Kombinationen mit Koch-Kurven / Combinations with Koch curves

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

1-10

2-8

3

A = (-3.643699156823245, -0.4397567947890124)
r = 10 (slider from 0 to 15, step 0.1)
c = Circle(A, r)
C = Point(c)
D = Mirror(C, A)
F = (18, 6)
p3 = Kronep3(C, D)
p4 = Kronep4(D, C)
list1 = Sequence(Kronep3[Element[p4, i], Element[p4, i + 1]], i, 1, 64)
list2 = Sequence(Kronep4[Element[p3, i], Element[p3, i + 1]], i, 1, 64)
V01 = Polygon(Join[p7[D, C]])
V02 = Polygon(Join[p7[C, D]])
b = 5 (slider from 0 to 5, step 1)

Tool: Reihe
Input: b
Output: list1

b: {{(1, 2), (2, 3), (4, 5), (6, 7), (7, 8)}, {(11, 12), (13, 14), (17, 15), (14, 12), (19, 21)}}
list1 = Sequence(If[Mod[k, 5] ≟ 1, Element[Element[b, ceil(k / 5)], 1], If[Mod[k, 5] ≟ 2, Element[Element[b, ceil(k / 5)], 2], If[Mod[k, 5] ≟ 3, Element[Element[b, ceil(k / 5)], 3], If[Mod[k, 5] ≟ 4, Element[Element[b, ceil(k / 5)], 4], If[Mod[k, 5] ≟ 0, Element[Element[b, ceil(k / 5)], 5]]]]]], k, 1, 5 * Length[b])

Tool: Krone p3 (command Kronep3)
Input: A, B
Output: list1

u: Vector[A, B] / 3
v1: Rotate[u, 300*°] / 3
v2: Rotate[u, 60*°] / 3
A3: A + u + Rotate[u, 300*°]
A4: A + 2 * u
B1: A + u / 3
B2: B1 + v1
B3: A + 2 * u / 3
A2: A + u
C1: A2 + v1
C2: C1 + Rotate[v1, 300*°]
C3: A2 + 2 * v1
D1: A3 + v2
D2: D1 + Rotate[v2, 300*°]
D3: A3 + 2 * v2
E1: A4 + u / 3
E2: E1 + v1
E3: A4 + 2 * u / 3
list1: {A, A + u / 3, B1 + v1, A + 2 * u / 3, A + u, A2 + v1, C1 + Rotate[v1, 300*°], A2 + 2 * v1, A + u + Rotate[u, 300*°], A3 + v2, D1 + Rotate[v2, 300*°], A3 + 2 * v2, A + 2 * u, A4 + u / 3, E1 + v1, A4 + 2 * u / 3, B}

Tool: Krone p2 (command Kronep2)
Input: A, B
Output: list2

u: Vector[A, B] / 3
A3: A + u + Rotate[u, 300*°]
A4: A + 2 * u
A2: A + u
list2: {A, A + u, A + u + Rotate[u, 300*°], A + 2 * u, B}

Tool: Krone p4 (command Kronep4)
Input: A, C
Output: list3

B: (x(A) + (x(C) - x(A)) / 3, y(A) + (y(C) - y(A)) / 3)
u: Vector[C, A] / 9
v1: Rotate[u, 300*°] / 3
v2: Rotate[u, 60*°] / 3
A3: B + u + Rotate[u, 300*°]
A4: B + 2 * u
B1: B + u / 3
B2: B1 + v1
B3: B + 2 * u / 3
A2: B + u
C1: A2 + v1
C2: C1 + Rotate[v1, 300*°]
C3: A2 + 2 * v1
D1: A3 + v2
D2: D1 + Rotate[v2, 300*°]
D3: A3 + 2 * v2
E1: A4 + u / 3
E2: E1 + v1
E3: A4 + 2 * u / 3
b = Line(A, B)
a = Line(B, v1)
A' = Mirror(A, a)
E3' = Mirror(E3, a)
E1' = Mirror(E1, a)
A4' = Mirror(A4, a)
A2' = Mirror(A2, a)
B3' = Mirror(B3, a)
B1' = Mirror(B1, a)
E2' = Mirror(E2, a)
D3' = Mirror(D3, a)
C1' = Mirror(C1, a)
B2' = Mirror(B2, a)
D2' = Mirror(D2, a)
D1' = Mirror(D1, a)
C3' = Mirror(C3, a)
C2' = Mirror(C2, a)
A3' = Mirror(A3, a)
c = OrthogonalLine(A', b)
E3'_1 = Mirror(E3, c)
E1'_1 = Mirror(E1, c)
A4'_1 = Mirror(A4, c)
A2'_1 = Mirror(A2, c)
B3'_1 = Mirror(B3, c)
B1'_1 = Mirror(B1, c)
E2'_1 = Mirror(E2, c)
D3'_1 = Mirror(D3, c)
C1'_1 = Mirror(C1, c)
B2'_1 = Mirror(B2, c)
B1'' = Mirror(B1', c)
D2'_1 = Mirror(D2, c)
D1'_1 = Mirror(D1, c)
C3'_2 = Mirror(C3, c)
C2'_1 = Mirror(C2, c)
B2'' = Mirror(B2', c)
B3'' = Mirror(B3', c)
A3'_1 = Mirror(A3, c)
A2'' = Mirror(A2', c)
C2'' = Mirror(C2', c)
C1'' = Mirror(C1', c)
C3'' = Mirror(C3', c)
A3'' = Mirror(A3', c)
D1'' = Mirror(D1', c)
D3'' = Mirror(D3', c)
A4'' = Mirror(A4', c)
D2'' = Mirror(D2', c)
E1'' = Mirror(E1', c)
E2'' = Mirror(E2', c)
E3'' = Mirror(E3', c)
B' = Mirror(B, c)
list3: {A, A4 + 2 * u / 3, E1 + v1, A4 + u / 3, B + 2 * u, A3 + 2 * v2, D1 + Rotate[v2, 300*°], A3 + v2, B + u + Rotate[u, 300*°], A2 + 2 * v1, C1 + Rotate[v1, 300*°], A2 + v1, B + u, B + 2 * u / 3, B1 + v1, B + u / 3, (x(A) + (x(C) - x(A)) / 3, y(A) + (y(C) - y(A)) / 3), Mirror[B1, a], Mirror[B2, a], Mirror[B3, a], Mirror[A2, a], Mirror[C1, a], Mirror[C2, a], Mirror[C3, a], Mirror[A3, a], Mirror[D1, a], Mirror[D2, a], Mirror[D3, a], Mirror[A4, a], Mirror[E1, a], Mirror[E2, a], Mirror[E3, a], Mirror[A, a], Mirror[E3', c], Mirror[E2', c], Mirror[E1', c], Mirror[A4', c], Mirror[D3', c], Mirror[D2', c], Mirror[D1', c], Mirror[A3', c], Mirror[C3', c], Mirror[C2', c], Mirror[C1', c], Mirror[A2', c], Mirror[B3', c], Mirror[B2', c], Mirror[B1', c], Mirror[B, c], Mirror[B1, c], Mirror[B2, c], Mirror[B3, c], Mirror[A2, c], Mirror[C1, c], Mirror[C2, c], Mirror[C3, c], Mirror[A3, c], Mirror[D1, c], Mirror[D2, c], Mirror[D3, c], Mirror[A4, c], Mirror[E1, c], Mirror[E2, c], Mirror[E3, c], C}

Tool: Strecke p4 (command Streckep4)
Input: A, C
Output: list4

B: (x(A) + (x(C) - x(A)) / 3, y(A) + (y(C) - y(A)) / 3)
u: Vector[C, A] / 9
v1: Rotate[u, 300*°] / 3
v2: Rotate[u, 60*°] / 3
A3: B + u + Rotate[u, 300*°]
A4: B + 2 * u
B1: B + u / 3
B2: B1 + v1
B3: B + 2 * u / 3
A2: B + u
C1: A2 + v1
C2: C1 + Rotate[v1, 300*°]
C3: A2 + 2 * v1
D1: A3 + v2
D2: D1 + Rotate[v2, 300*°]
D3: A3 + 2 * v2
E1: A4 + u / 3
E2: E1 + v1
E3: A4 + 2 * u / 3
b = Line(A, B)
a = Line(B, v1)
A' = Mirror(A, a)
E3' = Mirror(E3, a)
E1' = Mirror(E1, a)
A4' = Mirror(A4, a)
A2' = Mirror(A2, a)
B3' = Mirror(B3, a)
B1' = Mirror(B1, a)
E2' = Mirror(E2, a)
D3' = Mirror(D3, a)
C1' = Mirror(C1, a)
B2' = Mirror(B2, a)
D2' = Mirror(D2, a)
D1' = Mirror(D1, a)
C3' = Mirror(C3, a)
C2' = Mirror(C2, a)
A3' = Mirror(A3, a)
c = OrthogonalLine(A', b)
E3'_1 = Mirror(E3, c)
E1'_1 = Mirror(E1, c)
A4'_1 = Mirror(A4, c)
A2'_1 = Mirror(A2, c)
B3'_1 = Mirror(B3, c)
B1'_1 = Mirror(B1, c)
E2'_1 = Mirror(E2, c)
D3'_1 = Mirror(D3, c)
C1'_1 = Mirror(C1, c)
B2'_1 = Mirror(B2, c)
B1'' = Mirror(B1', c)
D2'_1 = Mirror(D2, c)
D1'_1 = Mirror(D1, c)
C3'_2 = Mirror(C3, c)
C2'_1 = Mirror(C2, c)
B2'' = Mirror(B2', c)
B3'' = Mirror(B3', c)
A3'_1 = Mirror(A3, c)
A2'' = Mirror(A2', c)
C2'' = Mirror(C2', c)
C1'' = Mirror(C1', c)
C3'' = Mirror(C3', c)
A3'' = Mirror(A3', c)
D1'' = Mirror(D1', c)
D3'' = Mirror(D3', c)
A4'' = Mirror(A4', c)
D2'' = Mirror(D2', c)
E1'' = Mirror(E1', c)
E2'' = Mirror(E2', c)
E3'' = Mirror(E3', c)
B' = Mirror(B, c)
list3: {A, A4 + 2 * u / 3, E1 + v1, A4 + u / 3, B + 2 * u, A3 + 2 * v2, D1 + Rotate[v2, 300*°], A3 + v2, B + u + Rotate[u, 300*°], A2 + 2 * v1, C1 + Rotate[v1, 300*°], A2 + v1, B + u, B + 2 * u / 3, B1 + v1, B + u / 3, (x(A) + (x(C) - x(A)) / 3, y(A) + (y(C) - y(A)) / 3), Mirror[B1, a], Mirror[B2, a], Mirror[B3, a], Mirror[A2, a], Mirror[C1, a], Mirror[C2, a], Mirror[C3, a], Mirror[A3, a], Mirror[D1, a], Mirror[D2, a], Mirror[D3, a], Mirror[A4, a], Mirror[E1, a], Mirror[E2, a], Mirror[E3, a], Mirror[A, a], Mirror[E3', c], Mirror[E2', c], Mirror[E1', c], Mirror[A4', c], Mirror[D3', c], Mirror[D2', c], Mirror[D1', c], Mirror[A3', c], Mirror[C3', c], Mirror[C2', c], Mirror[C1', c], Mirror[A2', c], Mirror[B3', c], Mirror[B2', c], Mirror[B1', c], Mirror[B, c], Mirror[B1, c], Mirror[B2, c], Mirror[B3, c], Mirror[A2, c], Mirror[C1, c], Mirror[C2, c], Mirror[C3, c], Mirror[A3, c], Mirror[D1, c], Mirror[D2, c], Mirror[D3, c], Mirror[A4, c], Mirror[E1, c], Mirror[E2, c], Mirror[E3, c], C}
list4 = Sequence(Segment[Element[list3, i], Element[list3, i + 1]], i, 1, 64)

Tool: p7
Input: A, B
Output: p7

p4 = Kronep4(A, B)
p7 = Sequence(Kronep4[Element[p4, i], Element[p4, i + 1]], i, 1, 64)