平面四重对称镶嵌艺术与数学
几何画板及绘图内容加载中,请稍等……
本范例展示了基于正方形基础的四重对称平面镶嵌艺术。它涉及对称性、几何作图等核心数学概念。通过使用Polygon、Rotate、Segment、Vector、Translate等GeoGebra命令,可以构造出具有四重对称性的镶嵌图案。适合初中及以上学生学习几何变换与对称性。
GeoGebra 命令及参数
Polygon(A, B, 4) → poly1, f, g, h, i, C, DPoint(f) → ERotate(G, (-(90°)), B) → G'Rotate(E, (-(90°)), B) → E'Point(f) → FRotate(F, (-(90°)), B) → F'Polygon(D, A, E, G, F, B, F', G', E', C) → poly2, d, c, a, e, g_1, f_1, b, f', g', e'Rotate(D, (-(90°)), B) → D'Rotate(A, (-(90°)), B) → A'Rotate(E, (-(90°)), B) → E'_1Rotate(G, (-(90°)), B) → G'_1Rotate(F, (-(90°)), B) → F'_1Rotate(B, (-(90°)), B) → B'Rotate(F', (-(90°)), B) → F''Rotate(G', (-(90°)), B) → G''Rotate(E', (-(90°)), B) → E''Rotate(C, (-(90°)), B) → C'Polygon(D', A', E'_1, G'_1, F'_1, B', F'', G'', E'', C') → poly2', d', c', a', e'_1, g'_1, f'_1, b', f'', g'', e''Rotate(D', (-(90°)), B) → D''Rotate(A', (-(90°)), B) → A''Rotate(E'_1, (-(90°)), B) → HRotate(G'_1, (-(90°)), B) → IRotate(F'_1, (-(90°)), B) → JRotate(B', (-(90°)), B) → B''Rotate(F'', (-(90°)), B) → F'''Rotate(G'', (-(90°)), B) → G'''Rotate(E'', (-(90°)), B) → E'''Rotate(C', (-(90°)), B) → C''Polygon(D'', A'', H, I, J, B'', F''', G''', E''', C'') → poly2'', d'', c'', a'', h_1, i_1, j, b'', f''', g''', e'''Rotate(D'', (-(90°)), B) → D'''Rotate(A'', (-(90°)), B) → A'''Rotate(H, (-(90°)), B) → H'Rotate(I, (-(90°)), B) → I'Rotate(J, (-(90°)), B) → J'Rotate(B'', (-(90°)), B) → B'''Rotate(F''', (-(90°)), B) → KRotate(G''', (-(90°)), B) → LRotate(E''', (-(90°)), B) → MRotate(C'', (-(90°)), B) → C'''Polygon(D''', A''', H', I', J', B''', K, L, M, C''') → poly2''', d''', c''', a''', h', i', j', b''', k, l, mRotate(D, (-(90°)), D'') → D'_1Rotate(A, (-(90°)), D'') → A'_1Rotate(E, (-(90°)), D'') → E'_2Rotate(G, (-(90°)), D'') → G'_2Rotate(F, (-(90°)), D'') → F'_2Rotate(B, (-(90°)), D'') → B'_1Rotate(F', (-(90°)), D'') → F''_1Rotate(G', (-(90°)), D'') → G''_1Rotate(E', (-(90°)), D'') → E''_1Rotate(C, (-(90°)), D'') → C'_1Polygon(D'_1, A'_1, E'_2, G'_2, F'_2, B'_1, F''_1, G''_1, E''_1, C'_1) → poly2'_1, d'_1, c'_1, a'_1, e'_2, g'_2, f'_2, b'_1, f''_1, g''_1, e''_1Rotate(D', (-(90°)), D'') → D''_1Rotate(A', (-(90°)), D'') → A''_1Rotate(E'_1, (-(90°)), D'') → NRotate(G'_1, (-(90°)), D'') → ORotate(F'_1, (-(90°)), D'') → PRotate(B', (-(90°)), D'') → B''_1Rotate(F'', (-(90°)), D'') → F'''_1Rotate(G'', (-(90°)), D'') → G'''_1Rotate(E'', (-(90°)), D'') → E'''_1Rotate(C', (-(90°)), D'') → C''_1Polygon(D''_1, A''_1, N, O, P, B''_1, F'''_1, G'''_1, E'''_1, C''_1) → poly2''_1, d''_1, c''_1, a''_1, n, o, p, b''_1, f'''_1, g'''_1, e'''_1Rotate(D'', (-(90°)), D'') → D'''_1Rotate(A'', (-(90°)), D'') → A'''_1Rotate(H, (-(90°)), D'') → H'_1Rotate(I, (-(90°)), D'') → I'_1Rotate(J, (-(90°)), D'') → J'_1Rotate(B'', (-(90°)), D'') → B'''_1Rotate(F''', (-(90°)), D'') → QRotate(G''', (-(90°)), D'') → RRotate(E''', (-(90°)), D'') → SRotate(C'', (-(90°)), D'') → C'''_1Polygon(D'''_1, A'''_1, H'_1, I'_1, J'_1, B'''_1, Q, R, S, C'''_1) → poly2'''_1, d'''_1, c'''_1, a'''_1, h'_1, i'_1, j'_1, b'''_1, q, r, sRotate(D''', (-(90°)), D'') → TRotate(A''', (-(90°)), D'') → URotate(H', (-(90°)), D'') → H''Rotate(I', (-(90°)), D'') → I''Rotate(J', (-(90°)), D'') → J''Rotate(B''', (-(90°)), D'') → VRotate(K, (-(90°)), D'') → K'Rotate(L, (-(90°)), D'') → L'Rotate(M, (-(90°)), D'') → M'Rotate(C''', (-(90°)), D'') → WPolygon(T, U, H'', I'', J'', V, K', L', M', W) → poly3, t, w, u, h'', i'', j'', v, k', l', m'Rotate(D'_1, (-(90°)), D'') → ZRotate(A'_1, (-(90°)), D'') → A_1Rotate(E'_2, (-(90°)), D'') → B_1Rotate(G'_2, (-(90°)), D'') → C_1Rotate(F'_2, (-(90°)), D'') → D_1Rotate(B'_1, (-(90°)), D'') → E_1Rotate(F''_1, (-(90°)), D'') → F_1Rotate(G''_1, (-(90°)), D'') → G_1Rotate(E''_1, (-(90°)), D'') → H_1Rotate(C'_1, (-(90°)), D'') → I_1Polygon(Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly4, z_1, i_2, a_1, b_1, c_1, d_1, e_1, f_2, g_2, h_2Rotate(D''_1, (-(90°)), D'') → J_1Rotate(A''_1, (-(90°)), D'') → K_1Rotate(N, (-(90°)), D'') → N'Rotate(O, (-(90°)), D'') → O'Rotate(P, (-(90°)), D'') → P'Rotate(B''_1, (-(90°)), D'') → L_1Rotate(F'''_1, (-(90°)), D'') → M_1Rotate(G'''_1, (-(90°)), D'') → N_1Rotate(E'''_1, (-(90°)), D'') → O_1Rotate(C''_1, (-(90°)), D'') → P_1Polygon(J_1, K_1, N', O', P', L_1, M_1, N_1, O_1, P_1) → poly5, j_1, p_1, k_1, n', o', p', l_1, m_1, n_1, o_1Rotate(D'''_1, (-(90°)), D'') → Q_1Rotate(A'''_1, (-(90°)), D'') → R_1Rotate(H'_1, (-(90°)), D'') → S_1Rotate(I'_1, (-(90°)), D'') → T_1Rotate(J'_1, (-(90°)), D'') → U_1Rotate(B'''_1, (-(90°)), D'') → V_1Rotate(Q, (-(90°)), D'') → Q'Rotate(R, (-(90°)), D'') → R'Rotate(S, (-(90°)), D'') → S'Rotate(C'''_1, (-(90°)), D'') → W_1Polygon(Q_1, R_1, S_1, T_1, U_1, V_1, Q', R', S', W_1) → poly6, q_1, w_1, r_1, s_1, t_1, u_1, v_1, q', r', s'Rotate(T, (-(90°)), D'') → T'Rotate(U, (-(90°)), D'') → U'Rotate(H'', (-(90°)), D'') → H'''Rotate(I'', (-(90°)), D'') → I'''Rotate(J'', (-(90°)), D'') → J'''Rotate(V, (-(90°)), D'') → V'Rotate(K', (-(90°)), D'') → K''Rotate(L', (-(90°)), D'') → L''Rotate(M', (-(90°)), D'') → M''Rotate(W, (-(90°)), D'') → W'Polygon(T', U', H''', I''', J''', V', K'', L'', M'', W') → poly3', t', w', u', h''', i''', j''', v', k'', l'', m''Rotate(C', (-(90°)), D'') → C''_2Rotate(D', (-(90°)), D'') → D''_2Segment(C''_2, D''_2) → c''_2Rotate(D'', (-(90°)), D'') → D'''_2Rotate(A'', (-(90°)), D'') → A'''_2Segment(D'''_2, A'''_2) → d'''_2Rotate(D, 90°, D'') → D'_2Rotate(A, 90°, D'') → A'_2Rotate(E, 90°, D'') → E'_3Rotate(G, 90°, D'') → G'_3Rotate(F, 90°, D'') → F'_3Rotate(B, 90°, D'') → B'_2Rotate(F', 90°, D'') → F''_2Rotate(G', 90°, D'') → G''_2Rotate(E', 90°, D'') → E''_2Rotate(C, 90°, D'') → C'_2Polygon(D'_2, A'_2, E'_3, G'_3, F'_3, B'_2, F''_2, G''_2, E''_2, C'_2) → poly2'_2, d'_2, c'_2, a'_2, e'_3, g'_3, f'_3, b'_2, f''_2, g''_2, e''_2Rotate(D', 90°, D'') → D''_3Rotate(A', 90°, D'') → A''_2Rotate(E'_1, 90°, D'') → Z_1Rotate(G'_1, 90°, D'') → A_2Rotate(F'_1, 90°, D'') → B_2Rotate(B', 90°, D'') → B''_2Rotate(F'', 90°, D'') → F'''_2Rotate(G'', 90°, D'') → G'''_2Rotate(E'', 90°, D'') → E'''_2Rotate(C', 90°, D'') → C''_3Polygon(D''_3, A''_2, Z_1, A_2, B_2, B''_2, F'''_2, G'''_2, E'''_2, C''_3) → poly2''_2, d''_3, c''_3, a''_2, z_2, a_2, b_2, b''_2, f'''_2, g'''_2, e'''_2Rotate(D'', 90°, D'') → D'''_3Rotate(A'', 90°, D'') → A'''_3Rotate(H, 90°, D'') → H'_2Rotate(I, 90°, D'') → I'_2Rotate(J, 90°, D'') → J'_2Rotate(B'', 90°, D'') → B'''_2Rotate(F''', 90°, D'') → C_2Rotate(G''', 90°, D'') → D_2Rotate(E''', 90°, D'') → E_2Rotate(C'', 90°, D'') → C'''_2Polygon(D'''_3, A'''_3, H'_2, I'_2, J'_2, B'''_2, C_2, D_2, E_2, C'''_2) → poly2'''_2, d'''_3, c'''_2, a'''_3, h'_2, i'_2, j'_2, b'''_2, c_2, d_2, e_2Rotate(D''', 90°, D'') → F_2Rotate(A''', 90°, D'') → G_2Rotate(H', 90°, D'') → H''_1Rotate(I', 90°, D'') → I''_1Rotate(J', 90°, D'') → J''_1Rotate(B''', 90°, D'') → H_2Rotate(K, 90°, D'') → K'_1Rotate(L, 90°, D'') → L'_1Rotate(M, 90°, D'') → M'_1Rotate(C''', 90°, D'') → I_2Polygon(F_2, G_2, H''_1, I''_1, J''_1, H_2, K'_1, L'_1, M'_1, I_2) → poly7, f_3, i_3, g_3, h''_1, i''_1, j''_1, h_3, k'_1, l'_1, m'_1Rotate(C'''_1, 90°, D'') → J_2Rotate(D'''_1, 90°, D'') → K_2Segment(J_2, K_2) → j_2Rotate(T, 90°, D'') → T'_1Rotate(U, 90°, D'') → U'_1Segment(T'_1, U'_1) → t'_1Rotate(Q_1, 90°, D'') → L_2Rotate(D'''_2, 90°, D'') → M_2Vector(D, D'_1) → u_2Vector(D'_2, D) → v_2Translate(D, v_2) → D'_3Translate(A, v_2) → A'_3Translate(E, v_2) → E'_4Translate(G, v_2) → G'_4Translate(F, v_2) → F'_4Translate(B, v_2) → B'_3Translate(F', v_2) → F''_3Translate(G', v_2) → G''_3Translate(E', v_2) → E''_3Translate(C, v_2) → C'_3Polygon(D'_3, A'_3, E'_4, G'_4, F'_4, B'_3, F''_3, G''_3, E''_3, C'_3) → poly2'_3, d'_3, c'_3, a'_3, e'_4, g'_4, f'_4, b'_3, f''_3, g''_3, e''_3Translate(D''', v_2) → N_2Translate(A''', v_2) → O_2Translate(H', v_2) → H''_2Translate(I', v_2) → I''_2Translate(J', v_2) → J''_2Translate(B''', v_2) → P_2Translate(K, v_2) → K'_2Translate(L, v_2) → L'_2Translate(M, v_2) → M'_2Translate(C''', v_2) → Q_2Polygon(N_2, O_2, H''_2, I''_2, J''_2, P_2, K'_2, L'_2, M'_2, Q_2) → poly8, n_2, q_2, o_2, h''_2, i''_2, j''_2, p_2, k'_2, l'_2, m'_2Translate(D'', v_2) → D'''_4Translate(A'', v_2) → A'''_4Translate(H, v_2) → H'_3Translate(I, v_2) → I'_3Translate(J, v_2) → J'_3Translate(B'', v_2) → B'''_3Translate(F''', v_2) → R_2Translate(G''', v_2) → S_2Translate(E''', v_2) → T_2Translate(C'', v_2) → C'''_3Polygon(D'''_4, A'''_4, H'_3, I'_3, J'_3, B'''_3, R_2, S_2, T_2, C'''_3) → poly2'''_3, d'''_4, c'''_3, a'''_4, h'_3, i'_3, j'_3, b'''_3, r_2, s_2, t_2Translate(D', v_2) → D''_4Translate(A', v_2) → A''_3Translate(E'_1, v_2) → U_2Translate(G'_1, v_2) → V_2Translate(F'_1, v_2) → W_2Translate(B', v_2) → B''_3Translate(F'', v_2) → F'''_3Translate(G'', v_2) → G'''_3Translate(E'', v_2) → E'''_3Translate(C', v_2) → C''_4Polygon(D''_4, A''_3, U_2, V_2, W_2, B''_3, F'''_3, G'''_3, E'''_3, C''_4) → poly2''_3, d''_4, c''_4, a''_3, u_3, v_3, w_2, b''_3, f'''_3, g'''_3, e'''_3Translate(D''_3, k_2) → Z_2Translate(A''_2, k_2) → A_3Translate(Z_1, k_2) → B_3Translate(A_2, k_2) → C_3Translate(B_2, k_2) → D_3Translate(B''_2, k_2) → E_3Translate(F'''_2, k_2) → F_3Translate(G'''_2, k_2) → G_3Translate(E'''_2, k_2) → H_3Translate(C''_3, k_2) → I_3Translate(Z_2, v_2) → J_3Translate(A_3, v_2) → K_3Translate(B_3, v_2) → L_3Translate(C_3, v_2) → M_3Translate(D_3, v_2) → N_3Translate(E_3, v_2) → O_3Translate(F_3, v_2) → P_3Translate(G_3, v_2) → Q_3Translate(H_3, v_2) → R_3Translate(I_3, v_2) → S_3Polygon(J_3, K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3) → poly9', j_3, s_3, k_3, l_3, m_3, n_3, o_3, p_3, q_3, r_3Translate(D'_2, v_2) → T_3Translate(A'_2, v_2) → U_3Translate(E'_3, v_2) → V_3Translate(G'_3, v_2) → W_3Translate(F'_3, v_2) → Z_3Translate(B'_2, v_2) → A_4Translate(F''_2, v_2) → B_4Translate(G''_2, v_2) → C_4Translate(E''_2, v_2) → D_4Translate(C'_2, v_2) → E_4Polygon(T_3, U_3, V_3, W_3, Z_3, A_4, B_4, C_4, D_4, E_4) → poly10, t_3, e_4, u_4, v_4, w_3, z_4, a_4, b_4, c_4, d_4Translate(F_2, v_2) → F_4Translate(G_2, v_2) → G_4Translate(H''_1, v_2) → H_4Translate(I''_1, v_2) → I_4Translate(J''_1, v_2) → J_4Translate(H_2, v_2) → K_4Translate(K'_1, v_2) → L_4Translate(L'_1, v_2) → M_4Translate(M'_1, v_2) → N_4Translate(I_2, v_2) → O_4Polygon(F_4, G_4, H_4, I_4, J_4, K_4, L_4, M_4, N_4, O_4) → poly7', f_5, o_4, g_5, h_5, i_5, j_4, k_4, l_4, m_4, n_4Translate(D'''_3, v_2) → P_4Translate(A'''_3, v_2) → Q_4Translate(H'_2, v_2) → R_4Translate(I'_2, v_2) → S_4Translate(J'_2, v_2) → T_4Translate(B'''_2, v_2) → U_4Translate(C_2, v_2) → V_4Translate(D_2, v_2) → W_4Translate(E_2, v_2) → Z_4Translate(C'''_2, v_2) → A_5Polygon(P_4, Q_4, R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5) → poly11, p_4, a_5, q_4, r_4, s_4, t_4, u_5, v_5, w_4, z_5Translate(D'''_1, v_2) → B_5Translate(A'''_1, v_2) → C_5Translate(H'_1, v_2) → D_5Translate(I'_1, v_2) → E_5Translate(J'_1, v_2) → F_5Translate(B'''_1, v_2) → G_5Translate(Q, v_2) → Q'_1Translate(R, v_2) → R'_1Translate(S, v_2) → S'_1Translate(C'''_1, v_2) → H_5Polygon(B_5, C_5, D_5, E_5, F_5, G_5, Q'_1, R'_1, S'_1, H_5) → poly12, b_5, h_6, c_5, d_5, e_5, f_6, g_6, q'_1, r'_1, s'_1Translate(Q_1, v_2) → I_5Translate(R_1, v_2) → J_5Translate(S_1, v_2) → K_5Translate(T_1, v_2) → L_5Translate(U_1, v_2) → M_5Translate(V_1, v_2) → N_5Translate(Q', v_2) → Q''Translate(R', v_2) → R''Translate(S', v_2) → S''Translate(W_1, v_2) → O_5Polygon(I_5, J_5, K_5, L_5, M_5, N_5, Q'', R'', S'', O_5) → poly6', i_6, o_5, j_5, k_5, l_5, m_5, n_5, q'', r'', s''Translate(J_1, v_2) → P_5Translate(K_1, v_2) → Q_5Translate(N', v_2) → N''Translate(O', v_2) → O''Translate(P', v_2) → P''Translate(L_1, v_2) → R_5Translate(M_1, v_2) → S_5Translate(N_1, v_2) → T_5Translate(O_1, v_2) → U_5Translate(P_1, v_2) → V_5Polygon(P_5, Q_5, N'', O'', P'', R_5, S_5, T_5, U_5, V_5) → poly5', p_5, v_6, q_5, n'', o'', p'', r_5, s_5, t_5, u_6Translate(T, v_2) → T'_2Translate(U, v_2) → U'_2Translate(H'', v_2) → H'''_1Translate(I'', v_2) → I'''_1Translate(J'', v_2) → J'''_1Translate(V, v_2) → V'_1Translate(K', v_2) → K''_1Translate(L', v_2) → L''_1Translate(M', v_2) → M''_1Translate(W, v_2) → W'_1Polygon(T'_2, U'_2, H'''_1, I'''_1, J'''_1, V'_1, K''_1, L''_1, M''_1, W'_1) → poly3'_1, t'_2, w'_1, u'_2, h'''_1, i'''_1, j'''_1, v'_1, k''_1, l''_1, m''_1Translate(T', v_2) → T''Translate(U', v_2) → U''Translate(H''', v_2) → W_5Translate(I''', v_2) → Z_5Translate(J''', v_2) → A_6Translate(V', v_2) → V''Translate(K'', v_2) → K'''Translate(L'', v_2) → L'''Translate(M'', v_2) → M'''Translate(W', v_2) → W''Polygon(T'', U'', W_5, Z_5, A_6, V'', K''', L''', M''', W'') → poly3'', t'', w'', u'', w_5, z_6, a_6, v'', k''', l''', m'''Translate(Z, v_2) → Z'Translate(A_1, v_2) → B_6Translate(B_1, v_2) → C_6Translate(C_1, v_2) → D_6Translate(D_1, v_2) → E_6Translate(E_1, v_2) → F_6Translate(F_1, v_2) → G_6Translate(G_1, v_2) → H_6Translate(H_1, v_2) → I_6Translate(I_1, v_2) → J_6Polygon(Z', B_6, C_6, D_6, E_6, F_6, G_6, H_6, I_6, J_6) → poly4', z', j_6, b_6, c_6, d_6, e_6, f_7, g_7, h_7, i_7Translate(D'_1, v_2) → K_6Translate(A'_1, v_2) → L_6Translate(E'_2, v_2) → M_6Translate(G'_2, v_2) → N_6Translate(F'_2, v_2) → O_6Translate(B'_1, v_2) → P_6Translate(F''_1, v_2) → Q_6Translate(G''_1, v_2) → R_6Translate(E''_1, v_2) → S_6Translate(C'_1, v_2) → T_6Polygon(K_6, L_6, M_6, N_6, O_6, P_6, Q_6, R_6, S_6, T_6) → poly13, k_6, t_6, l_6, m_6, n_6, o_6, p_6, q_6, r_6, s_6Translate(D''_1, v_2) → U_6Translate(A''_1, v_2) → V_6Translate(N, v_2) → N'_1Translate(O, v_2) → O'_1Translate(P, v_2) → P'_1Translate(B''_1, v_2) → W_6Translate(F'''_1, v_2) → Z_6Translate(G'''_1, v_2) → A_7Translate(E'''_1, v_2) → B_7Translate(C''_1, v_2) → C_7Polygon(U_6, V_6, N'_1, O'_1, P'_1, W_6, Z_6, A_7, B_7, C_7) → poly14, u_7, c_7, v_7, n'_1, o'_1, p'_1, w_6, z_7, a_7, b_7Translate(D', u_2) → D''_5Translate(A', u_2) → A''_4Translate(E'_1, u_2) → D_7Translate(G'_1, u_2) → E_7Translate(F'_1, u_2) → F_7Translate(B', u_2) → B''_4Translate(F'', u_2) → F'''_4Translate(G'', u_2) → G'''_4Translate(E'', u_2) → E'''_4Translate(C', u_2) → C''_5Polygon(D''_5, A''_4, D_7, E_7, F_7, B''_4, F'''_4, G'''_4, E'''_4, C''_5) → poly2''_4, d''_5, c''_5, a''_4, d_7, e_7, f_8, b''_4, f'''_4, g'''_4, e'''_4Translate(D, u_2) → D'_4Translate(A, u_2) → A'_4Translate(E, u_2) → E'_5Translate(G, u_2) → G'_5Translate(F, u_2) → F'_5Translate(B, u_2) → B'_4Translate(F', u_2) → F''_4Translate(G', u_2) → G''_4Translate(E', u_2) → E''_4Translate(C, u_2) → C'_4Polygon(D'_4, A'_4, E'_5, G'_5, F'_5, B'_4, F''_4, G''_4, E''_4, C'_4) → poly2'_4, d'_4, c'_4, a'_4, e'_5, g'_5, f'_5, b'_4, f''_4, g''_4, e''_4Translate(D''', u_2) → G_7Translate(A''', u_2) → H_7Translate(H', u_2) → H''_3Translate(I', u_2) → I''_3Translate(J', u_2) → J''_3Translate(B''', u_2) → I_7Translate(K, u_2) → K'_3Translate(L, u_2) → L'_3Translate(M, u_2) → M'_3Translate(C''', u_2) → J_7Polygon(G_7, H_7, H''_3, I''_3, J''_3, I_7, K'_3, L'_3, M'_3, J_7) → poly15, g_8, j_7, h_8, h''_3, i''_3, j''_3, i_8, k'_3, l'_3, m'_3Translate(D'', u_2) → D'''_5Translate(A'', u_2) → A'''_5Translate(H, u_2) → H'_4Translate(I, u_2) → I'_4Translate(J, u_2) → J'_4Translate(B'', u_2) → B'''_4Translate(F''', u_2) → K_7Translate(G''', u_2) → L_7Translate(E''', u_2) → M_7Translate(C'', u_2) → C'''_4Polygon(D'''_5, A'''_5, H'_4, I'_4, J'_4, B'''_4, K_7, L_7, M_7, C'''_4) → poly2'''_4, d'''_5, c'''_4, a'''_5, h'_4, i'_4, j'_4, b'''_4, k_7, l_7, m_7Translate(Z_2, u_2) → N_7Translate(A_3, u_2) → O_7Translate(B_3, u_2) → P_7Translate(C_3, u_2) → Q_7Translate(D_3, u_2) → R_7Translate(E_3, u_2) → S_7Translate(F_3, u_2) → T_7Translate(G_3, u_2) → U_7Translate(H_3, u_2) → V_7Translate(I_3, u_2) → W_7Polygon(N_7, O_7, P_7, Q_7, R_7, S_7, T_7, U_7, V_7, W_7) → poly9'_1, n_7, w_7, o_7, p_7, q_7, r_7, s_7, t_7, u_8, v_8Translate(D'_2, u_2) → Z_7Translate(A'_2, u_2) → A_8Translate(E'_3, u_2) → B_8Translate(G'_3, u_2) → C_8Translate(F'_3, u_2) → D_8Translate(B'_2, u_2) → E_8Translate(F''_2, u_2) → F_8Translate(G''_2, u_2) → G_8Translate(E''_2, u_2) → H_8Translate(C'_2, u_2) → I_8Polygon(Z_7, A_8, B_8, C_8, D_8, E_8, F_8, G_8, H_8, I_8) → poly16, z_8, i_9, a_8, b_8, c_8, d_8, e_8, f_9, g_9, h_9Translate(D'''_3, u_2) → J_8Translate(A'''_3, u_2) → K_8Translate(H'_2, u_2) → L_8Translate(I'_2, u_2) → M_8Translate(J'_2, u_2) → N_8Translate(B'''_2, u_2) → O_8Translate(C_2, u_2) → P_8Translate(D_2, u_2) → Q_8Translate(E_2, u_2) → R_8Translate(C'''_2, u_2) → S_8Polygon(J_8, K_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8, S_8) → poly17, j_8, s_8, k_8, l_8, m_8, n_8, o_8, p_8, q_8, r_8Translate(F_2, u_2) → T_8Translate(G_2, u_2) → U_8Translate(H''_1, u_2) → V_8Translate(I''_1, u_2) → W_8Translate(J''_1, u_2) → Z_8Translate(H_2, u_2) → A_9Translate(K'_1, u_2) → B_9Translate(L'_1, u_2) → C_9Translate(M'_1, u_2) → D_9Translate(I_2, u_2) → E_9Polygon(T_8, U_8, V_8, W_8, Z_8, A_9, B_9, C_9, D_9, E_9) → poly7'_1, t_8, e_9, u_9, v_9, w_8, z_9, a_9, b_9, c_9, d_9Translate(D'_3, u_2) → F_9Translate(A'_3, u_2) → G_9Translate(E'_4, u_2) → H_9Translate(G'_4, u_2) → I_9Translate(F'_4, u_2) → J_9Translate(B'_3, u_2) → K_9Translate(F''_3, u_2) → L_9Translate(G''_3, u_2) → M_9Translate(E''_3, u_2) → N_9Translate(C'_3, u_2) → O_9Polygon(F_9, G_9, H_9, I_9, J_9, K_9, L_9, M_9, N_9, O_9) → poly9, f_4, o_9, g_4, h_4, i_4, j_9, k_9, l_9, m_9, n_9Translate(D''_4, u_2) → P_9Translate(A''_3, u_2) → Q_9Translate(U_2, u_2) → R_9Translate(V_2, u_2) → S_9Translate(W_2, u_2) → T_9Translate(B''_3, u_2) → U_9Translate(F'''_3, u_2) → V_9Translate(G'''_3, u_2) → W_9Translate(E'''_3, u_2) → Z_9Translate(C''_4, u_2) → A_{10}Polygon(P_9, Q_9, R_9, S_9, T_9, U_9, V_9, W_9, Z_9, A_{10}) → poly18, p_9, a_{10}, q_9, r_9, s_9, t_9, u_{10}, v_{10}, w_9, z_3Translate(N_2, u_2) → B_{10}Translate(O_2, u_2) → C_{10}Translate(H''_2, u_2) → D_{10}Translate(I''_2, u_2) → E_{10}Translate(J''_2, u_2) → F_{10}Translate(P_2, u_2) → G_{10}Translate(K'_2, u_2) → H_{10}Translate(L'_2, u_2) → I_{10}Translate(M'_2, u_2) → J_{10}Translate(Q_2, u_2) → K_{10}Polygon(B_{10}, C_{10}, D_{10}, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, J_{10}, K_{10}) → poly8', b_{10}, k_{10}, c_{10}, d_{10}, e_{10}, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}Translate(D'''_4, u_2) → L_{10}Translate(A'''_4, u_2) → M_{10}Translate(H'_3, u_2) → N_{10}Translate(I'_3, u_2) → O_{10}Translate(J'_3, u_2) → P_{10}Translate(B'''_3, u_2) → Q_{10}Translate(R_2, u_2) → R_{10}Translate(S_2, u_2) → S_{10}Translate(T_2, u_2) → T_{10}Translate(C'''_3, u_2) → U_{10}Polygon(L_{10}, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, S_{10}, T_{10}, U_{10}) → poly19, l_{10}, u_{11}, m_{10}, n_{10}, o_{10}, p_{10}, q_{10}, r_{10}, s_{10}, t_{10}Translate(P_4, u_2) → V_{10}Translate(Q_4, u_2) → W_{10}Translate(R_4, u_2) → Z_{10}Translate(S_4, u_2) → A_{11}Translate(T_4, u_2) → B_{11}Translate(U_4, u_2) → C_{11}Translate(V_4, u_2) → D_{11}Translate(W_4, u_2) → E_{11}Translate(Z_4, u_2) → F_{11}Translate(A_5, u_2) → G_{11}Polygon(V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}) → poly11', v_{11}, g_{11}, w_{10}, z_{10}, a_{11}, b_{11}, c_{11}, d_{11}, e_{11}, f_{11}Translate(J_3, u_2) → H_{11}Translate(K_3, u_2) → I_{11}Translate(L_3, u_2) → J_{11}Translate(M_3, u_2) → K_{11}Translate(N_3, u_2) → L_{11}Translate(O_3, u_2) → M_{11}Translate(P_3, u_2) → N_{11}Translate(Q_3, u_2) → O_{11}Translate(R_3, u_2) → P_{11}Translate(S_3, u_2) → Q_{11}Polygon(H_{11}, I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}, Q_{11}) → poly9'', h_{11}, q_{11}, i_{11}, j_{11}, k_{11}, l_{11}, m_{11}, n_{11}, o_{11}, p_{11}Translate(F_4, u_2) → R_{11}Translate(G_4, u_2) → S_{11}Translate(H_4, u_2) → T_{11}Translate(I_4, u_2) → U_{11}Translate(J_4, u_2) → V_{11}Translate(K_4, u_2) → W_{11}Translate(L_4, u_2) → Z_{11}Translate(M_4, u_2) → A_{12}Translate(N_4, u_2) → B_{12}Translate(O_4, u_2) → C_{12}Polygon(R_{11}, S_{11}, T_{11}, U_{11}, V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}) → poly7'', r_{11}, c_{12}, s_{11}, t_{11}, u_{12}, v_{12}, w_{11}, z_{11}, a_{12}, b_{12}Translate(T_3, u_2) → D_{12}Translate(U_3, u_2) → E_{12}Translate(V_3, u_2) → F_{12}Translate(W_3, u_2) → G_{12}Translate(Z_3, u_2) → H_{12}Translate(A_4, u_2) → I_{12}Translate(B_4, u_2) → J_{12}Translate(C_4, u_2) → K_{12}Translate(D_4, u_2) → L_{12}Translate(E_4, u_2) → M_{12}Polygon(D_{12}, E_{12}, F_{12}, G_{12}, H_{12}, I_{12}, J_{12}, K_{12}, L_{12}, M_{12}) → poly10', d_{12}, m_{12}, e_{12}, f_{12}, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, l_{12}Translate(T'_2, u_2) → N_{12}Translate(U'_2, u_2) → O_{12}Translate(H'''_1, u_2) → P_{12}Translate(I'''_1, u_2) → Q_{12}Translate(J'''_1, u_2) → R_{12}Translate(V'_1, u_2) → S_{12}Translate(K''_1, u_2) → T_{12}Translate(L''_1, u_2) → U_{12}Translate(M''_1, u_2) → V_{12}Translate(W'_1, u_2) → W_{12}Polygon(N_{12}, O_{12}, P_{12}, Q_{12}, R_{12}, S_{12}, T_{12}, U_{12}, V_{12}, W_{12}) → poly20, n_{12}, w_{12}, o_{12}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}, u_{13}, v_{13}Translate(B_5, u_2) → Z_{12}Translate(C_5, u_2) → A_{13}Translate(D_5, u_2) → B_{13}Translate(E_5, u_2) → C_{13}Translate(F_5, u_2) → D_{13}Translate(G_5, u_2) → E_{13}Translate(Q'_1, u_2) → F_{13}Translate(R'_1, u_2) → G_{13}Translate(S'_1, u_2) → H_{13}Translate(H_5, u_2) → I_{13}Polygon(Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}, G_{13}, H_{13}, I_{13}) → poly12', z_{12}, i_{13}, a_{13}, b_{13}, c_{13}, d_{13}, e_{13}, f_{13}, g_{13}, h_{13}Translate(I_5, u_2) → J_{13}Translate(J_5, u_2) → K_{13}Translate(K_5, u_2) → L_{13}Translate(L_5, u_2) → M_{13}Translate(M_5, u_2) → N_{13}Translate(N_5, u_2) → O_{13}Translate(Q'', u_2) → Q'''Translate(R'', u_2) → R'''Translate(S'', u_2) → S'''Translate(O_5, u_2) → P_{13}Polygon(J_{13}, K_{13}, L_{13}, M_{13}, N_{13}, O_{13}, Q''', R''', S''', P_{13}) → poly6'', j_{13}, p_{13}, k_{13}, l_{13}, m_{13}, n_{13}, o_{13}, q''', r''', s'''Translate(P_5, u_2) → Q_{13}Translate(Q_5, u_2) → R_{13}Translate(N'', u_2) → N'''Translate(O'', u_2) → O'''Translate(P'', u_2) → P'''Translate(R_5, u_2) → S_{13}Translate(S_5, u_2) → T_{13}Translate(T_5, u_2) → U_{13}Translate(U_5, u_2) → V_{13}Translate(V_5, u_2) → W_{13}Polygon(Q_{13}, R_{13}, N''', O''', P''', S_{13}, T_{13}, U_{13}, V_{13}, W_{13}) → poly5'', q_{13}, w_{13}, r_{13}, n''', o''', p''', s_{13}, t_{13}, u_{14}, v_{14}Translate(T, u_2) → T'_3Translate(U, u_2) → U'_3Translate(H'', u_2) → H'''_2Translate(I'', u_2) → I'''_2Translate(J'', u_2) → J'''_2Translate(V, u_2) → V'_2Translate(K', u_2) → K''_2Translate(L', u_2) → L''_2Translate(M', u_2) → M''_2Translate(W, u_2) → W'_2Polygon(T'_3, U'_3, H'''_2, I'''_2, J'''_2, V'_2, K''_2, L''_2, M''_2, W'_2) → poly3'_2, t'_3, w'_2, u'_3, h'''_2, i'''_2, j'''_2, v'_2, k''_2, l''_2, m''_2Translate(D'''_1, u_2) → Z_{13}Translate(A'''_1, u_2) → A_{14}Translate(H'_1, u_2) → B_{14}Translate(I'_1, u_2) → C_{14}Translate(J'_1, u_2) → D_{14}Translate(B'''_1, u_2) → E_{14}Translate(Q, u_2) → Q'_2Translate(R, u_2) → R'_2Translate(S, u_2) → S'_2Translate(C'''_1, u_2) → F_{14}Polygon(Z_{13}, A_{14}, B_{14}, C_{14}, D_{14}, E_{14}, Q'_2, R'_2, S'_2, F_{14}) → poly21, z_{13}, f_{14}, a_{14}, b_{14}, c_{14}, d_{14}, e_{14}, q'_2, r'_2, s'_2Translate(Q_1, u_2) → G_{14}Translate(R_1, u_2) → H_{14}Translate(S_1, u_2) → I_{14}Translate(T_1, u_2) → J_{14}Translate(U_1, u_2) → K_{14}Translate(V_1, u_2) → L_{14}Translate(Q', u_2) → Q''_1Translate(R', u_2) → R''_1Translate(S', u_2) → S''_1Translate(W_1, u_2) → M_{14}Polygon(G_{14}, H_{14}, I_{14}, J_{14}, K_{14}, L_{14}, Q''_1, R''_1, S''_1, M_{14}) → poly6'_1, g_{14}, m_{14}, h_{14}, i_{14}, j_{14}, k_{14}, l_{14}, q''_1, r''_1, s''_1Translate(J_1, u_2) → N_{14}Translate(K_1, u_2) → O_{14}Translate(N', u_2) → N''_1Translate(O', u_2) → O''_1Translate(P', u_2) → P''_1Translate(L_1, u_2) → P_{14}Translate(M_1, u_2) → Q_{14}Translate(N_1, u_2) → R_{14}Translate(O_1, u_2) → S_{14}Translate(P_1, u_2) → T_{14}Polygon(N_{14}, O_{14}, N''_1, O''_1, P''_1, P_{14}, Q_{14}, R_{14}, S_{14}, T_{14}) → poly5'_1, n_{14}, t_{14}, o_{14}, n''_1, o''_1, p''_1, p_{14}, q_{14}, r_{14}, s_{14}Translate(K_6, u_2) → U_{14}Translate(L_6, u_2) → V_{14}Translate(M_6, u_2) → W_{14}Translate(N_6, u_2) → Z_{14}Translate(O_6, u_2) → A_{15}Translate(P_6, u_2) → B_{15}Translate(Q_6, u_2) → C_{15}Translate(R_6, u_2) → D_{15}Translate(S_6, u_2) → E_{15}Translate(T_6, u_2) → F_{15}Polygon(U_{14}, V_{14}, W_{14}, Z_{14}, A_{15}, B_{15}, C_{15}, D_{15}, E_{15}, F_{15}) → poly13', u_{15}, f_{15}, v_{15}, w_{14}, z_{14}, a_{15}, b_{15}, c_{15}, d_{15}, e_{15}Translate(U_6, u_2) → G_{15}Translate(V_6, u_2) → H_{15}Translate(N'_1, u_2) → I_{15}Translate(O'_1, u_2) → J_{15}Translate(P'_1, u_2) → K_{15}Translate(W_6, u_2) → L_{15}Translate(Z_6, u_2) → M_{15}Translate(A_7, u_2) → N_{15}Translate(B_7, u_2) → O_{15}Translate(C_7, u_2) → P_{15}Polygon(G_{15}, H_{15}, I_{15}, J_{15}, K_{15}, L_{15}, M_{15}, N_{15}, O_{15}, P_{15}) → poly14', g_{15}, p_{15}, h_{15}, i_{15}, j_{15}, k_{15}, l_{15}, m_{15}, n_{15}, o_{15}Translate(T'', u_2) → T'''Translate(U'', u_2) → U'''Translate(W_5, u_2) → Q_{15}Translate(Z_5, u_2) → R_{15}Translate(A_6, u_2) → S_{15}Translate(V'', u_2) → V'''Translate(K''', u_2) → T_{15}Translate(L''', u_2) → U_{15}Translate(M''', u_2) → V_{15}Translate(W'', u_2) → W'''Polygon(T''', U''', Q_{15}, R_{15}, S_{15}, V''', T_{15}, U_{15}, V_{15}, W''') → poly3''', t''', w''', u''', q_{15}, r_{15}, s_{15}, v''', t_{15}, u_{16}, v_{16}Translate(Z', u_2) → Z''Translate(B_6, u_2) → W_{15}Translate(C_6, u_2) → Z_{15}Translate(D_6, u_2) → A_{16}Translate(E_6, u_2) → B_{16}Translate(F_6, u_2) → C_{16}Translate(G_6, u_2) → D_{16}Translate(H_6, u_2) → E_{16}Translate(I_6, u_2) → F_{16}Translate(J_6, u_2) → G_{16}Polygon(Z'', W_{15}, Z_{15}, A_{16}, B_{16}, C_{16}, D_{16}, E_{16}, F_{16}, G_{16}) → poly4'', z'', g_{16}, w_{15}, z_{15}, a_{16}, b_{16}, c_{16}, d_{16}, e_{16}, f_{16}Translate(D'_1, u_2) → H_{16}Translate(A'_1, u_2) → I_{16}Translate(E'_2, u_2) → J_{16}Translate(G'_2, u_2) → K_{16}Translate(F'_2, u_2) → L_{16}Translate(B'_1, u_2) → M_{16}Translate(F''_1, u_2) → N_{16}Translate(G''_1, u_2) → O_{16}Translate(E''_1, u_2) → P_{16}Translate(C'_1, u_2) → Q_{16}Polygon(H_{16}, I_{16}, J_{16}, K_{16}, L_{16}, M_{16}, N_{16}, O_{16}, P_{16}, Q_{16}) → poly22, h_{16}, q_{16}, i_{16}, j_{16}, k_{16}, l_{16}, m_{16}, n_{16}, o_{16}, p_{16}Translate(D''_1, u_2) → R_{16}Translate(A''_1, u_2) → S_{16}Translate(N, u_2) → N'_2Translate(O, u_2) → O'_2Translate(P, u_2) → P'_2Translate(B''_1, u_2) → T_{16}Translate(F'''_1, u_2) → U_{16}Translate(G'''_1, u_2) → V_{16}Translate(E'''_1, u_2) → W_{16}Translate(C''_1, u_2) → Z_{16}Polygon(R_{16}, S_{16}, N'_2, O'_2, P'_2, T_{16}, U_{16}, V_{16}, W_{16}, Z_{16}) → poly23, r_{16}, z_{16}, s_{16}, n'_2, o'_2, p'_2, t_{16}, u_{17}, v_{17}, w_{16}Translate(T', u_2) → T''_1Translate(U', u_2) → U''_1Translate(H''', u_2) → A_{17}Translate(I''', u_2) → B_{17}Translate(J''', u_2) → C_{17}Translate(V', u_2) → V''_1Translate(K'', u_2) → K'''_1Translate(L'', u_2) → L'''_1Translate(M'', u_2) → M'''_1Translate(W', u_2) → W''_1Polygon(T''_1, U''_1, A_{17}, B_{17}, C_{17}, V''_1, K'''_1, L'''_1, M'''_1, W''_1) → poly3''_1, t''_1, w''_1, u''_1, a_{17}, b_{17}, c_{17}, v''_1, k'''_1, l'''_1, m'''_1Translate(Z, u_2) → Z'_1Translate(A_1, u_2) → D_{17}Translate(B_1, u_2) → E_{17}Translate(C_1, u_2) → F_{17}Translate(D_1, u_2) → G_{17}Translate(E_1, u_2) → H_{17}Translate(F_1, u_2) → I_{17}Translate(G_1, u_2) → J_{17}Translate(H_1, u_2) → K_{17}Translate(I_1, u_2) → L_{17}Polygon(Z'_1, D_{17}, E_{17}, F_{17}, G_{17}, H_{17}, I_{17}, J_{17}, K_{17}, L_{17}) → poly4'_1, z'_1, l_{17}, d_{17}, e_{17}, f_{17}, g_{17}, h_{17}, i_{17}, j_{17}, k_{17}