埃舍尔的飞马镶嵌图形探索
几何画板及绘图内容加载中,请稍等……
本范例旨在帮助你探索和总结M.C.埃舍尔使用的底层多边形镶嵌规则。它涉及多边形、向量、平移等核心概念。通过使用GeoGebra命令如Segment、Polygon、Vector、Translate等,你可以直观地理解镶嵌图形的构造过程。适合初中及以上学生或数学爱好者。
GeoGebra 命令及参数
Segment(A, B) → aPolygon(C, D, E) → poly1, e, c, dPolygon(F, G, H, I) → poly2, f, g, h, iPolygon(J, K, L, M, N, O) → poly3, j, k, l, m, n, oPolygon(P, Q, R, S, T) → poly4, p_1, q_1, r, s, tPolygon(A_1, H_1, G_1, B_1) → poly6, a_1, h_1, g_1, b_1Polygon(B_1, C_1, F_1, G_1) → poly7, b_2, c_1, f_1, g_2Polygon(C_1, D_1, E_1, F_1) → poly8, c_2, d_1, e_1, f_2Polygon(E_1, L_1, K_1, F_1) → poly9, e_2, l_1, k_1, f_3Polygon(K_1, J_1, G_1, F_1) → poly10, k_2, j_1, g_3, f_4Polygon(J_1, I_1, H_1, G_1) → poly11, j_2, i_1, h_2, g_4Polygon(I_1, P_1, O_1, J_1) → poly12, i_2, p_2, o_1, j_3Polygon(O_1, N_1, K_1, J_1) → poly13, o_2, n_1, k_3, j_4Polygon(N_1, M_1, L_1, K_1) → poly14, n_2, m_1, l_2, k_4Polygon(Q_1, R_1, C_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, S_1, M_2, N_2, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2, T_1, W_2, Z_2, A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3, U_1, I_3, J_3, K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, V_1, S_3, T_3, U_3, V_3, W_3, Z_3, A_4, B_4, C_4, D_4, W_1, E_4, F_4, G_4, H_4, I_4, J_4, K_4, L_4, M_4, N_4, Z_1, O_4, P_4, Q_4, R_4, S_4, T_4, U_4, A_2, B_2) → poly15, q_2, b_3, c_3, d_2, e_3, f_5, g_5, h_3, i_3, j_5, k_5, r_2, l_3, m_2, n_3, o_3, p_3, q_3, r_3, s_2, t_2, u_3, s_1, v_2, w_2, z_2, a_3, b_4, c_4, d_3, e_4, f_6, g_6, t_1, h_4, i_4, j_6, k_6, l_4, m_3, n_4, o_4, p_4, q_4, u_2, r_4, s_3, t_3, u_4, v_3, w_3, z_3, a_4, b_5, c_5, v_1, d_4, e_5, f_7, g_7, h_5, i_5, j_7, k_7, l_5, m_4, w_1, n_5, o_5, p_5, q_5, r_5, s_4, t_4, u_5, z_1, a_2Vector(V_4, W_4) → v_4Translate(Q_1, v_4) → Z_4Translate(R_1, v_4) → A_5Translate(S_1, v_4) → B_5Translate(T_1, v_4) → C_5Translate(U_1, v_4) → D_5Translate(V_1, v_4) → E_5Translate(W_1, v_4) → F_5Translate(Z_1, v_4) → G_5Translate(A_2, v_4) → H_5Translate(B_2, v_4) → I_5Translate(C_2, v_4) → J_5Translate(D_2, v_4) → K_5Translate(E_2, v_4) → L_5Translate(F_2, v_4) → M_5Translate(G_2, v_4) → N_5Translate(H_2, v_4) → O_5Translate(I_2, v_4) → P_5Translate(J_2, v_4) → Q_5Translate(K_2, v_4) → R_5Translate(L_2, v_4) → S_5Translate(M_2, v_4) → T_5Translate(N_2, v_4) → U_5Translate(O_2, v_4) → V_5Translate(P_2, v_4) → W_5Translate(Q_2, v_4) → Z_5Translate(R_2, v_4) → A_6Translate(S_2, v_4) → B_6Translate(T_2, v_4) → C_6Translate(U_2, v_4) → D_6Translate(V_2, v_4) → E_6Translate(W_2, v_4) → F_6Translate(Z_2, v_4) → G_6Translate(A_3, v_4) → H_6Translate(B_3, v_4) → I_6Translate(C_3, v_4) → J_6Translate(D_3, v_4) → K_6Translate(E_3, v_4) → L_6Translate(F_3, v_4) → M_6Translate(G_3, v_4) → N_6Translate(H_3, v_4) → O_6Translate(I_3, v_4) → P_6Translate(J_3, v_4) → Q_6Translate(K_3, v_4) → R_6Translate(L_3, v_4) → S_6Translate(M_3, v_4) → T_6Translate(N_3, v_4) → U_6Translate(O_3, v_4) → V_6Translate(P_3, v_4) → W_6Translate(Q_3, v_4) → Z_6Translate(R_3, v_4) → A_7Translate(S_3, v_4) → B_7Translate(T_3, v_4) → C_7Translate(U_3, v_4) → D_7Translate(V_3, v_4) → E_7Translate(W_3, v_4) → F_7Translate(Z_3, v_4) → G_7Translate(A_4, v_4) → H_7Translate(B_4, v_4) → I_7Translate(C_4, v_4) → J_7Translate(D_4, v_4) → K_7Translate(E_4, v_4) → L_7Translate(F_4, v_4) → M_7Translate(G_4, v_4) → N_7Translate(H_4, v_4) → O_7Translate(I_4, v_4) → P_7Translate(J_4, v_4) → Q_7Translate(K_4, v_4) → R_7Translate(L_4, v_4) → S_7Translate(M_4, v_4) → T_7Translate(N_4, v_4) → U_7Translate(O_4, v_4) → V_7Translate(P_4, v_4) → W_7Translate(Q_4, v_4) → Z_7Translate(R_4, v_4) → A_8Translate(S_4, v_4) → B_8Translate(T_4, v_4) → C_8Translate(U_4, v_4) → D_8Polygon(Z_4, A_5, J_5, K_5, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, S_5, B_5, T_5, U_5, V_5, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, C_5, F_6, G_6, H_6, I_6, J_6, K_6, L_6, M_6, N_6, O_6, D_5, P_6, Q_6, R_6, S_6, T_6, U_6, V_6, W_6, Z_6, A_7, E_5, B_7, C_7, D_7, E_7, F_7, G_7, H_7, I_7, J_7, K_7, F_5, L_7, M_7, N_7, O_7, P_7, Q_7, R_7, S_7, T_7, U_7, G_5, V_7, W_7, Z_7, A_8, B_8, C_8, D_8, H_5, I_5) → poly15', z_4, i_6, j_8, k_8, l_6, m_5, n_6, o_6, p_6, q_6, r_6, a_5, s_5, t_5, u_6, v_5, w_5, z_5, a_6, b_7, c_7, d_6, b_6, e_7, f_9, g_9, h_7, i_7, j_9, k_9, l_7, m_6, n_7, c_6, o_7, p_7, q_7, r_7, s_6, t_6, u_7, v_6, w_6, z_6, d_5, a_7, b_8, c_8, d_7, e_8, f_{10}, g_{10}, h_8, i_8, j_{10}, e_6, k_{10}, l_8, m_7, n_8, o_8, p_8, q_8, r_8, s_7, t_7, f_8, u_8, v_7, w_7, z_7, a_8, b_9, c_9, d_8, g_8, h_6Translate(pic1, v_4) → pic1'Polygon(S_9, E_8, N_8, O_8, P_8, Q_8, R_8, S_8, T_8, U_8, V_8, W_8, F_8, Z_8, A_9, B_9, C_9, D_9, E_9, F_9, G_9, H_9, I_9, G_8, J_9, K_9, L_9, M_9, N_9, O_9, P_9, Q_9, R_9, H_8, I_8, J_8, K_8, L_8, M_8) → poly16, s_9, m_9, n_9, o_9, p_9, q_9, r_9, s_8, t_8, u_9, v_8, e_9, w_8, z_8, a_9, b_{10}, c_{10}, d_9, e_{10}, f_{12}, g_{12}, h_{10}, f_{11}, i_{10}, j_{12}, k_{12}, l_{10}, m_{10}, n_{10}, o_{10}, p_{10}, q_{10}, r_{10}, g_{11}, h_9, i_9, j_{11}, k_{11}, l_9Polygon(T_9, U_9, F_{10}, V_9, W_9, Z_9, A_{10}, B_{10}, C_{10}, D_{10}, E_{10}) → poly17, t_9, e_{11}, f_{13}, u_{10}, v_9, w_9, z_9, a_{10}, b_{11}, c_{11}, d_{10}Polygon(G_{10}, H_{10}, Q_{10}, R_{10}, S_{10}, T_{10}, U_{10}, I_{10}, J_{10}, K_{10}, L_{10}, M_{10}, N_{10}, O_{10}, P_{10}) → poly18, g_{13}, p_{11}, q_{11}, r_{11}, s_{10}, t_{10}, u_{11}, h_{11}, i_{11}, j_{13}, k_{13}, l_{11}, m_{11}, n_{11}, o_{11}Polygon(V_{10}, W_{10}, H_{11}, I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}, Q_{11}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}) → poly19, v_{10}, g_{14}, h_{12}, i_{12}, j_{14}, k_{14}, l_{12}, m_{12}, n_{12}, o_{12}, p_{12}, w_{10}, q_{12}, z_{10}, a_{11}, b_{12}, c_{12}, d_{11}, e_{12}, f_{14}Polygon(R_{11}, S_{11}, T_{11}, U_{11}, V_{11}, W_{11}, Z_{11}) → poly20, r_{12}, s_{11}, t_{11}, u_{12}, v_{11}, w_{11}, z_{11}Polygon(A_{12}, B_{12}, K_{12}, L_{12}, M_{12}, N_{12}, C_{12}, D_{12}, E_{12}, F_{12}, G_{12}, H_{12}, I_{12}, J_{12}) → poly21, a_{12}, j_{15}, k_{15}, l_{13}, m_{13}, n_{13}, b_{13}, c_{13}, d_{12}, e_{13}, f_{15}, g_{15}, h_{13}, i_{13}Vector(J_1, O_{12}) → v_{12}Vector(J_1, P_{12}) → w_{12}Vector(J_1, Q_{12}) → z_{12}Vector(J_1, R_{12}) → s_{12}Translate(S_9, v_{12}) → S_{12}Translate(E_8, v_{12}) → T_{12}Translate(F_8, v_{12}) → U_{12}Translate(G_8, v_{12}) → V_{12}Translate(H_8, v_{12}) → W_{12}Translate(I_8, v_{12}) → Z_{12}Translate(J_8, v_{12}) → A_{13}Translate(K_8, v_{12}) → B_{13}Translate(L_8, v_{12}) → C_{13}Translate(M_8, v_{12}) → D_{13}Translate(N_8, v_{12}) → E_{13}Translate(O_8, v_{12}) → F_{13}Translate(P_8, v_{12}) → G_{13}Translate(Q_8, v_{12}) → H_{13}Translate(R_8, v_{12}) → I_{13}Translate(S_8, v_{12}) → J_{13}Translate(T_8, v_{12}) → K_{13}Translate(U_8, v_{12}) → L_{13}Translate(V_8, v_{12}) → M_{13}Translate(W_8, v_{12}) → N_{13}Translate(Z_8, v_{12}) → O_{13}Translate(A_9, v_{12}) → P_{13}Translate(B_9, v_{12}) → Q_{13}Translate(C_9, v_{12}) → R_{13}Translate(D_9, v_{12}) → S_{13}Translate(E_9, v_{12}) → T_{13}Translate(F_9, v_{12}) → U_{13}Translate(G_9, v_{12}) → V_{13}Translate(H_9, v_{12}) → W_{13}Translate(I_9, v_{12}) → Z_{13}Translate(J_9, v_{12}) → A_{14}Translate(K_9, v_{12}) → B_{14}Translate(L_9, v_{12}) → C_{14}Translate(M_9, v_{12}) → D_{14}Translate(N_9, v_{12}) → E_{14}Translate(O_9, v_{12}) → F_{14}Translate(P_9, v_{12}) → G_{14}Translate(Q_9, v_{12}) → H_{14}Translate(R_9, v_{12}) → I_{14}Polygon(S_{12}, T_{12}, E_{13}, F_{13}, G_{13}, H_{13}, I_{13}, J_{13}, K_{13}, L_{13}, M_{13}, N_{13}, U_{12}, O_{13}, P_{13}, Q_{13}, R_{13}, S_{13}, T_{13}, U_{13}, V_{13}, W_{13}, Z_{13}, V_{12}, A_{14}, B_{14}, C_{14}, D_{14}, E_{14}, F_{14}, G_{14}, H_{14}, I_{14}, W_{12}, Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}) → poly16', s_{13}, d_{13}, e_{14}, f_{16}, g_{16}, h_{14}, i_{14}, j_{16}, k_{16}, l_{14}, m_{14}, t_{12}, n_{14}, o_{13}, p_{13}, q_{13}, r_{13}, s_{14}, t_{13}, u_{14}, v_{14}, w_{14}, u_{13}, z_{14}, a_{14}, b_{15}, c_{15}, d_{14}, e_{15}, f_{17}, g_{17}, h_{15}, i_{15}, v_{13}, w_{13}, z_{13}, a_{13}, b_{14}, c_{14}Translate(V_{10}, s_{12}) → J_{14}Translate(W_{10}, s_{12}) → K_{14}Translate(Z_{10}, s_{12}) → L_{14}Translate(A_{11}, s_{12}) → M_{14}Translate(B_{11}, s_{12}) → N_{14}Translate(C_{11}, s_{12}) → O_{14}Translate(D_{11}, s_{12}) → P_{14}Translate(E_{11}, s_{12}) → Q_{14}Translate(F_{11}, s_{12}) → R_{14}Translate(G_{11}, s_{12}) → S_{14}Translate(H_{11}, s_{12}) → T_{14}Translate(I_{11}, s_{12}) → U_{14}Translate(J_{11}, s_{12}) → V_{14}Translate(K_{11}, s_{12}) → W_{14}Translate(L_{11}, s_{12}) → Z_{14}Translate(M_{11}, s_{12}) → A_{15}Translate(N_{11}, s_{12}) → B_{15}Translate(O_{11}, s_{12}) → C_{15}Translate(P_{11}, s_{12}) → D_{15}Translate(Q_{11}, s_{12}) → E_{15}Polygon(J_{14}, K_{14}, T_{14}, U_{14}, V_{14}, W_{14}, Z_{14}, A_{15}, B_{15}, C_{15}, D_{15}, E_{15}, L_{14}, M_{14}, N_{14}, O_{14}, P_{14}, Q_{14}, R_{14}, S_{14}) → poly19', j_{17}, s_{15}, t_{14}, u_{15}, v_{15}, w_{15}, z_{15}, a_{15}, b_{16}, c_{16}, d_{15}, k_{17}, e_{16}, l_{15}, m_{15}, n_{15}, o_{14}, p_{14}, q_{14}, r_{14}Translate(R_{11}, z_{12}) → F_{15}Translate(S_{11}, z_{12}) → G_{15}Translate(T_{11}, z_{12}) → H_{15}Translate(U_{11}, z_{12}) → I_{15}Translate(V_{11}, z_{12}) → J_{15}Translate(W_{11}, z_{12}) → K_{15}Translate(Z_{11}, z_{12}) → L_{15}Polygon(F_{15}, G_{15}, H_{15}, I_{15}, J_{15}, K_{15}, L_{15}) → poly20', f_{18}, g_{18}, h_{16}, i_{16}, j_{18}, k_{18}, l_{16}Translate(A_{12}, z_{12}) → M_{15}Translate(B_{12}, z_{12}) → N_{15}Translate(C_{12}, z_{12}) → O_{15}Translate(D_{12}, z_{12}) → P_{15}Translate(E_{12}, z_{12}) → Q_{15}Translate(F_{12}, z_{12}) → R_{15}Translate(G_{12}, z_{12}) → S_{15}Translate(H_{12}, z_{12}) → T_{15}Translate(I_{12}, z_{12}) → U_{15}Translate(J_{12}, z_{12}) → V_{15}Translate(K_{12}, z_{12}) → W_{15}Translate(L_{12}, z_{12}) → Z_{15}Translate(M_{12}, z_{12}) → A_{16}Translate(N_{12}, z_{12}) → B_{16}Polygon(M_{15}, N_{15}, W_{15}, Z_{15}, A_{16}, B_{16}, O_{15}, P_{15}, Q_{15}, R_{15}, S_{15}, T_{15}, U_{15}, V_{15}) → poly21', m_{16}, v_{16}, w_{16}, z_{16}, a_{16}, b_{17}, n_{16}, o_{15}, p_{15}, q_{15}, r_{15}, s_{16}, t_{15}, u_{16}Translate(G_{10}, w_{12}) → C_{16}Translate(H_{10}, w_{12}) → D_{16}Translate(I_{10}, w_{12}) → E_{16}Translate(J_{10}, w_{12}) → F_{16}Translate(K_{10}, w_{12}) → G_{16}Translate(L_{10}, w_{12}) → H_{16}Translate(M_{10}, w_{12}) → I_{16}Translate(N_{10}, w_{12}) → J_{16}Translate(O_{10}, w_{12}) → K_{16}Translate(P_{10}, w_{12}) → L_{16}Translate(Q_{10}, w_{12}) → M_{16}Translate(R_{10}, w_{12}) → N_{16}Translate(S_{10}, w_{12}) → O_{16}Translate(T_{10}, w_{12}) → P_{16}Translate(U_{10}, w_{12}) → Q_{16}Polygon(C_{16}, D_{16}, M_{16}, N_{16}, O_{16}, P_{16}, Q_{16}, E_{16}, F_{16}, G_{16}, H_{16}, I_{16}, J_{16}, K_{16}, L_{16}) → poly18', c_{17}, l_{17}, m_{17}, n_{17}, o_{16}, p_{16}, q_{16}, d_{16}, e_{17}, f_{19}, g_{19}, h_{17}, i_{17}, j_{19}, k_{19}Translate(T_9, w_{12}) → R_{16}Translate(U_9, w_{12}) → S_{16}Translate(V_9, w_{12}) → T_{16}Translate(W_9, w_{12}) → U_{16}Translate(Z_9, w_{12}) → V_{16}Translate(A_{10}, w_{12}) → W_{16}Translate(B_{10}, w_{12}) → Z_{16}Translate(C_{10}, w_{12}) → A_{17}Translate(D_{10}, w_{12}) → B_{17}Translate(E_{10}, w_{12}) → C_{17}Translate(F_{10}, w_{12}) → D_{17}Polygon(R_{16}, S_{16}, D_{17}, T_{16}, U_{16}, V_{16}, W_{16}, Z_{16}, A_{17}, B_{17}, C_{17}) → poly17', r_{16}, c_{18}, d_{17}, s_{17}, t_{16}, u_{17}, v_{17}, w_{17}, z_{17}, a_{17}, b_{18}Point(z_{12}) → U_{18}Point(z_{12}) → V_{18}Point(z_{12}) → W_{18}Polygon(E_{17}, F_{17}, O_{17}, P_{17}, Q_{17}, R_{17}, S_{17}, T_{17}, U_{17}, V_{17}, W_{17}, Z_{17}, G_{17}, A_{18}, B_{18}, C_{18}, D_{18}, E_{18}, F_{18}, G_{18}, H_{18}, I_{18}, J_{18}, H_{17}, K_{18}, L_{18}, M_{18}, N_{18}, O_{18}, P_{18}, Q_{18}, R_{18}, S_{18}, T_{18}, I_{17}, Z_{18}, A_{19}, B_{19}, C_{19}, D_{19}, E_{19}, F_{19}, G_{19}, H_{19}, I_{19}, J_{17}, J_{19}, K_{19}, L_{19}, M_{19}, N_{19}, O_{19}, P_{19}, Q_{19}, R_{19}, S_{19}, K_{17}, T_{19}, U_{19}, V_{19}, W_{19}, Z_{19}, A_{20}, B_{20}, C_{20}, D_{20}, E_{20}, L_{17}, F_{20}, G_{20}, H_{20}, I_{20}, J_{20}, K_{20}, L_{20}, M_{20}, M_{17}, N_{17}) → poly22, e_{18}, n_{18}, o_{17}, p_{17}, q_{17}, r_{17}, s_{18}, t_{17}, u_{18}, v_{18}, w_{18}, f_{20}, z_{18}, a_{18}, b_{19}, c_{19}, d_{18}, e_{19}, f_{21}, g_{21}, h_{19}, i_{19}, g_{20}, j_{21}, k_{21}, l_{19}, m_{19}, n_{19}, o_{18}, p_{18}, q_{18}, r_{18}, s_{19}, h_{18}, t_{18}, z_{19}, a_{19}, b_{20}, c_{20}, d_{19}, e_{20}, f_{22}, g_{22}, h_{20}, i_{18}, i_{20}, j_{22}, k_{22}, l_{20}, m_{20}, n_{20}, o_{19}, p_{19}, q_{19}, r_{19}, j_{20}, s_{20}, t_{19}, u_{19}, v_{19}, w_{19}, z_{20}, a_{20}, b_{21}, c_{21}, d_{20}, k_{20}, e_{21}, f_{23}, g_{23}, h_{21}, i_{21}, j_{23}, k_{23}, l_{21}, m_{21}, l_{18}, m_{18}Polygon(N_{20}, O_{20}, P_{20}, Q_{20}, R_{20}, S_{20}) → poly23, n_{21}, o_{20}, p_{20}, q_{20}, r_{20}, s_{21}Vector(J_1, T_{20}) → u_{20}Translate(N_{20}, u_{20}) → U_{20}Translate(O_{20}, u_{20}) → V_{20}Translate(P_{20}, u_{20}) → W_{20}Translate(Q_{20}, u_{20}) → Z_{20}Translate(R_{20}, u_{20}) → A_{21}Translate(S_{20}, u_{20}) → B_{21}Polygon(U_{20}, V_{20}, W_{20}, Z_{20}, A_{21}, B_{21}) → poly23', u_{21}, v_{20}, w_{20}, z_{21}, a_{21}, b_{22}Polygon(C_{21}, D_{21}, E_{21}, F_{21}) → poly24, c_{22}, d_{21}, e_{22}, f_{24}Point(u_{20}) → L_{21}Point(u_{20}) → M_{21}Polygon(G_{21}, H_{21}, I_{21}, J_{21}, K_{21}, N_{21}, O_{21}, P_{21}, Q_{21}) → poly25, g_{24}, h_{22}, i_{22}, j_{24}, k_{24}, n_{22}, o_{21}, p_{21}, q_{21}Translate(pic1, v_{12}) → pic1'_1Translate(pic1, z_{12}) → pic1'_2Polygon(R_{21}, S_{21}, T_{21}, U_{21}) → poly5, r_{21}, s_{22}, t_{21}, u_1Polygon(V_{21}, W_{21}, Z_{21}, A_{22}) → poly26, v_{21}, w_{21}, z_{22}, a_{22}
AI 绘图提示语参考
绘制一个由多个多边形组成的复杂镶嵌图案,这些多边形通过共享边和顶点相互连接,形成类似马赛克的视觉效果。图案中包含不同边数的多边形,它们以特定的方式排列和组合,展现出几何对称性和重复性。首先绘制几个基本的多边形,然后通过平移和旋转这些多边形,逐步构建出整个镶嵌图案,注意保持多边形之间的连续性和无缝拼接。
以上提示语为根据原始绘图命令整理而来,仅供参考。