
Liste2 = Sequence(Rotate[{lVV2, lVV2', v02, v021, v0211, v0211', v0212}, π / 5 * i], i, 0, 9)
VZ1, WA1, WB1, WC1, WD1, WE1, WF1 = Polygon(Element[lPV5, 1], Element[lPV5, 3], Element[lPV5, 5], Element[lPV5, 2], Element[lPV5, 4], Element[lPV5, 1])
lrZz = Sequence(Dilate[VZ1, 1 / f^(i), An], i, 0, n_{ext})
Liste8 = Sequence(Rotate[lrZz, π / 5 * i], i, 0, 9)
a = true
b = false
Name2: "" + (LaTeX[Autor]) + ""
p = Circle(An, Element[Element[lrPV, 5], 3])