Tessellations Using Reflection

Loading the math board and drawing, please wait…

2017-08-03 06:28

In collections Tessellations Using Reflections, Rotations, and Translations

This example demonstrates how to create tessellations using reflection. It involves concepts of trigonometric functions and geometric transformations. Using GeoGebra commands such as Polygon, Line, Segment, and Mirror, complex tessellations can be constructed. Suitable for high school mathematics education and students learning geometric transformations.

GeoGebra Commands & Parameters

Polygon(A, B, K, L, M, N, O, P, Q, R, S, T, C, U, V, W, Z, D, E, F, G, H, I, J) → poly1, a, j, k, l, m, n, o, p, q, r, s, b, t, u, v, w, z, c, d, e, f, g, h, i
Polygon(A_1, B_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, C_1, U_1, V_1, W_1, Z_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1) → poly1_1, a_1, j_1, k_1, l_1, m_1, n_1, o_1, p_1, q_1, r_1, s_1, b_1, t_1, u_1, v_1, w_1, z_1, c_1, d_1, e_1, f_1, g_1, h_1, i_1
Polygon(A_2, B_2, K_2, L_2, M_2, N_2, O_2, P_2, Q_2, R_2, S_2, T_2, C_2, U_2, V_2, W_2, Z_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2) → poly1_2, a_2, j_2, k_2, l_2, m_2, n_2, o_2, p_2, q_2, r_2, s_2, b_2, t_2, u_2, v_2, w_2, z_2, c_2, d_2, e_2, f_2, g_2, h_2, i_2
Polygon(A_3, B_3, C_3, D_3, E_3, F_3, G_3) → poly2, a_3, b_3, c_3, d_3, e_3, f_3, g_3
Line(H_3, I_3) → h_3
Polygon(A_4, B_4, K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, C_4, U_3, V_3, W_3, Z_3, D_4, E_4, F_4, G_4, H_4, I_4, J_3) → poly1_3, a_4, j_3, k_3, l_3, m_3, n_3, o_3, p_3, q_3, r_3, s_3, b_4, t_3, u_3, v_3, w_3, z_3, c_4, d_4, e_4, f_4, g_4, h_4, i_3
Segment(J_4, K_4) → i_4
Mirror(A_4, i_4) → L_4
Mirror(B_4, i_4) → M_4
Mirror(C_4, i_4) → N_4
Mirror(D_4, i_4) → O_4
Mirror(E_4, i_4) → P_4
Mirror(F_4, i_4) → Q_4
Mirror(G_4, i_4) → R_4
Mirror(H_4, i_4) → S_4
Mirror(I_4, i_4) → T_4
Mirror(J_3, i_4) → U_4
Mirror(K_3, i_4) → V_4
Mirror(L_3, i_4) → W_4
Mirror(M_3, i_4) → Z_4
Mirror(N_3, i_4) → A_5
Mirror(O_3, i_4) → B_5
Mirror(P_3, i_4) → C_5
Mirror(Q_3, i_4) → D_5
Mirror(R_3, i_4) → E_5
Mirror(S_3, i_4) → F_5
Mirror(T_3, i_4) → G_5
Mirror(U_3, i_4) → H_5
Mirror(V_3, i_4) → I_5
Mirror(W_3, i_4) → J_5
Mirror(Z_3, i_4) → K_5
Polygon(L_4, M_4, V_4, W_4, Z_4, A_5, B_5, C_5, D_5, E_5, F_5, G_5, N_4, H_5, I_5, J_5, K_5, O_4, P_4, Q_4, R_4, S_4, T_4, U_4) → poly3, l_4, u_4, v_4, w_4, z_4, a_5, b_5, c_5, d_5, e_5, f_5, m_4, g_5, h_5, i_5, j_5, k_5, n_4, o_4, p_4, q_4, r_4, s_4, t_4
Polygon(A_8, B_8, C_8, D_8, E_8, F_8, G_8) → poly2_1, a_8, b_8, c_8, d_8, e_8, f_8, g_8
Line(P_7, Q_7) → l_5
Polygon(A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}, G_{13}) → poly2_2, a_9, b_{13}, c_9, d_{13}, e_{13}, f_9, g_{13}
Segment(J_9, K_9) → i_9
Segment(L_{10}, M_{10}) → j_{10}
Segment(N_{11}, O_{11}) → k_{11}
Polygon(A_6, B_6, K_6, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, S_5, T_5, C_6, U_5, V_5, W_5, Z_5, D_6, E_6, F_6, G_6, H_6, I_6, J_6) → poly1_4, a_6, j_4, k_4, l_6, m_5, n_5, o_5, p_5, q_5, r_5, s_5, b_6, t_5, u_5, v_5, w_5, z_5, c_6, d_6, e_6, f_6, g_6, h_6, i_6
Segment(J_7, K_7) → i_7
Mirror(A_6, i_7) → L_6
Mirror(B_6, i_7) → M_6
Mirror(C_6, i_7) → N_6
Mirror(D_6, i_7) → O_6
Mirror(E_6, i_7) → P_6
Mirror(F_6, i_7) → Q_6
Mirror(G_6, i_7) → R_6
Mirror(H_6, i_7) → S_6
Mirror(I_6, i_7) → T_6
Mirror(J_6, i_7) → U_6
Mirror(K_6, i_7) → V_6
Mirror(L_5, i_7) → W_6
Mirror(M_5, i_7) → Z_6
Mirror(N_5, i_7) → A_7
Mirror(O_5, i_7) → B_7
Mirror(P_5, i_7) → C_7
Mirror(Q_5, i_7) → D_7
Mirror(R_5, i_7) → E_7
Mirror(S_5, i_7) → F_7
Mirror(T_5, i_7) → G_7
Mirror(U_5, i_7) → H_7
Mirror(V_5, i_7) → I_7
Mirror(W_5, i_7) → J_8
Mirror(Z_5, i_7) → K_8
Polygon(L_6, M_6, V_6, W_6, Z_6, A_7, B_7, C_7, D_7, E_7, F_7, G_7, N_6, H_7, I_7, J_8, K_8, O_6, P_6, Q_6, R_6, S_6, T_6, U_6) → poly3_1, l_7, u_6, v_6, w_6, z_6, a_7, b_7, c_7, d_7, e_7, f_7, m_6, g_7, h_7, i_8, j_6, k_6, n_6, o_6, p_6, q_6, r_6, s_6, t_6
Polygon(A_9, B_{10}, C_9, D_9, E_9, F_9, G_9) → poly2_3, a_{10}, b_9, c_{10}, d_9, e_9, f_{10}, g_9
Line(L_7, M_7) → j_7
Polygon(A_{10}, B_{11}, K_{10}, L_8, M_8, N_7, O_7, P_8, Q_8, R_7, S_7, T_7, C_{10}, U_7, V_7, W_7, Z_7, D_{10}, E_{10}, F_{10}, G_{10}, H_8, I_8, J_{10}) → poly1_5, a_{11}, j_8, k_7, l_8, m_7, n_7, o_7, p_7, q_7, r_7, s_7, b_{10}, t_7, u_7, v_7, w_7, z_7, c_{11}, d_{10}, e_{10}, f_{11}, g_{10}, h_8, i_{10}
Segment(J_{12}, K_{12}) → i_{12}
Mirror(A_{10}, i_{12}) → L_9
Mirror(B_{11}, i_{12}) → M_9
Mirror(C_{10}, i_{12}) → N_8
Mirror(D_{10}, i_{12}) → O_8
Mirror(E_{10}, i_{12}) → P_9
Mirror(F_{10}, i_{12}) → Q_9
Mirror(G_{10}, i_{12}) → R_8
Mirror(H_8, i_{12}) → S_8
Mirror(I_8, i_{12}) → T_8
Mirror(J_{10}, i_{12}) → U_8
Mirror(K_{10}, i_{12}) → V_8
Mirror(L_8, i_{12}) → W_8
Mirror(M_8, i_{12}) → Z_8
Mirror(N_7, i_{12}) → A_{11}
Mirror(O_7, i_{12}) → B_{12}
Mirror(P_8, i_{12}) → C_{11}
Mirror(Q_8, i_{12}) → D_{11}
Mirror(R_7, i_{12}) → E_{11}
Mirror(S_7, i_{12}) → F_{11}
Mirror(T_7, i_{12}) → G_{11}
Mirror(U_7, i_{12}) → H_9
Mirror(V_7, i_{12}) → I_9
Mirror(W_7, i_{12}) → J_{11}
Mirror(Z_7, i_{12}) → K_{11}
Polygon(L_9, M_9, V_8, W_8, Z_8, A_{11}, B_{12}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}, N_8, H_9, I_9, J_{11}, K_{11}, O_8, P_9, Q_9, R_8, S_8, T_8, U_8) → poly3_2, l_9, u_8, v_8, w_8, z_8, a_{12}, b_{11}, c_{12}, d_{11}, e_{11}, f_{12}, m_8, g_{11}, h_9, i_{11}, j_9, k_8, n_8, o_8, p_8, q_8, r_8, s_8, t_8
Polygon(A_{12}, B_{15}, K_{13}, L_{11}, M_{11}, N_9, O_9, P_{10}, Q_{10}, R_9, S_9, T_9, C_{12}, U_9, V_9, W_9, Z_9, D_{12}, E_{12}, F_{12}, G_{12}, H_{10}, I_{10}, J_{13}) → poly1_6, a_{13}, j_{11}, k_9, l_{10}, m_9, n_9, o_9, p_9, q_9, r_9, s_9, b_{12}, t_9, u_9, v_9, w_9, z_9, c_{13}, d_{12}, e_{12}, f_{13}, g_{12}, h_{10}, i_{13}
Segment(J_{15}, K_{15}) → i_{15}
Mirror(A_{12}, i_{15}) → L_{12}
Mirror(B_{15}, i_{15}) → M_{12}
Mirror(C_{12}, i_{15}) → N_{10}
Mirror(D_{12}, i_{15}) → O_{10}
Mirror(E_{12}, i_{15}) → P_{11}
Mirror(F_{12}, i_{15}) → Q_{11}
Mirror(G_{12}, i_{15}) → R_{10}
Mirror(H_{10}, i_{15}) → S_{10}
Mirror(I_{10}, i_{15}) → T_{10}
Mirror(J_{13}, i_{15}) → U_{10}
Mirror(K_{13}, i_{15}) → V_{10}
Mirror(L_{11}, i_{15}) → W_{10}
Mirror(M_{11}, i_{15}) → Z_{10}
Mirror(N_9, i_{15}) → A_{14}
Mirror(O_9, i_{15}) → B_{16}
Mirror(P_{10}, i_{15}) → C_{14}
Mirror(Q_{10}, i_{15}) → D_{14}
Mirror(R_9, i_{15}) → E_{14}
Mirror(S_9, i_{15}) → F_{14}
Mirror(T_9, i_{15}) → G_{14}
Mirror(U_9, i_{15}) → H_{11}
Mirror(V_9, i_{15}) → I_{11}
Mirror(W_9, i_{15}) → J_{14}
Mirror(Z_9, i_{15}) → K_{14}
Polygon(L_{12}, M_{12}, V_{10}, W_{10}, Z_{10}, A_{14}, B_{16}, C_{14}, D_{14}, E_{14}, F_{14}, G_{14}, N_{10}, H_{11}, I_{11}, J_{14}, K_{14}, O_{10}, P_{11}, Q_{11}, R_{10}, S_{10}, T_{10}, U_{10}) → poly3_3, l_{11}, u_{10}, v_{10}, w_{10}, z_{10}, a_{14}, b_{14}, c_{14}, d_{14}, e_{14}, f_{14}, m_{10}, g_{14}, h_{11}, i_{14}, j_{12}, k_{10}, n_{10}, o_{10}, p_{10}, q_{10}, r_{10}, s_{10}, t_{10}
Polygon(A_{15}, B_{18}, K_{16}, L_{13}, M_{13}, N_{12}, O_{12}, P_{12}, Q_{12}, R_{11}, S_{11}, T_{11}, C_{15}, U_{11}, V_{11}, W_{11}, Z_{11}, D_{15}, E_{15}, F_{15}, G_{15}, H_{12}, I_{12}, J_{16}) → poly1_7, a_{15}, j_{13}, k_{12}, l_{12}, m_{11}, n_{11}, o_{11}, p_{11}, q_{11}, r_{11}, s_{11}, b_{15}, t_{11}, u_{11}, v_{11}, w_{11}, z_{11}, c_{15}, d_{15}, e_{15}, f_{15}, g_{15}, h_{12}, i_{16}
Segment(J_{18}, K_{18}) → i_{18}
Mirror(A_{15}, i_{18}) → L_{14}
Mirror(B_{18}, i_{18}) → M_{14}
Mirror(C_{15}, i_{18}) → N_{13}
Mirror(D_{15}, i_{18}) → O_{13}
Mirror(E_{15}, i_{18}) → P_{13}
Mirror(F_{15}, i_{18}) → Q_{13}
Mirror(G_{15}, i_{18}) → R_{12}
Mirror(H_{12}, i_{18}) → S_{12}
Mirror(I_{12}, i_{18}) → T_{12}
Mirror(J_{16}, i_{18}) → U_{12}
Mirror(K_{16}, i_{18}) → V_{12}
Mirror(L_{13}, i_{18}) → W_{12}
Mirror(M_{13}, i_{18}) → Z_{12}
Mirror(N_{12}, i_{18}) → A_{16}
Mirror(O_{12}, i_{18}) → B_{19}
Mirror(P_{12}, i_{18}) → C_{16}
Mirror(Q_{12}, i_{18}) → D_{16}
Mirror(R_{11}, i_{18}) → E_{16}
Mirror(S_{11}, i_{18}) → F_{16}
Mirror(T_{11}, i_{18}) → G_{16}
Mirror(U_{11}, i_{18}) → H_{13}
Mirror(V_{11}, i_{18}) → I_{13}
Mirror(W_{11}, i_{18}) → J_{17}
Mirror(Z_{11}, i_{18}) → K_{17}
Polygon(L_{14}, M_{14}, V_{12}, W_{12}, Z_{12}, A_{16}, B_{19}, C_{16}, D_{16}, E_{16}, F_{16}, G_{16}, N_{13}, H_{13}, I_{13}, J_{17}, K_{17}, O_{13}, P_{13}, Q_{13}, R_{12}, S_{12}, T_{12}, U_{12}) → poly3_4, l_{13}, u_{12}, v_{12}, w_{12}, z_{12}, a_{16}, b_{16}, c_{16}, d_{16}, e_{16}, f_{16}, m_{12}, g_{16}, h_{13}, i_{17}, j_{14}, k_{13}, n_{12}, o_{12}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}
Polygon(A_{17}, B_{21}, K_{19}, L_{15}, M_{15}, N_{14}, O_{14}, P_{14}, Q_{14}, R_{13}, S_{13}, T_{13}, C_{17}, U_{13}, V_{13}, W_{13}, Z_{13}, D_{17}, E_{17}, F_{17}, G_{17}, H_{14}, I_{14}, J_{19}) → poly1_8, a_{17}, j_{15}, k_{14}, l_{14}, m_{13}, n_{13}, o_{13}, p_{13}, q_{13}, r_{13}, s_{13}, b_{17}, t_{13}, u_{13}, v_{13}, w_{13}, z_{13}, c_{17}, d_{17}, e_{17}, f_{17}, g_{17}, h_{14}, i_{19}
Segment(J_{21}, K_{21}) → i_{21}
Mirror(A_{17}, i_{21}) → L_{16}
Mirror(B_{21}, i_{21}) → M_{16}
Mirror(C_{17}, i_{21}) → N_{15}
Mirror(D_{17}, i_{21}) → O_{15}
Mirror(E_{17}, i_{21}) → P_{15}
Mirror(F_{17}, i_{21}) → Q_{15}
Mirror(G_{17}, i_{21}) → R_{14}
Mirror(H_{14}, i_{21}) → S_{14}
Mirror(I_{14}, i_{21}) → T_{14}
Mirror(J_{19}, i_{21}) → U_{14}
Mirror(K_{19}, i_{21}) → V_{14}
Mirror(L_{15}, i_{21}) → W_{14}
Mirror(M_{15}, i_{21}) → Z_{14}
Mirror(N_{14}, i_{21}) → A_{18}
Mirror(O_{14}, i_{21}) → B_{22}
Mirror(P_{14}, i_{21}) → C_{18}
Mirror(Q_{14}, i_{21}) → D_{18}
Mirror(R_{13}, i_{21}) → E_{18}
Mirror(S_{13}, i_{21}) → F_{18}
Mirror(T_{13}, i_{21}) → G_{18}
Mirror(U_{13}, i_{21}) → H_{15}
Mirror(V_{13}, i_{21}) → I_{15}
Mirror(W_{13}, i_{21}) → J_{20}
Mirror(Z_{13}, i_{21}) → K_{20}
Polygon(L_{16}, M_{16}, V_{14}, W_{14}, Z_{14}, A_{18}, B_{22}, C_{18}, D_{18}, E_{18}, F_{18}, G_{18}, N_{15}, H_{15}, I_{15}, J_{20}, K_{20}, O_{15}, P_{15}, Q_{15}, R_{14}, S_{14}, T_{14}, U_{14}) → poly3_5, l_{15}, u_{14}, v_{14}, w_{14}, z_{14}, a_{18}, b_{18}, c_{18}, d_{18}, e_{18}, f_{18}, m_{14}, g_{18}, h_{15}, i_{20}, j_{16}, k_{15}, n_{14}, o_{14}, p_{14}, q_{14}, r_{14}, s_{14}, t_{14}
Polygon(A_{19}, B_{24}, K_{22}, L_{17}, M_{17}, N_{16}, O_{16}, P_{16}, Q_{16}, R_{15}, S_{15}, T_{15}, C_{19}, U_{15}, V_{15}, W_{15}, Z_{15}, D_{19}, E_{19}, F_{19}, G_{19}, H_{16}, I_{16}, J_{22}) → poly1_9, a_{19}, j_{17}, k_{16}, l_{16}, m_{15}, n_{15}, o_{15}, p_{15}, q_{15}, r_{15}, s_{15}, b_{19}, t_{15}, u_{15}, v_{15}, w_{15}, z_{15}, c_{19}, d_{19}, e_{19}, f_{19}, g_{19}, h_{16}, i_{22}
Segment(J_{24}, K_{24}) → i_{24}
Mirror(A_{19}, i_{24}) → L_{18}
Mirror(B_{24}, i_{24}) → M_{18}
Mirror(C_{19}, i_{24}) → N_{17}
Mirror(D_{19}, i_{24}) → O_{17}
Mirror(E_{19}, i_{24}) → P_{17}
Mirror(F_{19}, i_{24}) → Q_{17}
Mirror(G_{19}, i_{24}) → R_{16}
Mirror(H_{16}, i_{24}) → S_{16}
Mirror(I_{16}, i_{24}) → T_{16}
Mirror(J_{22}, i_{24}) → U_{16}
Mirror(K_{22}, i_{24}) → V_{16}
Mirror(L_{17}, i_{24}) → W_{16}
Mirror(M_{17}, i_{24}) → Z_{16}
Mirror(N_{16}, i_{24}) → A_{20}
Mirror(O_{16}, i_{24}) → B_{25}
Mirror(P_{16}, i_{24}) → C_{20}
Mirror(Q_{16}, i_{24}) → D_{20}
Mirror(R_{15}, i_{24}) → E_{20}
Mirror(S_{15}, i_{24}) → F_{20}
Mirror(T_{15}, i_{24}) → G_{20}
Mirror(U_{15}, i_{24}) → H_{17}
Mirror(V_{15}, i_{24}) → I_{17}
Mirror(W_{15}, i_{24}) → J_{23}
Mirror(Z_{15}, i_{24}) → K_{23}
Polygon(L_{18}, M_{18}, V_{16}, W_{16}, Z_{16}, A_{20}, B_{25}, C_{20}, D_{20}, E_{20}, F_{20}, G_{20}, N_{17}, H_{17}, I_{17}, J_{23}, K_{23}, O_{17}, P_{17}, Q_{17}, R_{16}, S_{16}, T_{16}, U_{16}) → poly3_6, l_{17}, u_{16}, v_{16}, w_{16}, z_{16}, a_{20}, b_{20}, c_{20}, d_{20}, e_{20}, f_{20}, m_{16}, g_{20}, h_{17}, i_{23}, j_{18}, k_{17}, n_{16}, o_{16}, p_{16}, q_{16}, r_{16}, s_{16}, t_{16}
Polygon(A_{21}, B_{27}, K_{25}, L_{19}, M_{19}, N_{18}, O_{18}, P_{18}, Q_{18}, R_{17}, S_{17}, T_{17}, C_{21}, U_{17}, V_{17}, W_{17}, Z_{17}, D_{21}, E_{21}, F_{21}, G_{21}, H_{18}, I_{18}, J_{25}) → poly1_{10}, a_{21}, j_{19}, k_{18}, l_{18}, m_{17}, n_{17}, o_{17}, p_{17}, q_{17}, r_{17}, s_{17}, b_{21}, t_{17}, u_{17}, v_{17}, w_{17}, z_{17}, c_{21}, d_{21}, e_{21}, f_{21}, g_{21}, h_{18}, i_{25}
Segment(J_{27}, K_{27}) → i_{27}
Mirror(A_{21}, i_{27}) → L_{20}
Mirror(B_{27}, i_{27}) → M_{20}
Mirror(C_{21}, i_{27}) → N_{19}
Mirror(D_{21}, i_{27}) → O_{19}
Mirror(E_{21}, i_{27}) → P_{19}
Mirror(F_{21}, i_{27}) → Q_{19}
Mirror(G_{21}, i_{27}) → R_{18}
Mirror(H_{18}, i_{27}) → S_{18}
Mirror(I_{18}, i_{27}) → T_{18}
Mirror(J_{25}, i_{27}) → U_{18}
Mirror(K_{25}, i_{27}) → V_{18}
Mirror(L_{19}, i_{27}) → W_{18}
Mirror(M_{19}, i_{27}) → Z_{18}
Mirror(N_{18}, i_{27}) → A_{22}
Mirror(O_{18}, i_{27}) → B_{28}
Mirror(P_{18}, i_{27}) → C_{22}
Mirror(Q_{18}, i_{27}) → D_{22}
Mirror(R_{17}, i_{27}) → E_{22}
Mirror(S_{17}, i_{27}) → F_{22}
Mirror(T_{17}, i_{27}) → G_{22}
Mirror(U_{17}, i_{27}) → H_{19}
Mirror(V_{17}, i_{27}) → I_{19}
Mirror(W_{17}, i_{27}) → J_{26}
Mirror(Z_{17}, i_{27}) → K_{26}
Polygon(L_{20}, M_{20}, V_{18}, W_{18}, Z_{18}, A_{22}, B_{28}, C_{22}, D_{22}, E_{22}, F_{22}, G_{22}, N_{19}, H_{19}, I_{19}, J_{26}, K_{26}, O_{19}, P_{19}, Q_{19}, R_{18}, S_{18}, T_{18}, U_{18}) → poly3_7, l_{19}, u_{18}, v_{18}, w_{18}, z_{18}, a_{22}, b_{22}, c_{22}, d_{22}, e_{22}, f_{22}, m_{18}, g_{22}, h_{19}, i_{26}, j_{20}, k_{19}, n_{18}, o_{18}, p_{18}, q_{18}, r_{18}, s_{18}, t_{18}

AI Drawing Prompt Reference

Create a tessellation pattern composed of multiple irregular polygons, each formed by a series of vertices with wavy or zigzag edges. Start with one base polygon, then use a straight line as a mirror axis to reflect the entire polygon to the other side, creating a symmetric tessellation unit. Repeat this process by mirroring along different directions to extend the pattern across the plane. Include several smaller heptagons as separate elements within the overall structure. Emphasize symmetry and periodicity, with colors chosen to highlight each polygonal region.

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