Example 20a. Mosaics with Half-Squares

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2023-03-22 03:18

In collections Realidades de GeoGebra

This GeoGebra example demonstrates how to create mosaics using half-squares. Key mathematical concepts include geometric construction, reflection transformation, and square. Using commands like Polygon, Midpoint, Point, Ray, Intersect, Vector, Translate, and Rotate, complex mosaics can be constructed. Suitable for middle school and above students to learn geometric transformations and mosaic creation.

GeoGebra Commands & Parameters

Polygon(B, C, 4) → poly1, a, b, f, g, F, G
Polygon(D, E, 4) → poly2, c, d, e, i, H, I
Polygon(A, J, 4) → poly3, h, j, k, l, K, L
Midpoint(f) → M
Midpoint(b) → N
Midpoint(a) → O
Midpoint(g) → P
Midpoint(F, B) → Q
Polygon(M, Q, N, F) → poly4, m, q, n, f_1
Point(f_1) → Move
Midpoint(Q, F) → S
Ray(Move, S) → p
Intersect(q, p) → T
Vector(G, I) → u
Polygon(Move, T, Q, M) → poly5, r, t, q_1, m_1
Translate(Move, u) → R'
Translate(T, u) → T'
Translate(Q, u) → Q'
Translate(M, u) → M'
Polygon(R', T', Q', M') → poly5', r', t', q', m'
Rotate(R', 90°, Q') → R''
Rotate(T', 90°, Q') → T''
Rotate(Q', 90°, Q') → Q''
Rotate(M', 90°, Q') → M''
Polygon(R'', T'', Q'', M'') → poly5'', r'', t'', q'', m''
Rotate(R'', 90°, Q'') → R'''
Rotate(T'', 90°, Q'') → T'''
Rotate(Q'', 90°, Q'') → Q'''
Rotate(M'', 90°, Q'') → M'''
Polygon(R''', T''', Q''', M''') → poly5''', r''', t''', q''', m'''
Rotate(R''', 90°, Q''') → U
Rotate(T''', 90°, Q''') → V
Rotate(Q''', 90°, Q''') → W
Rotate(M''', 90°, Q''') → Z
Polygon(U, V, W, Z) → poly6, u_1, v, w, z
Midpoint(K, A) → A_1
Midpoint(k) → B_1
Midpoint(l) → C_1
Midpoint(h) → D_1
Midpoint(j) → E_1
Translate(R', u) → R''_1
Translate(R'', u) → R'''_1
Translate(R''', u) → F_1
Translate(U, u) → U'
Translate(T'', u) → T'''_1
Translate(T''', u) → G_1
Translate(V, u) → V'
Translate(T', u) → T''_1
Polygon(R''_1, B_1, E_1, T''_1, T'''_1, R'''_1, C_1, G_1, F_1, D_1, V', U') → poly7, r''_1, u', e_1, t''_1, b_1, t'''_1, r'''_1, c_1, g_1, f_2, d_1, v'
Vector(A, I_1) → s
Vector(I_1, J_1) → v_1
Vector(I_1, K_1) → w_1
Translate(R''_1, s) → L_1
Translate(B_1, s) → M_1
Translate(T'''_1, s) → N_1
Translate(R'''_1, s) → O_1
Translate(C_1, s) → P_1
Translate(G_1, s) → Q_1
Translate(F_1, s) → R_1
Translate(D_1, s) → S_1
Translate(V', s) → V''
Translate(U', s) → U''
Translate(E_1, s) → T_1
Translate(T''_1, s) → U_1
Polygon(L_1, M_1, T_1, U_1, N_1, O_1, P_1, Q_1, R_1, S_1, V'', U'') → poly7', l_1, u'', t_1, u_2, m_2, n_1, o_1, p_1, q_2, r_1, s_1, v''
Translate(L_1, v_1) → V_1
Translate(M_1, v_1) → W_1
Translate(N_1, v_1) → Z_1
Translate(O_1, v_1) → A_2
Translate(P_1, v_1) → B_2
Translate(Q_1, v_1) → C_2
Translate(R_1, v_1) → D_2
Translate(S_1, v_1) → E_2
Translate(V'', v_1) → V'''
Translate(U'', v_1) → U'''
Translate(T_1, v_1) → F_2
Translate(U_1, v_1) → G_2
Polygon(V_1, W_1, F_2, G_2, Z_1, A_2, B_2, C_2, D_2, E_2, V''', U''') → poly7'', v_2, u''', f_3, g_2, w_2, z_1, a_2, b_2, c_2, d_2, e_2, v'''
Translate(L_1, w_1) → H_2
Translate(M_1, w_1) → I_2
Translate(N_1, w_1) → J_2
Translate(O_1, w_1) → K_2
Translate(P_1, w_1) → L_2
Translate(Q_1, w_1) → M_2
Translate(R_1, w_1) → N_2
Translate(S_1, w_1) → O_2
Translate(V'', w_1) → V'''_1
Translate(U'', w_1) → U'''_1
Translate(T_1, w_1) → P_2
Translate(U_1, w_1) → Q_2
Polygon(H_2, I_2, P_2, Q_2, J_2, K_2, L_2, M_2, N_2, O_2, V'''_1, U'''_1) → poly7''_1, h_2, u'''_1, p_2, q_3, i_2, j_2, k_2, l_2, m_3, n_2, o_2, v'''_1
Translate(V_1, v_1) → R_2
Translate(W_1, v_1) → S_2
Translate(Z_1, v_1) → T_2
Translate(A_2, v_1) → U_2
Translate(B_2, v_1) → V_2
Translate(C_2, v_1) → W_2
Translate(D_2, v_1) → Z_2
Translate(E_2, v_1) → A_3
Translate(V''', v_1) → B_3
Translate(U''', v_1) → C_3
Translate(F_2, v_1) → D_3
Translate(G_2, v_1) → E_3
Polygon(R_2, S_2, D_3, E_3, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3) → poly7''', r_2, c_3, d_3, e_3, s_2, t_2, u_3, v_3, w_3, z_2, a_3, b_3
Translate(R_2, v_1) → F_3
Translate(S_2, v_1) → G_3
Translate(T_2, v_1) → H_3
Translate(U_2, v_1) → I_3
Translate(V_2, v_1) → J_3
Translate(W_2, v_1) → K_3
Translate(Z_2, v_1) → L_3
Translate(A_3, v_1) → M_3
Translate(B_3, v_1) → N_3
Translate(C_3, v_1) → O_3
Translate(D_3, v_1) → P_3
Translate(E_3, v_1) → Q_3
Polygon(F_3, G_3, P_3, Q_3, H_3, I_3, J_3, K_3, L_3, M_3, N_3, O_3) → poly8, f_4, o_3, p_3, q_4, g_3, h_3, i_3, j_3, k_3, l_3, m_4, n_3
Translate(M_3, v_1) → A_4
Translate(A_4, w_1) → M_4
Translate(F_3, w_1) → R_4
Translate(G_3, w_1) → S_4
Translate(H_3, w_1) → T_4
Translate(I_3, w_1) → U_4
Translate(J_3, w_1) → V_4
Translate(K_3, w_1) → W_4
Translate(L_3, w_1) → Z_4
Translate(M_3, w_1) → A_5
Translate(N_3, w_1) → B_5
Translate(O_3, w_1) → C_5
Translate(P_3, w_1) → D_5
Translate(Q_3, w_1) → E_5
Polygon(R_4, S_4, D_5, E_5, T_4, U_4, V_4, W_4, Z_4, A_5, B_5, C_5) → poly8'_1, r_4, c_5, d_5, e_5, s_4, t_4, u_5, v_5, w_5, z_4, a_5, b_5
Translate(R_2, w_1) → F_5
Translate(S_2, w_1) → G_5
Translate(T_2, w_1) → H_5
Translate(U_2, w_1) → I_5
Translate(V_2, w_1) → J_5
Translate(W_2, w_1) → K_5
Translate(Z_2, w_1) → L_5
Translate(A_3, w_1) → M_5
Translate(B_3, w_1) → N_5
Translate(C_3, w_1) → O_5
Translate(D_3, w_1) → P_5
Translate(E_3, w_1) → Q_5
Polygon(F_5, G_5, P_5, Q_5, H_5, I_5, J_5, K_5, L_5, M_5, N_5, O_5) → poly9, f_6, o_5, p_5, q_6, g_5, h_5, i_5, j_5, k_5, l_5, m_6, n_5
Translate(V_1, w_1) → R_5
Translate(W_1, w_1) → S_5
Translate(Z_1, w_1) → T_5
Translate(A_2, w_1) → U_5
Translate(B_2, w_1) → V_5
Translate(C_2, w_1) → W_5
Translate(D_2, w_1) → Z_5
Translate(E_2, w_1) → A_6
Translate(V''', w_1) → B_6
Translate(U''', w_1) → C_6
Translate(F_2, w_1) → D_6
Translate(G_2, w_1) → E_6
Polygon(R_5, S_5, D_6, E_6, T_5, U_5, V_5, W_5, Z_5, A_6, B_6, C_6) → poly7'''_1, r_5, c_6, d_6, e_6, s_5, t_5, u_6, v_6, w_6, z_5, a_6, b_6
Translate(R_5, w_1) → F_6
Translate(S_5, w_1) → G_6
Translate(T_5, w_1) → H_6
Translate(U_5, w_1) → I_6
Translate(V_5, w_1) → J_6
Translate(W_5, w_1) → K_6
Translate(Z_5, w_1) → L_6
Translate(A_6, w_1) → M_6
Translate(B_6, w_1) → N_6
Translate(C_6, w_1) → O_6
Translate(D_6, w_1) → P_6
Translate(E_6, w_1) → Q_6
Polygon(F_6, G_6, P_6, Q_6, H_6, I_6, J_6, K_6, L_6, M_6, N_6, O_6) → poly10, f_7, o_6, p_6, q_7, g_6, h_6, i_6, j_6, k_6, l_6, m_7, n_6
Translate(H_2, w_1) → R_6
Translate(I_2, w_1) → S_6
Translate(J_2, w_1) → T_6
Translate(K_2, w_1) → U_6
Translate(L_2, w_1) → V_6
Translate(M_2, w_1) → W_6
Translate(N_2, w_1) → Z_6
Translate(O_2, w_1) → A_7
Translate(V'''_1, w_1) → B_7
Translate(U'''_1, w_1) → C_7
Translate(P_2, w_1) → D_7
Translate(Q_2, w_1) → E_7
Polygon(R_6, S_6, D_7, E_7, T_6, U_6, V_6, W_6, Z_6, A_7, B_7, C_7) → poly11, r_6, c_7, d_7, e_7, s_6, t_6, u_7, v_7, w_7, z_6, a_7, b_7
Translate(R_6, w_1) → F_7
Translate(S_6, w_1) → G_7
Translate(T_6, w_1) → H_7
Translate(U_6, w_1) → I_7
Translate(V_6, w_1) → J_7
Translate(W_6, w_1) → K_7
Translate(Z_6, w_1) → L_7
Translate(A_7, w_1) → M_7
Translate(B_7, w_1) → N_7
Translate(C_7, w_1) → O_7
Translate(D_7, w_1) → P_7
Translate(E_7, w_1) → Q_7
Polygon(F_7, G_7, P_7, Q_7, H_7, I_7, J_7, K_7, L_7, M_7, N_7, O_7) → poly11', f_8, o_7, p_7, q_8, g_7, h_7, i_7, j_7, k_7, l_7, m_8, n_7
Translate(F_6, w_1) → R_7
Translate(G_6, w_1) → S_7
Translate(H_6, w_1) → T_7
Translate(I_6, w_1) → U_7
Translate(J_6, w_1) → V_7
Translate(K_6, w_1) → W_7
Translate(L_6, w_1) → Z_7
Translate(M_6, w_1) → A_8
Translate(N_6, w_1) → B_8
Translate(O_6, w_1) → C_8
Translate(P_6, w_1) → D_8
Translate(Q_6, w_1) → E_8
Polygon(R_7, S_7, D_8, E_8, T_7, U_7, V_7, W_7, Z_7, A_8, B_8, C_8) → poly10', r_7, c_8, d_8, e_8, s_7, t_7, u_8, v_8, w_8, z_7, a_8, b_8
Translate(F_7, w_1) → F_8
Translate(G_7, w_1) → G_8
Translate(H_7, w_1) → H_8
Translate(I_7, w_1) → I_8
Translate(J_7, w_1) → J_8
Translate(K_7, w_1) → K_8
Translate(L_7, w_1) → L_8
Translate(M_7, w_1) → M_8
Translate(N_7, w_1) → N_8
Translate(O_7, w_1) → O_8
Translate(P_7, w_1) → P_8
Translate(Q_7, w_1) → Q_8
Polygon(F_8, G_8, P_8, Q_8, H_8, I_8, J_8, K_8, L_8, M_8, N_8, O_8) → poly11'', f_9, o_8, p_8, q_9, g_8, h_8, i_8, j_8, k_8, l_8, m_9, n_8
Translate(F_8, w_1) → R_8
Translate(G_8, w_1) → S_8
Translate(H_8, w_1) → T_8
Translate(I_8, w_1) → U_8
Translate(J_8, w_1) → V_8
Translate(K_8, w_1) → W_8
Translate(L_8, w_1) → Z_8
Translate(M_8, w_1) → A_9
Translate(N_8, w_1) → B_9
Translate(O_8, w_1) → C_9
Translate(P_8, w_1) → D_9
Translate(Q_8, w_1) → E_9
Polygon(R_8, S_8, D_9, E_9, T_8, U_8, V_8, W_8, Z_8, A_9, B_9, C_9) → poly11''', r_8, c_9, d_9, e_9, s_8, t_8, u_9, v_9, w_9, z_8, a_9, b_9
Translate(R_7, w_1) → F_9
Translate(S_7, w_1) → G_9
Translate(T_7, w_1) → H_9
Translate(U_7, w_1) → I_9
Translate(V_7, w_1) → J_9
Translate(W_7, w_1) → K_9
Translate(Z_7, w_1) → L_9
Translate(A_8, w_1) → M_9
Translate(B_8, w_1) → N_9
Translate(C_8, w_1) → O_9
Translate(D_8, w_1) → P_9
Translate(E_8, w_1) → Q_9
Polygon(F_9, G_9, P_9, Q_9, H_9, I_9, J_9, K_9, L_9, M_9, N_9, O_9) → poly10'', f_{10}, o_9, p_9, q_{10}, g_9, h_9, i_9, j_9, k_9, l_9, m_{10}, n_9
Translate(F_9, w_1) → R_9
Translate(G_9, w_1) → S_9
Translate(H_9, w_1) → T_9
Translate(I_9, w_1) → U_9
Translate(J_9, w_1) → V_9
Translate(K_9, w_1) → W_9
Translate(L_9, w_1) → Z_9
Translate(M_9, w_1) → A_{10}
Translate(N_9, w_1) → B_{10}
Translate(O_9, w_1) → C_{10}
Translate(P_9, w_1) → D_{10}
Translate(Q_9, w_1) → E_{10}
Polygon(R_9, S_9, D_{10}, E_{10}, T_9, U_9, V_9, W_9, Z_9, A_{10}, B_{10}, C_{10}) → poly10''', r_9, c_{10}, d_{10}, e_{10}, s_9, t_9, u_{10}, v_{10}, w_{10}, z_9, a_{10}, b_{10}
Translate(F_5, w_1) → F_{10}
Translate(G_5, w_1) → G_{10}
Translate(H_5, w_1) → H_{10}
Translate(I_5, w_1) → I_{10}
Translate(J_5, w_1) → J_{10}
Translate(K_5, w_1) → K_{10}
Translate(L_5, w_1) → L_{10}
Translate(M_5, w_1) → M_{10}
Translate(N_5, w_1) → N_{10}
Translate(O_5, w_1) → O_{10}
Translate(P_5, w_1) → P_{10}
Translate(Q_5, w_1) → Q_{10}
Polygon(F_{10}, G_{10}, P_{10}, Q_{10}, H_{10}, I_{10}, J_{10}, K_{10}, L_{10}, M_{10}, N_{10}, O_{10}) → poly9', f_{11}, o_{10}, p_{10}, q_{11}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}, m_{11}, n_{10}
Translate(F_{10}, w_1) → S_{10}
Translate(G_{10}, w_1) → T_{10}
Translate(H_{10}, w_1) → U_{10}
Translate(I_{10}, w_1) → V_{10}
Translate(J_{10}, w_1) → W_{10}
Translate(K_{10}, w_1) → Z_{10}
Translate(L_{10}, w_1) → A_{11}
Translate(M_{10}, w_1) → B_{11}
Translate(N_{10}, w_1) → C_{11}
Translate(O_{10}, w_1) → D_{11}
Translate(P_{10}, w_1) → E_{11}
Translate(Q_{10}, w_1) → F_{11}
Polygon(S_{10}, T_{10}, E_{11}, F_{11}, U_{10}, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}) → poly9'', s_{10}, d_{11}, e_{11}, f_{12}, t_{10}, u_{11}, v_{11}, w_{11}, z_{10}, a_{11}, b_{11}, c_{11}
Translate(S_{10}, w_1) → G_{11}
Translate(T_{10}, w_1) → H_{11}
Translate(U_{10}, w_1) → I_{11}
Translate(V_{10}, w_1) → J_{11}
Translate(W_{10}, w_1) → K_{11}
Translate(Z_{10}, w_1) → L_{11}
Translate(A_{11}, w_1) → M_{11}
Translate(B_{11}, w_1) → N_{11}
Translate(C_{11}, w_1) → O_{11}
Translate(D_{11}, w_1) → P_{11}
Translate(E_{11}, w_1) → Q_{11}
Translate(F_{11}, w_1) → R_{11}
Polygon(G_{11}, H_{11}, Q_{11}, R_{11}, I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}) → poly9''', g_{11}, p_{11}, q_{12}, r_{11}, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, m_{12}, n_{11}, o_{11}
Translate(G_{11}, w_1) → S_{11}
Translate(H_{11}, w_1) → T_{11}
Translate(I_{11}, w_1) → U_{11}
Translate(J_{11}, w_1) → V_{11}
Translate(K_{11}, w_1) → W_{11}
Translate(L_{11}, w_1) → Z_{11}
Translate(M_{11}, w_1) → A_{12}
Translate(N_{11}, w_1) → B_{12}
Translate(O_{11}, w_1) → C_{12}
Translate(P_{11}, w_1) → D_{12}
Translate(Q_{11}, w_1) → E_{12}
Translate(R_{11}, w_1) → F_{12}
Polygon(S_{11}, T_{11}, E_{12}, F_{12}, U_{11}, V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, D_{12}) → poly12, s_{11}, d_{12}, e_{12}, f_{13}, t_{11}, u_{12}, v_{12}, w_{12}, z_{11}, a_{12}, b_{12}, c_{12}
Translate(R_4, w_1) → R_3
Translate(S_4, w_1) → S_3
Translate(T_4, w_1) → T_3
Translate(U_4, w_1) → U_3
Translate(V_4, w_1) → V_3
Translate(W_4, w_1) → W_3
Translate(Z_4, w_1) → Z_3
Translate(A_5, w_1) → B_4
Translate(B_5, w_1) → C_4
Translate(C_5, w_1) → D_4
Translate(D_5, w_1) → E_4
Translate(E_5, w_1) → F_4
Polygon(R_3, S_3, E_4, F_4, T_3, U_3, V_3, W_3, Z_3, B_4, C_4, D_4) → poly13, r_3, d_4, e_4, f_5, s_3, t_3, u_4, v_4, w_4, z_3, b_4, c_4
Translate(R_3, w_1) → G_4
Translate(S_3, w_1) → H_4
Translate(T_3, w_1) → I_4
Translate(U_3, w_1) → J_4
Translate(V_3, w_1) → K_4
Translate(W_3, w_1) → L_4
Translate(Z_3, w_1) → N_4
Translate(B_4, w_1) → O_4
Translate(C_4, w_1) → P_4
Translate(D_4, w_1) → Q_4
Translate(E_4, w_1) → G_{12}
Translate(F_4, w_1) → H_{12}
Polygon(G_4, H_4, G_{12}, H_{12}, I_4, J_4, K_4, L_4, N_4, O_4, P_4, Q_4) → poly13', g_4, q_5, g_{12}, h_{12}, h_4, i_4, j_4, k_4, l_4, n_4, o_4, p_4
Translate(G_4, w_1) → I_{12}
Translate(H_4, w_1) → J_{12}
Translate(I_4, w_1) → K_{12}
Translate(J_4, w_1) → L_{12}
Translate(K_4, w_1) → M_{12}
Translate(L_4, w_1) → N_{12}
Translate(N_4, w_1) → O_{12}
Translate(O_4, w_1) → P_{12}
Translate(P_4, w_1) → Q_{12}
Translate(Q_4, w_1) → R_{12}
Translate(G_{12}, w_1) → S_{12}
Translate(H_{12}, w_1) → T_{12}
Polygon(I_{12}, J_{12}, S_{12}, T_{12}, K_{12}, L_{12}, M_{12}, N_{12}, O_{12}, P_{12}, Q_{12}, R_{12}) → poly13'', i_{12}, r_{12}, s_{12}, t_{12}, j_{12}, k_{12}, l_{12}, m_5, n_{12}, o_{12}, p_{12}, q_{13}
Translate(I_{12}, w_1) → U_{12}
Translate(J_{12}, w_1) → V_{12}
Translate(K_{12}, w_1) → W_{12}
Translate(L_{12}, w_1) → Z_{12}
Translate(M_{12}, w_1) → A_{13}
Translate(N_{12}, w_1) → B_{13}
Translate(O_{12}, w_1) → C_{13}
Translate(P_{12}, w_1) → D_{13}
Translate(Q_{12}, w_1) → E_{13}
Translate(R_{12}, w_1) → F_{13}
Translate(S_{12}, w_1) → G_{13}
Translate(T_{12}, w_1) → H_{13}
Polygon(U_{12}, V_{12}, G_{13}, H_{13}, W_{12}, Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}) → poly13''', u_{13}, f_{14}, g_{13}, h_{13}, v_{13}, w_{13}, z_{12}, a_{13}, b_{13}, c_{13}, d_{13}, e_{13}

AI Drawing Prompt Reference

This is a mosaic made of half-squares. The mosaic consists of squares connected through reflection transformations. To construct it, first draw several squares. Then find the centers of these squares. Next, use these centers as starting points and move along a vector to create new square positions. Finally, combine these new positions with the original squares to form the complex mosaic. The sizes and positions of the squares need careful adjustment to ensure a beautiful mosaic effect.

The prompt above was compiled from the original drawing commands and is for reference only.