Külgede arv

Külje pikkus

Ümbermõõt

S_n

Indeks n

Tähised:\
K_i - i-nda \;  hulknurga \;  külgede \;  arv;\
L_i - i-nda \;  hulknurga \;  külje \;  pikkus;\
Ü_i - i-nda \;  hulknurga \; ümbermõõt;\
S_i - i-nda \;  hulknurga \;  pindala.

\ovalbox{K_0=3; \;  \; L_0=1; \;  \;Ü_0=3;\
S_0=\frac{1}{2}·1·1\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}.}

\ovalbox{
K_1=4·K_0=12; \;  \;  \;L_1=\frac{1}{3}·L_0=\frac{1}{3}; \; \ \;  \;Ü_1=\frac{4}{3}·Ü_0=4;\
 S_1=S_0+\frac{K_0}{9}·S_0=\frac{\sqrt{3}}{4} +\frac{3}{9}·\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{3}.}

\ovalbox{
K_n=4·K_{n-1}; \;  \;  \;L_n=\frac{1}{3}·L_{n-1}; \; \;  \;Ü_n=\frac{4}{3}·Ü_{n-1};\
 S_n=S_{n-1}+\frac{K_{n-1}}{9^n}·S_0.}

Kochi lumehelbe ümbermõõt ja pindala

\ovalbox{\lim_{n \to \infty }S_n =\frac{8}{5}S_0=\frac{8}{5}·\frac{\sqrt{3}}{4}=\frac{2\sqrt{3}}{5}≈0,692820323... }

B2: 3
C2: 1
D2: 3
A2: 0
A3: 1
E2: sqrt(3) / 4
E3: E2 + (B2 * $E$2 / 9^(A3))
A4 = 2
A5 = 3
A6 = 4
A7 = 5
A8 = 6
A9 = 7
A10 = 8
A11 = 9
A12 = 10
A13 = 11
A14 = 12
A15 = 13
B3: (B2 * 4)
B4: (B3 * 4)
B5: (B4 * 4)
B6: (B5 * 4)
B7: (B6 * 4)
B8: (B7 * 4)
B9: (B8 * 4)
B10: (B9 * 4)
B11: (B10 * 4)
B12: (B11 * 4)
B13: (B12 * 4)
B14: (B13 * 4)
B15: (B14 * 4)
B16: (B15 * 4)
C3: C2 / 3
C4: C3 / 3
C5: C4 / 3
C6: C5 / 3
C7: C6 / 3
C8: C7 / 3
C9: C8 / 3
C10: C9 / 3
C11: C10 / 3
C12: C11 / 3
C13: C12 / 3
C14: C13 / 3
C15: C14 / 3
C16: C15 / 3
D3: (D2 * 4 / 3)
D4: (D3 * 4 / 3)
D5: (D4 * 4 / 3)
D6: (D5 * 4 / 3)
D7: (D6 * 4 / 3)
D8: (D7 * 4 / 3)
D9: (D8 * 4 / 3)
D10: (D9 * 4 / 3)
D11: (D10 * 4 / 3)
D12: (D11 * 4 / 3)
D13: (D12 * 4 / 3)
D14: (D13 * 4 / 3)
D15: (D14 * 4 / 3)
D16: (D15 * 4 / 3)
E4: E3 + (B3 * $E$2 / 9^(A4))
E5: E4 + (B4 * $E$2 / 9^(A5))
E6: E5 + (B5 * $E$2 / 9^(A6))
E7: E6 + (B6 * $E$2 / 9^(A7))
E8: E7 + (B7 * $E$2 / 9^(A8))
E9: E8 + (B8 * $E$2 / 9^(A9))
E10: E9 + (B9 * $E$2 / 9^(A10))
E11: E10 + (B10 * $E$2 / 9^(A11))
E12: E11 + (B11 * $E$2 / 9^(A12))
E13: E12 + (B12 * $E$2 / 9^(A13))
E14: E13 + (B13 * $E$2 / 9^(A14))
E15: E14 + (B14 * $E$2 / 9^(A15))
A16 = 14
A17 = 15
B17: (B16 * 4)
C17: C16 / 3
D17: (D16 * 4 / 3)
E16: E15 + (B15 * $E$2 / 9^(A16))
E17: E16 + (B16 * $E$2 / 9^(A17))
u = Vector((0, 0), (12 / 9, 0))