三角形密铺几何演示：利用中点与镜像变换实现平面镶嵌

几何画板及绘图内容加载中，请稍等……

2016-05-06 05:13

本范例演示了三角形如何通过几何变换实现平面密铺现象。涉及的核心数学概念包括全等三角形、平面图形性质及面积计算。作图要点在于利用 GeoGebra 的 Midpoint 命令确定关键点，通过 Mirror 命令生成对称图形，并结合 Polygon 命令闭合多边形区域。该工具适合中学阶段学生学习几何变换，或教师用于课堂展示密铺原理。

GeoGebra 命令及参数

Midpoint(A, B) → D
Midpoint(B, C) → E
Midpoint(C, A) → F
Mirror(H, D) → H'
Mirror(G, D) → G'
Mirror(K, F) → K'
Mirror(L, F) → L'
Mirror(I, E) → I'
Mirror(J, E) → J'
Polygon(A, G', C, L', K', K, L, H', H, G, B, J', I', I, J) → poly1, a, j, c, l', k', k, l, g', h', h, g, b, j', i', i
Mirror(A, F) → A'
Mirror(G', F) → G''
Mirror(H', F) → H''
Mirror(H, F) → H'_1
Mirror(G, F) → G'_1
Mirror(B, F) → B'
Mirror(J', F) → J''
Mirror(I', F) → I''
Mirror(I, F) → I'_1
Mirror(J, F) → J'_1
Mirror(C, F) → C'
Mirror(L', F) → L''
Mirror(K', F) → K''
Mirror(K, F) → K'_1
Mirror(L, F) → L'_1
Polygon(A', G'', C', L'', K'', K'_1, L'_1, H'', H'_1, G'_1, B', J'', I'', I'_1, J'_1) → poly1', a', j'_1, c', l'', k'', k'_1, l'_1, g'', h'', h'_1, g'_1, b', j'', i'', i'_1
Mirror(A, D) → A'_1
Mirror(G', D) → G''_1
Mirror(H', D) → H''_1
Mirror(H, D) → H'_2
Mirror(G, D) → G'_2
Mirror(B, D) → B'_1
Mirror(J', D) → J''_1
Mirror(I', D) → I''_1
Mirror(I, D) → I'_2
Mirror(J, D) → J'_2
Mirror(C, D) → C'_1
Mirror(L', D) → L''_1
Mirror(K', D) → K''_1
Mirror(K, D) → K'_2
Mirror(L, D) → L'_2
Polygon(A'_1, G''_1, C'_1, L''_1, K''_1, K'_2, L'_2, H''_1, H'_2, G'_2, B'_1, J''_1, I''_1, I'_2, J'_2) → poly1'_1, a'_1, j'_2, c'_1, l''_1, k''_1, k'_2, l'_2, g''_1, h''_1, h'_2, g'_2, b'_1, j''_1, i''_1, i'_2
Mirror(A, E) → A'_2
Mirror(G', E) → G''_2
Mirror(H', E) → H''_2
Mirror(H, E) → H'_3
Mirror(G, E) → G'_3
Mirror(B, E) → B'_2
Mirror(J', E) → J''_2
Mirror(I', E) → I''_2
Mirror(I, E) → I'_3
Mirror(J, E) → J'_3
Mirror(C, E) → C'_2
Mirror(L', E) → L''_2
Mirror(K', E) → K''_2
Mirror(K, E) → K'_3
Mirror(L, E) → L'_3
Polygon(A'_2, G''_2, C'_2, L''_2, K''_2, K'_3, L'_3, H''_2, H'_3, G'_3, B'_2, J''_2, I''_2, I'_3, J'_3) → poly1'_2, a'_2, j'_3, c'_2, l''_2, k''_2, k'_3, l'_3, g''_2, h''_2, h'_3, g'_3, b'_2, j''_2, i''_2, i'_3
Mirror(A', D) → A''
Mirror(G'', D) → G'''
Mirror(H'', D) → H'''
Mirror(H'_1, D) → M
Mirror(G'_1, D) → N
Mirror(B', D) → B''
Mirror(J'', D) → J'''
Mirror(I'', D) → I'''
Mirror(I'_1, D) → O
Mirror(J'_1, D) → P
Mirror(C', D) → C''
Mirror(L'', D) → L'''
Mirror(K'', D) → K'''
Mirror(K'_1, D) → Q
Mirror(L'_1, D) → R
Polygon(A'', G''', C'', L''', K''', Q, R, H''', M, N, B'', J''', I''', O, P) → poly1'', a'', p, c'', l''', k''', q, r, g''', h''', m, n, b'', j''', i''', o
Mirror(A'', F) → A'''
Mirror(G''', F) → S
Mirror(H''', F) → T
Mirror(M, F) → M'
Mirror(N, F) → N'
Mirror(B'', F) → B'''
Mirror(J''', F) → U
Mirror(I''', F) → V
Mirror(O, F) → O'
Mirror(P, F) → P'
Mirror(C'', F) → C'''
Mirror(L''', F) → W
Mirror(K''', F) → Z
Mirror(Q, F) → Q'
Mirror(R, F) → R'
Polygon(A''', S, C''', W, Z, Q', R', T, M', N', B''', U, V, O', P') → poly1''', a''', p', c''', w, z_1, q', r', s, t, m', n', b''', u, v, o'
Mirror(A'_2, F) → A_1
Mirror(G''_2, F) → B_1
Mirror(H''_2, F) → C_1
Mirror(H'_3, F) → D_1
Mirror(G'_3, F) → E_1
Mirror(B'_2, F) → F_1
Mirror(J''_2, F) → G_1
Mirror(I''_2, F) → H_1
Mirror(I'_3, F) → I_1
Mirror(J'_3, F) → J_1
Mirror(C'_2, F) → K_1
Mirror(L''_2, F) → L_1
Mirror(K''_2, F) → M_1
Mirror(K'_3, F) → N_1
Mirror(L'_3, F) → O_1
Polygon(A_1, B_1, K_1, L_1, M_1, N_1, O_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1) → poly2, a_1, j_1, k_1, l_1, m_1, n_1, o_1, b_1, c_1, d_1, e_1, f_1, g_1, h_1, i_1
Mirror(A_1, D) → P_1
Mirror(B_1, D) → Q_1
Mirror(C_1, D) → R_1
Mirror(D_1, D) → S_1
Mirror(E_1, D) → T_1
Mirror(F_1, D) → U_1
Mirror(G_1, D) → V_1
Mirror(H_1, D) → W_1
Mirror(I_1, D) → Z_1
Mirror(J_1, D) → A_2
Mirror(K_1, D) → B_2
Mirror(L_1, D) → C_2
Mirror(M_1, D) → D_2
Mirror(N_1, D) → E_2
Mirror(O_1, D) → F_2
Polygon(P_1, Q_1, B_2, C_2, D_2, E_2, F_2, R_1, S_1, T_1, U_1, V_1, W_1, Z_1, A_2) → poly2', p_1, a_2, b_2, c_2, d_2, e_2, f_2, q_1, r_1, s_1, t_1, u_1, v_1, w_1, z_2
Mirror(A'', E) → A'''_1
Mirror(G''', E) → G_2
Mirror(H''', E) → H_2
Mirror(M, E) → M'_1
Mirror(N, E) → N'_1
Mirror(B'', E) → B'''_1
Mirror(J''', E) → I_2
Mirror(I''', E) → J_2
Mirror(O, E) → O'_1
Mirror(P, E) → P'_1
Mirror(C'', E) → C'''_1
Mirror(L''', E) → K_2
Mirror(K''', E) → L_2
Mirror(Q, E) → Q'_1
Mirror(R, E) → R'_1
Polygon(A'''_1, G_2, C'''_1, K_2, L_2, Q'_1, R'_1, H_2, M'_1, N'_1, B'''_1, I_2, J_2, O'_1, P'_1) → poly1'''_1, a'''_1, p'_1, c'''_1, k_2, l_2, q'_1, r'_1, g_2, h_2, m'_1, n'_1, b'''_1, i_2, j_2, o'_1
Mirror(A'_1, E) → M_2
Mirror(G''_1, E) → N_2
Mirror(H''_1, E) → O_2
Mirror(H'_2, E) → P_2
Mirror(G'_2, E) → Q_2
Mirror(B'_1, E) → R_2
Mirror(J''_1, E) → S_2
Mirror(I''_1, E) → T_2
Mirror(I'_2, E) → U_2
Mirror(J'_2, E) → V_2
Mirror(C'_1, E) → W_2
Mirror(L''_1, E) → Z_2
Mirror(K''_1, E) → A_3
Mirror(K'_2, E) → B_3
Mirror(L'_2, E) → C_3
Polygon(M_2, N_2, W_2, Z_2, A_3, B_3, C_3, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2) → poly3, m_2, v_2, w_2, z_3, a_3, b_3, c_3, n_2, o_2, p_2, q_2, r_2, s_2, t_2, u_2
Mirror(A'_1, F) → D_3
Mirror(G''_1, F) → E_3
Mirror(H''_1, F) → F_3
Mirror(H'_2, F) → G_3
Mirror(G'_2, F) → H_3
Mirror(B'_1, F) → I_3
Mirror(J''_1, F) → J_3
Mirror(I''_1, F) → K_3
Mirror(I'_2, F) → L_3
Mirror(J'_2, F) → M_3
Mirror(C'_1, F) → N_3
Mirror(L''_1, F) → O_3
Mirror(K''_1, F) → P_3
Mirror(K'_2, F) → Q_3
Mirror(L'_2, F) → R_3
Polygon(D_3, E_3, N_3, O_3, P_3, Q_3, R_3, F_3, G_3, H_3, I_3, J_3, K_3, L_3, M_3) → poly4, d_3, m_3, n_3, o_3, p_3, q_3, r_3, e_3, f_3, g_3, h_3, i_3, j_3, k_3, l_3
Mirror(A', E) → A''_1
Mirror(G'', E) → G'''_1
Mirror(H'', E) → H'''_1
Mirror(H'_1, E) → S_3
Mirror(G'_1, E) → T_3
Mirror(B', E) → B''_1
Mirror(J'', E) → J'''_1
Mirror(I'', E) → I'''_1
Mirror(I'_1, E) → U_3
Mirror(J'_1, E) → V_3
Mirror(C', E) → C''_1
Mirror(L'', E) → L'''_1
Mirror(K'', E) → K'''_1
Mirror(K'_1, E) → W_3
Mirror(L'_1, E) → Z_3
Polygon(A''_1, G'''_1, C''_1, L'''_1, K'''_1, W_3, Z_3, H'''_1, S_3, T_3, B''_1, J'''_1, I'''_1, U_3, V_3) → poly1''_1, a''_1, v_3, c''_1, l'''_1, k'''_1, w_3, z_4, g'''_1, h'''_1, s_3, t_3, b''_1, j'''_1, i'''_1, u_3
Mirror(A'_2, D) → A_4
Mirror(G''_2, D) → B_4
Mirror(H''_2, D) → C_4
Mirror(H'_3, D) → D_4
Mirror(G'_3, D) → E_4
Mirror(B'_2, D) → F_4
Mirror(J''_2, D) → G_4
Mirror(I''_2, D) → H_4
Mirror(I'_3, D) → I_4
Mirror(J'_3, D) → J_4
Mirror(C'_2, D) → K_4
Mirror(L''_2, D) → L_4
Mirror(K''_2, D) → M_4
Mirror(K'_3, D) → N_4
Mirror(L'_3, D) → O_4
Polygon(A_4, B_4, K_4, L_4, M_4, N_4, O_4, C_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4) → poly5, a_4, j_4, k_4, l_4, m_4, n_4, o_4, b_4, c_4, d_4, e_4, f_4, g_4, h_4, i_4
Mirror(A''_1, D) → P_4
Mirror(G'''_1, D) → Q_4
Mirror(H'''_1, D) → R_4
Mirror(S_3, D) → S_4
Mirror(T_3, D) → T_4
Mirror(B''_1, D) → U_4
Mirror(J'''_1, D) → V_4
Mirror(I'''_1, D) → W_4
Mirror(U_3, D) → Z_4
Mirror(V_3, D) → A_5
Mirror(C''_1, D) → B_5
Mirror(L'''_1, D) → C_5
Mirror(K'''_1, D) → D_5
Mirror(W_3, D) → E_5
Mirror(Z_3, D) → F_5
Polygon(P_4, Q_4, B_5, C_5, D_5, E_5, F_5, R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5) → poly6, p_4, a_5, b_5, c_5, d_5, e_5, f_5, q_4, r_4, s_4, t_4, u_4, v_4, w_4, z_5
Mirror(P_1, F) → G_5
Mirror(Q_1, F) → H_5
Mirror(R_1, F) → I_5
Mirror(S_1, F) → J_5
Mirror(T_1, F) → K_5
Mirror(U_1, F) → L_5
Mirror(V_1, F) → M_5
Mirror(W_1, F) → N_5
Mirror(Z_1, F) → O_5
Mirror(A_2, F) → P_5
Mirror(B_2, F) → Q_5
Mirror(C_2, F) → R_5
Mirror(D_2, F) → S_5
Mirror(E_2, F) → T_5
Mirror(F_2, F) → U_5
Polygon(G_5, H_5, Q_5, R_5, S_5, T_5, U_5, I_5, J_5, K_5, L_5, M_5, N_5, O_5, P_5) → poly2'', g_5, p_5, q_5, r_5, s_5, t_5, u_5, h_5, i_5, j_5, k_5, l_5, m_5, n_5, o_5
Mirror(A''_1, F) → V_5
Mirror(G'''_1, F) → W_5
Mirror(H'''_1, F) → Z_5
Mirror(S_3, F) → A_6
Mirror(T_3, F) → B_6
Mirror(B''_1, F) → C_6
Mirror(J'''_1, F) → D_6
Mirror(I'''_1, F) → E_6
Mirror(U_3, F) → F_6
Mirror(V_3, F) → G_6
Mirror(C''_1, F) → H_6
Mirror(L'''_1, F) → I_6
Mirror(K'''_1, F) → J_6
Mirror(W_3, F) → K_6
Mirror(Z_3, F) → L_6
Polygon(V_5, W_5, H_6, I_6, J_6, K_6, L_6, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, G_6) → poly7, v_5, g_6, h_6, i_6, j_6, k_6, l_6, w_5, z_6, a_6, b_6, c_6, d_6, e_6, f_6
Mirror(D_3, D) → M_6
Mirror(E_3, D) → N_6
Mirror(F_3, D) → O_6
Mirror(G_3, D) → P_6
Mirror(H_3, D) → Q_6
Mirror(I_3, D) → R_6
Mirror(J_3, D) → S_6
Mirror(K_3, D) → T_6
Mirror(L_3, D) → U_6
Mirror(M_3, D) → V_6
Mirror(N_3, D) → W_6
Mirror(O_3, D) → Z_6
Mirror(P_3, D) → A_7
Mirror(Q_3, D) → B_7
Mirror(R_3, D) → C_7
Polygon(M_6, N_6, W_6, Z_6, A_7, B_7, C_7, O_6, P_6, Q_6, R_6, S_6, T_6, U_6, V_6) → poly4', m_6, v_6, w_6, z_7, a_7, b_7, c_7, n_6, o_6, p_6, q_6, r_6, s_6, t_6, u_6
Mirror(A'''_1, D) → D_7
Mirror(G_2, D) → E_7
Mirror(H_2, D) → F_7
Mirror(M'_1, D) → G_7
Mirror(N'_1, D) → H_7
Mirror(B'''_1, D) → I_7
Mirror(I_2, D) → J_7
Mirror(J_2, D) → K_7
Mirror(O'_1, D) → L_7
Mirror(P'_1, D) → M_7
Mirror(C'''_1, D) → N_7
Mirror(K_2, D) → O_7
Mirror(L_2, D) → P_7
Mirror(Q'_1, D) → Q_7
Mirror(R'_1, D) → R_7
Polygon(D_7, E_7, N_7, O_7, P_7, Q_7, R_7, F_7, G_7, H_7, I_7, J_7, K_7, L_7, M_7) → poly8, d_7, m_7, n_7, o_7, p_7, q_7, r_7, e_7, f_7, g_7, h_7, i_7, j_7, k_7, l_7
Mirror(M_2, D) → S_7
Mirror(N_2, D) → T_7
Mirror(O_2, D) → U_7
Mirror(P_2, D) → V_7
Mirror(Q_2, D) → W_7
Mirror(R_2, D) → Z_7
Mirror(S_2, D) → A_8
Mirror(T_2, D) → B_8
Mirror(U_2, D) → C_8
Mirror(V_2, D) → D_8
Mirror(W_2, D) → E_8
Mirror(Z_2, D) → F_8
Mirror(A_3, D) → G_8
Mirror(B_3, D) → H_8
Mirror(C_3, D) → I_8
Polygon(S_7, T_7, E_8, F_8, G_8, H_8, I_8, U_7, V_7, W_7, Z_7, A_8, B_8, C_8, D_8) → poly3', s_7, d_8, e_8, f_8, g_8, h_8, i_8, t_7, u_7, v_7, w_7, z_8, a_8, b_8, c_8
Mirror(G_5, D) → J_8
Mirror(H_5, D) → K_8
Mirror(I_5, D) → L_8
Mirror(J_5, D) → M_8
Mirror(K_5, D) → N_8
Mirror(L_5, D) → O_8
Mirror(M_5, D) → P_8
Mirror(N_5, D) → Q_8
Mirror(O_5, D) → R_8
Mirror(P_5, D) → S_8
Mirror(Q_5, D) → T_8
Mirror(R_5, D) → U_8
Mirror(S_5, D) → V_8
Mirror(T_5, D) → W_8
Mirror(U_5, D) → Z_8
Polygon(J_8, K_8, T_8, U_8, V_8, W_8, Z_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8, S_8) → poly2''', j_8, s_8, t_8, u_8, v_8, w_8, z_9, k_8, l_8, m_8, n_8, o_8, p_8, q_8, r_8
Mirror(A''', D) → A_9
Mirror(S, D) → S'
Mirror(T, D) → T'
Mirror(M', D) → M''
Mirror(N', D) → N''
Mirror(B''', D) → B_9
Mirror(U, D) → U'
Mirror(V, D) → V'
Mirror(O', D) → O''
Mirror(P', D) → P''
Mirror(C''', D) → C_9
Mirror(W, D) → W'
Mirror(Z, D) → Z'
Mirror(Q', D) → Q''
Mirror(R', D) → R''
Polygon(A_9, S', C_9, W', Z', Q'', R'', T', M'', N'', B_9, U', V', O'', P'') → poly9, a_9, p'', c_9, w', z', q'', r'', s', t', m'', n'', b_9, u', v', o''
Mirror(V_5, D) → D_9
Mirror(W_5, D) → E_9
Mirror(Z_5, D) → F_9
Mirror(A_6, D) → G_9
Mirror(B_6, D) → H_9
Mirror(C_6, D) → I_9
Mirror(D_6, D) → J_9
Mirror(E_6, D) → K_9
Mirror(F_6, D) → L_9
Mirror(G_6, D) → M_9
Mirror(H_6, D) → N_9
Mirror(I_6, D) → O_9
Mirror(J_6, D) → P_9
Mirror(K_6, D) → Q_9
Mirror(L_6, D) → R_9
Polygon(D_9, E_9, N_9, O_9, P_9, Q_9, R_9, F_9, G_9, H_9, I_9, J_9, K_9, L_9, M_9) → poly7', d_9, m_9, n_9, o_9, p_9, q_9, r_9, e_9, f_9, g_9, h_9, i_9, j_9, k_9, l_9
Mirror(D_7, F) → S_9
Mirror(E_7, F) → T_9
Mirror(F_7, F) → U_9
Mirror(G_7, F) → V_9
Mirror(H_7, F) → W_9
Mirror(I_7, F) → Z_9
Mirror(J_7, F) → A_{10}
Mirror(K_7, F) → B_{10}
Mirror(L_7, F) → C_{10}
Mirror(M_7, F) → D_{10}
Mirror(N_7, F) → E_{10}
Mirror(O_7, F) → F_{10}
Mirror(P_7, F) → G_{10}
Mirror(Q_7, F) → H_{10}
Mirror(R_7, F) → I_{10}
Polygon(S_9, T_9, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, U_9, V_9, W_9, Z_9, A_{10}, B_{10}, C_{10}, D_{10}) → poly8', s_9, d_{10}, e_{10}, f_{10}, g_{10}, h_{10}, i_{10}, t_9, u_9, v_9, w_9, z_{10}, a_{10}, b_{10}, c_{10}
Mirror(M_6, F) → J_{10}
Mirror(N_6, F) → K_{10}
Mirror(O_6, F) → L_{10}
Mirror(P_6, F) → M_{10}
Mirror(Q_6, F) → N_{10}
Mirror(R_6, F) → O_{10}
Mirror(S_6, F) → P_{10}
Mirror(T_6, F) → Q_{10}
Mirror(U_6, F) → R_{10}
Mirror(V_6, F) → S_{10}
Mirror(W_6, F) → T_{10}
Mirror(Z_6, F) → U_{10}
Mirror(A_7, F) → V_{10}
Mirror(B_7, F) → W_{10}
Mirror(C_7, F) → Z_{10}
Polygon(J_{10}, K_{10}, T_{10}, U_{10}, V_{10}, W_{10}, Z_{10}, L_{10}, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, S_{10}) → poly4'', j_{10}, s_{10}, t_{10}, u_{10}, v_{10}, w_{10}, z_{11}, k_{10}, l_{10}, m_{10}, n_{10}, o_{10}, p_{10}, q_{10}, r_{10}
Mirror(S_7, F) → A_{11}
Mirror(T_7, F) → B_{11}
Mirror(U_7, F) → C_{11}
Mirror(V_7, F) → D_{11}
Mirror(W_7, F) → E_{11}
Mirror(Z_7, F) → F_{11}
Mirror(A_8, F) → G_{11}
Mirror(B_8, F) → H_{11}
Mirror(C_8, F) → I_{11}
Mirror(D_8, F) → J_{11}
Mirror(E_8, F) → K_{11}
Mirror(F_8, F) → L_{11}
Mirror(G_8, F) → M_{11}
Mirror(H_8, F) → N_{11}
Mirror(I_8, F) → O_{11}
Polygon(A_{11}, B_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}, H_{11}, I_{11}, J_{11}) → poly3'', a_{11}, j_{11}, k_{11}, l_{11}, m_{11}, n_{11}, o_{11}, b_{11}, c_{11}, d_{11}, e_{11}, f_{11}, g_{11}, h_{11}, i_{11}

AI 绘图提示语参考

绘制一个由三角形密铺构成的平面图案。先取一个任意三角形，连接三边中点，将原三角形分割为四个全等的小三角形。以各边中点为对称中心，将原三角形及其内部的小三角形分别进行镜像反射，生成多个相邻的三角形，使它们围绕原三角形的三个顶点和三条边对称排列。最终形成一个由多个全等三角形无缝拼接而成的密铺图形，覆盖整个平面区域，展示三角形通过平移、旋转和镜像变换形成的周期性镶嵌图案。

以上提示语为根据原始绘图命令整理而来，仅供参考。