spirale
Author: une élève de seconde

agrandir le dessin pour  voir la spirale.

En effet la spirale 
a une longueur finie. 
La preuve : 

 10 x  \frac{ sin(22,5°) }{1 - cos(22,5°) }

On démontre au lycée
que la longueur totale  
est environ 50,3 cm

Faux, la spirale 
a une longueur finie.

Quelle est votre conjecture concernant la longueur de la spirale ?

Valeur exacte

la longueur est infinie

la longueur est finie

A_1 = (-1.6, 2)
B_1 = (18, 2)
a = Line(A_1, B_1)
C_1 = (8, 9)
b = OrthogonalLine(C_1, a)
O = Intersect(a, b)
c, d = AngularBisector(a, b)
e, f = AngularBisector(a, d)
A = (18, 2)
g, h = AngularBisector(a, c)
i = OrthogonalLine(A, a)
j = OrthogonalLine(A, g)
A1 = Intersect(g, j)
k = OrthogonalLine(A1, c)
D_1 = Intersect(c, k)
l = OrthogonalLine(D_1, e)
E = Intersect(e, l)
m = OrthogonalLine(E, b)
F = Intersect(b, m)
n = OrthogonalLine(F, h)
G = Intersect(h, n)
p = OrthogonalLine(G, d)
H = Intersect(d, p)
q = OrthogonalLine(H, f)
I = Intersect(f, q)
r = OrthogonalLine(I, a)
J = Intersect(a, r)
s = OrthogonalLine(J, g)
K = Intersect(g, s)
t = OrthogonalLine(K, c)
L = Intersect(c, t)
a1 = OrthogonalLine(L, e)
M = Intersect(e, a1)
b1 = OrthogonalLine(M, b)
N = Intersect(b, b1)
c1 = OrthogonalLine(N, h)
P = Intersect(h, c1)
d1 = OrthogonalLine(P, d)
Q = Intersect(d, d1)
e1 = OrthogonalLine(Q, f)
R = Intersect(f, e1)
f1 = OrthogonalLine(R, a)
S = Intersect(a, f1)
g1 = OrthogonalLine(S, g)
T = Intersect(g, g1)
h1 = OrthogonalLine(T, c)
U = Intersect(c, h1)
i1 = OrthogonalLine(U, e)
V = Intersect(e, i1)
j1 = OrthogonalLine(V, b)
W = Intersect(b, j1)
k1 = OrthogonalLine(W, h)
Z = Intersect(h, k1)
l1 = OrthogonalLine(Z, d)
B1 = Intersect(d, l1)
m1 = OrthogonalLine(B1, f)
C1 = Intersect(f, m1)
n1 = OrthogonalLine(C1, a)
D1 = Intersect(a, n1)
p1 = OrthogonalLine(D1, g)
E1 = Intersect(g, p1)
B = (8, 12)
J1 = Intersect(d, f)
C = (-2, 2)
d2 = OrthogonalLine(B, h)
k2 = OrthogonalLine(B, b)
l2 = OrthogonalLine(C, a)
D = (8, -8)
m2 = OrthogonalLine(D, b)
n2 = OrthogonalLine(C, g)
s2 = OrthogonalLine(D, h)
e4 = Segment(D1, C1)
f4 = Segment(C1, B1)
g4 = Segment(B1, Z)
h4 = Segment(Z, W)
i4 = Segment(W, V)
j4 = Segment(V, U)
k4 = Segment(U, T)
l4 = Segment(T, S)
m4 = Segment(S, R)
n4 = Segment(R, Q)
p4 = Segment(Q, P)
q4 = Segment(P, N)
r4 = Segment(N, M)
s4 = Segment(M, L)
t4 = Segment(L, K)
a5 = Segment(K, J)
b5 = Segment(J, I)
c5 = Segment(H, I)
d5 = Segment(H, G)
e5 = Segment(G, F)
f5 = Segment(F, E)
g5 = Segment(E, D_1)
h5 = Segment(D_1, A1)
E2 = Intersect(a, i)
i5 = Segment(A1, A)
G2 = (18, 12)
F2 = (-2, 12)
j5 = Segment(F2, G2)
I2 = (18, -8)
k5 = Segment(G2, I2)
H2 = (-2, -8)
l5 = Segment(I2, H2)
m5 = Segment(H2, F2)
n5 = Segment(H2, O)
J2 = Point(m5)
p5 = Segment(O, J2)
K2 = Intersect(f, l2)
M2 = Intersect(h, k2)
N2 = Intersect(k2, e)
O2 = Intersect(i, k5)
P2 = Intersect(g, i)
R2 = Intersect(e, l5)
T2 = Intersect(h, m2)
U2 = Intersect(i, k5)
V2 = Intersect(f, i)
q5 = Segment(K2, O)
r5 = Segment(F2, O)
s5 = Segment(M2, O)
t5 = Segment(B, O)
a6 = Segment(N2, O)
b6 = Segment(C, O)
c6 = Segment(O, A)
d6 = Segment(O, P2)
e6 = Segment(G2, O)
f6 = Segment(R2, O)
g6 = Segment(D, O)
h6 = Segment(T2, O)
i6 = Segment(I2, O)
j6 = Segment(V2, O)
W2 = Intersect(e4, b6)
n3 = OrthogonalLine(W2, p5)
Z2 = Intersect(p5, n3)
p3 = OrthogonalLine(Z2, n5)
A3 = Intersect(n5, p3)
q3 = OrthogonalLine(A3, f6)
B3 = Intersect(f6, q3)
r3 = OrthogonalLine(B3, g6)
C3 = Intersect(g6, r3)
s3 = OrthogonalLine(C3, h6)
D3 = Intersect(h6, s3)
t3 = OrthogonalLine(D3, i6)
E3 = Intersect(i6, t3)
a4 = OrthogonalLine(E3, j6)
F3 = Intersect(j6, a4)
b4 = OrthogonalLine(F3, c6)
G3 = Intersect(c6, b4)
c4 = OrthogonalLine(G3, d6)
H3 = Intersect(d6, c4)
d4 = OrthogonalLine(H3, e6)
I3 = Intersect(e6, d4)
k6 = OrthogonalLine(I3, a6)
J3 = Intersect(a6, k6)
l6 = OrthogonalLine(J3, t5)
K3 = Intersect(t5, l6)
m6 = OrthogonalLine(K3, s5)
L3 = Intersect(e4, b6)
n6 = OrthogonalLine(L3, p5)
M3 = Intersect(p5, n6)
p6 = OrthogonalLine(M3, n5)
N3 = Intersect(n5, p6)
q6 = OrthogonalLine(N3, f6)
O3 = Intersect(f6, q6)
r6 = OrthogonalLine(O3, g6)
P3 = Intersect(g6, r6)
s6 = OrthogonalLine(P3, h6)
Q3 = Intersect(h6, s6)
t6 = OrthogonalLine(Q3, i6)
R3 = Intersect(i6, t6)
a7 = OrthogonalLine(R3, j6)
S3 = Intersect(j6, a7)
b7 = OrthogonalLine(S3, c6)
T3 = Intersect(c6, b7)
c7 = OrthogonalLine(T3, d6)
U3 = Intersect(d6, c7)
d7 = OrthogonalLine(U3, e6)
V3 = Intersect(e6, d7)
e7 = OrthogonalLine(V3, a6)
W3 = Intersect(a6, e7)
f7 = OrthogonalLine(W3, t5)
Z3 = Intersect(t5, f7)
g7 = OrthogonalLine(Z3, s5)
A4 = Intersect(s5, g7)
h7 = OrthogonalLine(A4, r5)
B4 = Intersect(r5, h7)
i7 = OrthogonalLine(B4, q5)
C4 = Intersect(q5, i7)
j7 = OrthogonalLine(C4, b6)
D4 = Intersect(b6, j7)
k7 = OrthogonalLine(D4, p5)
E4 = Intersect(p5, k7)
l7 = OrthogonalLine(E4, n5)
F4 = Intersect(n5, l7)
m7 = OrthogonalLine(F4, f6)
G4 = Intersect(f6, m7)
n7 = OrthogonalLine(G4, g6)
H4 = Intersect(g6, n7)
p7 = OrthogonalLine(H4, h6)
I4 = Intersect(h6, p7)
q7 = OrthogonalLine(I4, i6)
J4 = Intersect(i6, q7)
r7 = OrthogonalLine(J4, j6)
K4 = Intersect(j6, r7)
s7 = OrthogonalLine(K4, c6)
L4 = Intersect(c6, s7)
t7 = OrthogonalLine(L4, d6)
M4 = Intersect(d6, t7)
a8 = OrthogonalLine(M4, e6)
b8 = OrthogonalLine(M4, e6)
N4 = Intersect(e6, b8)
c8 = OrthogonalLine(N4, a6)
O4 = Intersect(a6, c8)
d8 = OrthogonalLine(O4, t5)
P4 = Intersect(t5, d8)
e8 = OrthogonalLine(P4, s5)
Q4 = Intersect(s5, e8)
f8 = OrthogonalLine(Q4, r5)
R4 = Intersect(r5, f8)
g8 = OrthogonalLine(R4, q5)
S4 = Intersect(q5, g8)
h8 = OrthogonalLine(S4, b6)
T4 = Intersect(b6, h8)
i8 = OrthogonalLine(T4, p5)
U4 = Intersect(p5, i8)
j8 = OrthogonalLine(U4, n5)
V4 = Intersect(n5, j8)
k8 = OrthogonalLine(V4, f6)
W4 = Intersect(f6, k8)
l8 = OrthogonalLine(W4, g6)
Z4 = Intersect(g6, l8)
m8 = OrthogonalLine(Z4, h6)
A5 = Intersect(h6, m8)
n8 = OrthogonalLine(A5, i6)
B5 = Intersect(i6, n8)
p8 = OrthogonalLine(B5, j6)
C5 = Intersect(j6, p8)
q8 = OrthogonalLine(C5, c6)
D5 = Intersect(c6, q8)
r8 = OrthogonalLine(D5, d6)
E5 = Intersect(d6, r8)
s8 = OrthogonalLine(E5, e6)
F5 = Intersect(e6, s8)
t8 = OrthogonalLine(F5, a6)
G5 = Intersect(a6, t8)
a9 = OrthogonalLine(G5, t5)
H5 = Intersect(t5, a9)
b9 = OrthogonalLine(H5, s5)
I5 = Intersect(s5, b9)
c9 = OrthogonalLine(I5, r5)
J5 = Intersect(r5, c9)
d9 = OrthogonalLine(J5, q5)
K5 = Intersect(q5, d9)
e9 = OrthogonalLine(K5, b6)
L5 = Intersect(b6, e9)
f9 = OrthogonalLine(L5, p5)
M5 = Intersect(p5, f9)
g9 = OrthogonalLine(M5, n5)
N5 = Intersect(n5, g9)
h9 = OrthogonalLine(N5, f6)
O5 = Intersect(f6, h9)
i9 = OrthogonalLine(O5, g6)
P5 = Intersect(g6, i9)
j9 = OrthogonalLine(P5, h6)
Q5 = Intersect(h6, j9)
k9 = OrthogonalLine(Q5, i6)
R5 = Intersect(i6, k9)
l9 = OrthogonalLine(R5, j6)
S5 = Intersect(j6, l9)
m9 = OrthogonalLine(S5, c6)
T5 = Intersect(c6, m9)
n9 = OrthogonalLine(T5, d6)
U5 = Intersect(d6, n9)
p9 = OrthogonalLine(U5, e6)
V5 = Intersect(e6, p9)
q9 = OrthogonalLine(V5, a6)
W5 = Intersect(a6, q9)
r9 = OrthogonalLine(W5, t5)
Z5 = Intersect(t5, r9)
s9 = OrthogonalLine(Z5, s5)
A6 = Intersect(s5, s9)
t9 = OrthogonalLine(A6, r5)
B6 = Intersect(r5, t9)
a10 = OrthogonalLine(B6, q5)
C6 = Intersect(q5, a10)
b10 = OrthogonalLine(C6, b6)
D6 = Intersect(b6, b10)
c10 = OrthogonalLine(D6, p5)
E6 = Intersect(p5, c10)
d10 = OrthogonalLine(E6, n5)
a_1 = Segment(L3, M3)
b_1 = Segment(M3, N3)
c_1 = Segment(N3, O3)
d_1 = Segment(O3, P3)
e_1 = Segment(P3, Q3)
f_1 = Segment(Q3, R3)
g_1 = Segment(R3, S3)
h_1 = Segment(S3, T3)
i_1 = Segment(T3, U3)
j_1 = Segment(U3, V3)
k_1 = Segment(V3, W3)
l_1 = Segment(W3, Z3)
m_1 = Segment(Z3, A4)
n_1 = Segment(A4, B4)
p_1 = Segment(B4, C4)
q_1 = Segment(C4, D4)
r_1 = Segment(D4, E4)
s_1 = Segment(E4, F4)
t_1 = Segment(F4, G4)
a_2 = Segment(G4, H4)
b_2 = Segment(H4, I4)
c_2 = Segment(I4, J4)
d_2 = Segment(J4, K4)
e_2 = Segment(K4, L4)
f_2 = Segment(L4, M4)
g_2 = Segment(M4, N4)
h_2 = Segment(N4, O4)
i_2 = Segment(P4, O4)
j_2 = Segment(P4, Q4)
k_2 = Segment(Q4, R4)
l_2 = Segment(R4, S4)
m_2 = Segment(S4, T4)
n_2 = Segment(T4, U4)
p_2 = Segment(U4, V4)
q_2 = Segment(V4, W4)
r_2 = Segment(W4, Z4)
s_2 = Segment(Z4, A5)
t_2 = Segment(A5, B5)
a_3 = Segment(B5, C5)
b_3 = Segment(C5, D5)
c_3 = Segment(D5, E5)
d_3 = Segment(E5, F5)
e_3 = Segment(F5, G5)
f_3 = Segment(G5, H5)
g_3 = Segment(H5, I5)
h_3 = Segment(I5, J5)
i_3 = Segment(J5, K5)
j_3 = Segment(K5, L5)
k_3 = Segment(L5, M5)
l_3 = Segment(M5, N5)
m_3 = Segment(N5, O5)
n_3 = Segment(O5, P5)
p_3 = Segment(P5, Q5)
q_3 = Segment(Q5, R5)
r_3 = Segment(R5, S5)
s_3 = Segment(S5, T5)
t_3 = Segment(T5, U5)
a_4 = Segment(U5, V5)
b_4 = Segment(V5, W5)
c_4 = Segment(W5, Z5)
d_4 = Segment(Z5, A6)
e_4 = Segment(A6, B6)
f_4 = Segment(B6, C6)
g_4 = Segment(C6, D6)
h_4 = Segment(D6, E6)
L20: i5 + h5 + g5 + f5 + e5 + d5 + c5 + b5 + a5 + i4 + j4 + k4 + l4 + m4 + n4 + p4 + q4 + r4 + s4 + t4
L40: L20 + h4 + g4 + f4 + e4 + a_1 + b_1 + c_1 + d_1 + e_1 + f_1 + g_1 + h_1 + i_1 + j_1 + k_1 + l_1 + m_1 + n_1 + p_1 + q_1
texte2: "longueur de la spirale 
pour les 40 premiers segments
L40 =" + L40
L60: L40 + r_1 + s_1 + t_1 + a_2 + b_2 + c_2 + d_2 + e_2 + f_2 + g_2 + h_2 + i_2 + j_2 + k_2 + l_2 + m_2 + n_2 + p_2 + q_2 + r_2
texte1: "longueur de la spirale 
pour les 20 premiers segments 
(bleus)   L20 =" + L20
texte3: "longueur de la spirale 
pour les 60 premiers segments
L60 =" + L60
v = false
o = false
u = false