Indeks n

Külje pikkus

Külgede arv

Ümbermõõt

S_n

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

Kochi lumehelbe ümbermõõt ja pindala

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_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 }

\ovalbox{\lim_{ n \to \infty }S_n=\frac{8}{5}S_0=\frac{8}{5}*\frac{\sqrt{3}}{4}=\frac{2\sqrt{3}}{5}\approx0,69220323...}

A2: 0
A3: 1
A4 = 2
A5 = 3
A6 = 4
A7 = 5
A8 = 6
A9 = 7
A10 = 8
A11 = 9
A12 = 10
A13 = 11
A14 = 12
A15 = 13
A16 = 14
A17 = 15
B2: 3
B3: (B2 * 4)
C2: 1
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)
B17: (B16 * 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
C17: C16 / 3
D2: 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)
D17: (D16 * 4 / 3)
E2: sqrt(3) / 4
E3: E2 + (B2 * $E$2 / 9^(A3))
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))
E16: E15 + (B15 * $E$2 / 9^(A16))
E17: E16 + (B16 * $E$2 / 9^(A17))
A = Point(yAxis)
B = (7.794228634059947, 4.5)
C = (7.794228634059947, 13.5)
t1, c, a, b = Polygon(A, B, C)
D = (8.660254037844386, 9)
u = Vector(A, D)
A' = Translate(A, u)
B' = Translate(B, u)
C' = Translate(C, u)
E = (11.258330249197702, 7.5)
F = (13.856406460551018, 6)
G = (11.258330249197702, 4.5)
H = (16.454482671904334, 7.5)
I = (19.05255888325765, 9)
J = (16.454482671904334, 10.5)
K = (13.856406460551018, 12)
L = (11.258330249197702, 13.5)
M = (11.258330249197702, 10.5)
hulknurk1, a', e, g, f, b', h, i, j, c', k, l, m = Polygon(A', E, G, F, B', H, I, J, C', K, L, M)
t2, n, p, q = Polygon(A', B', C')
N = (22.516660498395403, 9)
v = Vector(A', N)
A'' = Translate(A', v)
E' = Translate(E, v)
G' = Translate(G, v)
F' = Translate(F, v)
B'' = Translate(B', v)
H' = Translate(H, v)
I' = Translate(I, v)
J' = Translate(J, v)
C'' = Translate(C', v)
K' = Translate(K, v)
L' = Translate(L, v)
M' = Translate(M, v)
hulknurk1', a'', e', g', f', b'', h', i', j', c'', k', l', m' = Polygon(A'', E', G', F', B'', H', I', J', C'', K', L', M')
O = (23.38268590217984, 8.5)
P = (24.24871130596428, 8)
Q = (25.11473670974872, 6.5)
R = (25.11473670974872, 5.5)
S = (25.980762113533157, 5)
T = (26.846787517317598, 5.5)
U = (28.578838324886473, 5.5)
V = (29.444863728670914, 5)
W = (30.31088913245535, 5.5)
Z = (30.31088913245535, 6.5)
A_1 = (31.17691453623979, 8)
B_1 = (32.042939940024226, 8.5)
C_1 = (32.042939940024226, 9.5)
D_1 = (31.17691453623979, 10)
E_1 = (30.31088913245535, 11.5)
F_1 = (30.31088913245535, 12.5)
G_1 = (29.444863728670914, 13)
H_1 = (28.578838324886473, 12.5)
I_1 = (26.846787517317598, 12.5)
J_1 = (25.980762113533157, 13)
K_1 = (25.11473670974872, 12.5)
L_1 = (25.11473670974872, 11.5)
M_1 = (24.24871130596428, 10)
N_1 = (23.38268590217984, 9.5)
O_1 = (23.38268590217984, 10.5)
P_1 = (23.38268590217984, 7.5)
Q_1 = (24.24871130596428, 6)
R_1 = (26.846787517317598, 4.5)
S_1 = (28.578838324886473, 4.5)
T_1 = (31.17691453623979, 6)
U_1 = (32.042939940024226, 7.5)
V_1 = (32.042939940024226, 10.5)
W_1 = (31.17691453623979, 12)
Z_1 = (28.578838324886473, 13.5)
A_2 = (26.846787517317598, 13.5)
B_2 = (24.24871130596428, 12)
hulknurk2, n_1, o, p_1, p_2, r, q_1, s, r_1, t, s_1, d, t_1, f_1, u_1, g_1, v_1, h_1, w, i_1, z_1, j_1, a_1, k_1, b_1, l_1, c_1, m_1, d_1, e_1, f_2, w_1, g_2, h_2, i_2, j_2, k_2, l_2, m_2, a_2, n_2, q_2, r_2, b_2, s_2, t_2, c_2, o_1, d_2 = Polygon(N, O, P_1, P, E', Q, Q_1, R, G', S, R_1, T, F', U, S_1, V, B'', W, T_1, Z, H', A_1, U_1, B_1, I', C_1, V_1, D_1, J', E_1, W_1, F_1, C'', G_1, Z_1, H_1, K', I_1, A_2, J_1, L', K_1, B_2, L_1, M', M_1, O_1, N_1)
A''_{1} = Translate(A', v)
E'_{1} = Translate(E, v)
G'_{1} = Translate(G, v)
F'_{1} = Translate(F, v)
B''_{1} = Translate(B', v)
H'_{1} = Translate(H, v)
I'_{1} = Translate(I, v)
J'_{1} = Translate(J, v)
C''_{1} = Translate(C', v)
K'_{1} = Translate(K, v)
L'_{1} = Translate(L, v)
M'_{1} = Translate(M, v)
hulknurk1'_{1}, a''_{1}, e'_{1}, g'_{1}, f'_{1}, b''_{1}, h'_{1}, i'_{1}, j'_{1}, c''_{1}, k'_{1}, l'_{1}, m'_{1} = Polygon(A''_{1}, E'_{1}, G'_{1}, F'_{1}, B''_{1}, H'_{1}, I'_{1}, J'_{1}, C''_{1}, K'_{1}, L'_{1}, M'_{1})