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Class 10 : Geometry  >> Missing Square !!! a Mathmagic !

Explanation

Benny Varghese
PGT Mathematics
JNV Pathanamthitta
Kerala-686511

Plz click the Strat Animation Button and 
observe what happens !!! Can you find the 
reason for that missing square???
In case that you are at your wit's end
 plz click the “explanation ”button

A_2 = (-1, 2)
B = (-1, -3)
C = (12, -3)
D = (12, 2)
q1, a, b_2, c, d = Polygon(A_2, B, C, D)
E_2 = (16, 1)
F = (21, 1)
G_2 = (21, 3)
t1, g, e, f = Polygon(E_2, F, G_2)
H = (11, 3)
I = (19, 3)
J = (19, 6)
t2, j, h, i = Polygon(H, I, J)
W = (14, -2)
Z = (14, -4)
N = (19, -4)
B_1 = (19, -3)
C_1 = (16, -3)
D_1 = (16, -2)
poly3, w, z_1, a_1, b_1, c_1, d_1 = Polygon(W, Z, N, B_1, C_1, D_1)
Q = (15, 0)
R_1 = (20, 0)
S_1 = (20, -2)
T_1 = (17, -2)
U = (17, -1)
V_1 = (15, -1)
poly2, q, r, s, t, u, v_2 = Polygon(Q, R_1, S_1, T_1, U, V_1)
k = 0 (slider from 0 to 17, step 0.001)
H' = Dilate(H, If[1 < k < 2, 2 - k, 0], B)
l = Vector(H, H')
H'_1 = Translate(H, l)
V = Translate(I, l)
J' = Translate(J, l)
t2', j', h'_1, i' = Polygon(H'_1, V, J')
K = (7, 0)
H'_2 = Dilate(H, If[1 < k < 2, 2 - k, 0], B)
E' = Dilate(E_2, If[2 < k < 3, 3 - k, 0], K)
m = Vector(E_2, E')
E'_1 = Translate(E_2, m)
E = Translate(F, m)
G' = Translate(G_2, m)
t1', g', e'_1, f' = Polygon(E'_1, E, G')
Q' = Dilate(Q, If[3 < k < 4, 4 - k, 0], K)
n = Vector(Q, Q')
Q'_1 = Translate(Q, n)
R' = Translate(R_1, n)
T = Translate(S_1, n)
T' = Translate(T_1, n)
U' = Translate(U, n)
V' = Translate(V_1, n)
poly2', q'_1, r', s', t', u', v' = Polygon(Q'_1, R', T, T', U', V')
L = Point(b_2)
Z' = Dilate(Z, If[4 < k < 5, 5 - k, 0], L)
p = Vector(Z, Z')
W' = Translate(W, p)
Z'_1 = Translate(Z, p)
M_2 = Translate(N, p)
N_1 = Translate(B_1, p)
O = Translate(C_1, p)
P_1 = Translate(D_1, p)
poly3', w', z'_1, m_1, n_1, o, p_1 = Polygon(W', Z'_1, M_2, N_1, O, P_1)
R = Dilate(M_2, If[6 < k < 7, 7 - k, 0], N)
u_1 = Vector(M_2, R)
W'' = Translate(W', u_1)
A_1 = Translate(Z'_1, u_1)
M' = Translate(M_2, u_1)
E_1 = Translate(N_1, u_1)
O' = Translate(O, u_1)
P' = Translate(P_1, u_1)
poly3'', w'', a_2, m', e_1, o', p' = Polygon(W'', A_1, M', E_1, O', P')
S = Dilate(T, If[7 < k < 8, 8 - k, 0], S_1)
v_1 = Vector(T, S)
F_1 = Translate(Q'_1, v_1)
R'' = Translate(R', v_1)
T'_1 = Translate(T, v_1)
T'' = Translate(T', v_1)
U'' = Translate(U', v_1)
V'' = Translate(V', v_1)
poly2'', f_1, r'', t'_1, t'', u'', v'' = Polygon(F_1, R'', T'_1, T'', U'', V'')
E'_2 = Dilate(E, If[8 < k < 9, 9 - k, 0], F)
w_1 = Vector(E, E'_2)
G = Translate(E'_1, w_1)
E'_3 = Translate(E, w_1)
G'' = Translate(G', w_1)
t1'', g'', g_1, e'_3 = Polygon(G, E'_3, G'')
A = Dilate(V, If[9 < k < 10, 10 - k, 0], I)
v = Vector(V, A)
H_1 = Translate(H'_1, v)
V'_1 = Translate(V, v)
J'' = Translate(J', v)
t2'', j'', h_1, v'_1 = Polygon(H_1, V'_1, J'')
P_2 = Intersect(a, b_2)
G'_1 = Dilate(G, If[11 < k < 12, 12 - k, 0], P_2)
i_1 = Vector(G, G'_1)
G_1 = Translate(E_2, i_1)
F' = Translate(F, i_1)
I_1 = Translate(G_2, i_1)
t1'_1, i_2, g_2, f'_1 = Polygon(G_1, F', I_1)
P = (4, -1)
H'_3 = Dilate(H, If[12 < k < 13, 13 - k, 0], P)
j_1 = Vector(H, H'_3)
H'_4 = Translate(H, j_1)
I' = Translate(I, j_1)
J'_1 = Translate(J, j_1)
t2'_1, j'_1, h'_4, i'_1 = Polygon(H'_4, I', J'_1)
W'_1 = Dilate(W, If[13 < k < 14, 14 - k, 0], P)
k_1 = Vector(W, W'_1)
W'_2 = Translate(W, k_1)
Z'_2 = Translate(Z, k_1)
N' = Translate(N, k_1)
J_1 = Translate(B_1, k_1)
K_1 = Translate(C_1, k_1)
L_1 = Translate(D_1, k_1)
poly3'_1, w'_2, z'_2, n', j_2, k_2, l_1 = Polygon(W'_2, Z'_2, N', J_1, K_1, L_1)
M = Intersect(b_2, c)
S' = Dilate(S, If[14 < k < 15, 15 - k, 0], M)
q_1 = Vector(S, S')
Q'_2 = Translate(Q, q_1)
M_1 = Translate(R_1, q_1)
O_1 = Translate(S_1, q_1)
Q_1 = Translate(T_1, q_1)
U'_1 = Translate(U, q_1)
U_1 = Translate(V_1, q_1)
poly2'_1, q'_2, m_2, o_1, q_2, u'_1, u_2 = Polygon(Q'_2, M_1, O_1, Q_1, U'_1, U_1)
B_2 = (-2, -4)
C_2 = (20, -4)
b = false
W_1 = (-2, -4)
Z_1 = (19.74, -4)