Symmetric Wallpaper Construction

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2014-10-23 21:43

In collections MTH 595 Book

This example demonstrates the creation of a symmetric wallpaper pattern using Dn symmetry. It involves core mathematical concepts such as symmetry, rotation, reflection, and angle bisectors. The construction utilizes GeoGebra commands and objects like Line, Rotate, Angle, Mirror, AngularBisector, Point, and OrthogonalLine. Suitable for high school students and geometry enthusiasts to understand the application of symmetry and geometric transformations in art.

GeoGebra Commands & Parameters

Line(A, B) → a
Rotate(B, α, A) → B'
Angle(B, A, B') → β
Line(A, B') → b
Mirror(a, b) → a'
Mirror(b, a') → b'
Mirror(a', b') → a''
Mirror(b', a'') → b''
Mirror(a'', b'') → a'''
Mirror(b'', a''') → b'''
Mirror(a''', b''') → c
Mirror(b''', c) → d
Mirror(c, d) → c'
Mirror(d, c') → d'
Mirror(c', d') → c''
Mirror(d', c'') → d''
Mirror(c'', d'') → c'''
Mirror(d'', c''') → d'''
Mirror(c''', d''') → e
Mirror(d''', e) → f
Mirror(C, b) → C'
Mirror(C', a') → C''
Mirror(C'', b') → C'''
Mirror(C''', a'') → D
Mirror(D, b'') → D'
Mirror(D', a''') → D''
Mirror(D'', b''') → D'''
Mirror(D''', c) → E
Mirror(E, d) → E'
Mirror(E', c') → E''
Mirror(E'', d') → E'''
Mirror(E''', c'') → F
Mirror(F, d'') → F'
Mirror(F', c''') → F''
Mirror(F'', d''') → F'''
Mirror(F''', e) → G
Mirror(G, f) → G'
Mirror(C, a) → C'_1
Mirror(C', a) → C''_1
Mirror(C'', a) → C'''_1
Mirror(C''', a) → H
Mirror(D, a) → D'_1
Mirror(D', a) → D''_1
Mirror(D'', a) → D'''_1
Mirror(D''', a) → I
Mirror(E, a) → E'_1
Mirror(E', a) → E''_1
Mirror(E'', a) → E'''_1
Mirror(E''', a) → J
Mirror(F, a) → F'_1
Mirror(F', a) → F''_1
Mirror(F'', a) → F'''_1
Mirror(F''', a) → K
Mirror(G, a) → G'_1
Mirror(G', a) → G''
AngularBisector(B, A, B') → g
Point(g) → L
OrthogonalLine(L, g) → h
Mirror(b, h) → b'_1
Mirror(h, a) → h'
Mirror(h', a') → h''
Mirror(h'', b) → h'''
Mirror(a, h''') → a'_1
Mirror(a'_1, a') → i
Mirror(h''', b'_1) → j
Mirror(a, j) → a'_2
Mirror(b'_1, a'_2) → k
Mirror(h''', k) → l
Mirror(j, a') → j'
Mirror(h', l) → h''_1
Mirror(h, b) → h'_1
Mirror(b'_1, h') → m
Mirror(h'', a'_1) → h'''_1
Mirror(h'', j) → h'''_2
Mirror(h'', k) → h'''_3
Mirror(h'', a') → h'''_4
Mirror(a', h'') → a''_1
Mirror(h'', h) → h'''_5
Mirror(h'', m) → h'''_6
Mirror(h'', h''_1) → h'''_7
Mirror(C', b'_1) → C''_2
Mirror(C'', b'_1) → C'''_2
Mirror(C''', b'_1) → M
Mirror(D, b'_1) → D'_2
Mirror(D', b'_1) → D''_2
Mirror(D'', b'_1) → D'''_2
Mirror(D''', b'_1) → N
Mirror(E, b'_1) → E'_2
Mirror(E', b'_1) → E''_2
Mirror(E'', b'_1) → E'''_2
Mirror(E''', b'_1) → O
Mirror(F, b'_1) → F'_2
Mirror(F', b'_1) → F''_2
Mirror(F'', b'_1) → F'''_2
Mirror(F''', b'_1) → P
Mirror(G, b'_1) → G'_2
Mirror(G', b'_1) → G''_1
Mirror(C'_1, b'_1) → Q
Mirror(C''_1, b'_1) → R
Mirror(C'''_1, b'_1) → S
Mirror(H, b'_1) → H'
Mirror(D'_1, b'_1) → T
Mirror(D''_1, b'_1) → U
Mirror(D'''_1, b'_1) → V
Mirror(I, b'_1) → I'
Mirror(E'_1, b'_1) → W
Mirror(E''_1, b'_1) → Z
Mirror(E'''_1, b'_1) → A_1
Mirror(J, b'_1) → J'
Mirror(F'_1, b'_1) → B_1
Mirror(F''_1, b'_1) → C_1
Mirror(F'''_1, b'_1) → D_1
Mirror(K, b'_1) → K'
Mirror(G'_1, b'_1) → E_1
Mirror(G'', b'_1) → G'''
Mirror(a', h'''_5) → a''_2
Mirror(a', h'''_6) → a''_3
Mirror(a', h'''_7) → a''_4
Mirror(l, h'''_5) → l'
Mirror(C', h') → C''_3
Mirror(C'', h') → C'''_3
Mirror(C''', h') → F_1
Mirror(D, h') → D'_3
Mirror(D', h') → D''_3
Mirror(D'', h') → D'''_3
Mirror(D''', h') → G_1
Mirror(E, h') → E'_3
Mirror(E', h') → E''_3
Mirror(E'', h') → E'''_3
Mirror(E''', h') → H_1
Mirror(F, h') → F'_3
Mirror(F', h') → F''_3
Mirror(F'', h') → F'''_3
Mirror(F''', h') → I_1
Mirror(G, h') → G'_3
Mirror(G', h') → G''_2
Mirror(C'_1, h') → J_1
Mirror(C''_1, h') → K_1
Mirror(C'''_1, h') → L_1
Mirror(H, h') → H'_1
Mirror(D'_1, h') → M_1
Mirror(D''_1, h') → N_1
Mirror(D'''_1, h') → O_1
Mirror(I, h') → I'_1
Mirror(E'_1, h') → P_1
Mirror(E''_1, h') → Q_1
Mirror(E'''_1, h') → R_1
Mirror(J, h') → J'_1
Mirror(F'_1, h') → S_1
Mirror(F''_1, h') → T_1
Mirror(F'''_1, h') → U_1
Mirror(K, h') → K'_1
Mirror(G'_1, h') → V_1
Mirror(G'', h') → G'''_1
Mirror(C''_3, a) → W_1
Mirror(C'''_3, a) → Z_1
Mirror(F_1, a) → A_2
Mirror(D'_3, a) → B_2
Mirror(D''_3, a) → C_2
Mirror(D'''_3, a) → D_2
Mirror(G_1, a) → E_2
Mirror(E'_3, a) → F_2
Mirror(E''_3, a) → G_2
Mirror(E'''_3, a) → H_2
Mirror(H_1, a) → I_2
Mirror(F'_3, a) → J_2
Mirror(F''_3, a) → K_2
Mirror(F'''_3, a) → L_2
Mirror(I_1, a) → M_2
Mirror(G'_3, a) → N_2
Mirror(G''_2, a) → O_2
Mirror(J_1, a) → P_2
Mirror(K_1, a) → Q_2
Mirror(L_1, a) → R_2
Mirror(H'_1, a) → S_2
Mirror(M_1, a) → T_2
Mirror(N_1, a) → U_2
Mirror(O_1, a) → V_2
Mirror(I'_1, a) → W_2
Mirror(P_1, a) → Z_2
Mirror(Q_1, a) → A_3
Mirror(R_1, a) → B_3
Mirror(J'_1, a) → C_3
Mirror(S_1, a) → D_3
Mirror(T_1, a) → E_3
Mirror(U_1, a) → F_3
Mirror(K'_1, a) → G_3
Mirror(V_1, a) → H_3
Mirror(G'''_1, a) → I_3
Mirror(C''_2, m) → J_3
Mirror(C'''_2, m) → K_3
Mirror(M, m) → M'
Mirror(D'_2, m) → L_3
Mirror(D''_2, m) → M_3
Mirror(D'''_2, m) → N_3
Mirror(N, m) → N'
Mirror(E'_2, m) → O_3
Mirror(E''_2, m) → P_3
Mirror(E'''_2, m) → Q_3
Mirror(O, m) → O'
Mirror(F'_2, m) → R_3
Mirror(F''_2, m) → S_3
Mirror(F'''_2, m) → T_3
Mirror(P, m) → P'
Mirror(G'_2, m) → U_3
Mirror(G''_1, m) → V_3
Mirror(Q, m) → Q'
Mirror(R, m) → R'
Mirror(S, m) → S'
Mirror(H', m) → H''
Mirror(T, m) → T'
Mirror(U, m) → U'
Mirror(V, m) → V'
Mirror(I', m) → I''
Mirror(W, m) → W'
Mirror(Z, m) → Z'
Mirror(A_1, m) → W_3
Mirror(J', m) → J''
Mirror(B_1, m) → Z_3
Mirror(C_1, m) → A_4
Mirror(D_1, m) → B_4
Mirror(K', m) → K''
Mirror(E_1, m) → C_4
Mirror(G''', m) → D_4
Mirror(C''_2, a') → E_4
Mirror(C'''_2, a') → F_4
Mirror(M, a') → M'_1
Mirror(D'_2, a') → G_4
Mirror(D''_2, a') → H_4
Mirror(D'''_2, a') → I_4
Mirror(N, a') → N'_1
Mirror(E'_2, a') → J_4
Mirror(E''_2, a') → K_4
Mirror(E'''_2, a') → L_4
Mirror(O, a') → O'_1
Mirror(F'_2, a') → M_4
Mirror(F''_2, a') → N_4
Mirror(F'''_2, a') → O_4
Mirror(P, a') → P'_1
Mirror(G'_2, a') → P_4
Mirror(G''_1, a') → Q_4
Mirror(Q, a') → Q'_1
Mirror(R, a') → R'_1
Mirror(S, a') → S'_1
Mirror(H', a') → H''_1
Mirror(T, a') → T'_1
Mirror(U, a') → U'_1
Mirror(V, a') → V'_1
Mirror(I', a') → I''_1
Mirror(W, a') → W'_1
Mirror(Z, a') → Z'_1
Mirror(A_1, a') → R_4
Mirror(J', a') → J''_1
Mirror(B_1, a') → S_4
Mirror(C_1, a') → T_4
Mirror(D_1, a') → U_4
Mirror(K', a') → K''_1
Mirror(E_1, a') → V_4
Mirror(G''', a') → W_4
Mirror(C', h''') → C''_4
Mirror(C'', h''') → C'''_4
Mirror(C''', h''') → Z_4
Mirror(D, h''') → D'_4
Mirror(D', h''') → D''_4
Mirror(D'', h''') → D'''_4
Mirror(D''', h''') → A_5
Mirror(E, h''') → E'_4
Mirror(E', h''') → E''_4
Mirror(E'', h''') → E'''_4
Mirror(E''', h''') → B_5
Mirror(F, h''') → F'_4
Mirror(F', h''') → F''_4
Mirror(F'', h''') → F'''_4
Mirror(F''', h''') → C_5
Mirror(G, h''') → G'_4
Mirror(G', h''') → G''_3
Mirror(C'_1, h''') → D_5
Mirror(C''_1, h''') → E_5
Mirror(C'''_1, h''') → F_5
Mirror(H, h''') → H'_2
Mirror(D'_1, h''') → G_5
Mirror(D''_1, h''') → H_5
Mirror(D'''_1, h''') → I_5
Mirror(I, h''') → I'_2
Mirror(E'_1, h''') → J_5
Mirror(E''_1, h''') → K_5
Mirror(E'''_1, h''') → L_5
Mirror(J, h''') → J'_2
Mirror(F'_1, h''') → M_5
Mirror(F''_1, h''') → N_5
Mirror(F'''_1, h''') → O_5
Mirror(K, h''') → K'_2
Mirror(G'_1, h''') → P_5
Mirror(G'', h''') → G'''_2
Mirror(C', h'') → C''_5
Mirror(C'', h'') → C'''_5
Mirror(C''', h'') → Q_5
Mirror(D, h'') → D'_5
Mirror(D', h'') → D''_5
Mirror(D'', h'') → D'''_5
Mirror(D''', h'') → R_5
Mirror(E, h'') → E'_5
Mirror(E', h'') → E''_5
Mirror(E'', h'') → E'''_5
Mirror(E''', h'') → S_5
Mirror(F, h'') → F'_5
Mirror(F', h'') → F''_5
Mirror(F'', h'') → F'''_5
Mirror(F''', h'') → T_5
Mirror(G, h'') → G'_5
Mirror(G', h'') → G''_4
Mirror(C'_1, h'') → U_5
Mirror(C''_1, h'') → V_5
Mirror(C'''_1, h'') → W_5
Mirror(H, h'') → H'_3
Mirror(D'_1, h'') → Z_5
Mirror(D''_1, h'') → A_6
Mirror(D'''_1, h'') → B_6
Mirror(I, h'') → I'_3
Mirror(E'_1, h'') → C_6
Mirror(E''_1, h'') → D_6
Mirror(E'''_1, h'') → E_6
Mirror(J, h'') → J'_3
Mirror(F'_1, h'') → F_6
Mirror(F''_1, h'') → G_6
Mirror(F'''_1, h'') → H_6
Mirror(K, h'') → K'_3
Mirror(G'_1, h'') → I_6
Mirror(G'', h'') → G'''_3
Mirror(J_3, h') → J_6
Mirror(K_3, h') → K_6
Mirror(M', h') → M''
Mirror(L_3, h') → L_6
Mirror(M_3, h') → M_6
Mirror(N_3, h') → N_6
Mirror(N', h') → N''
Mirror(O_3, h') → O_6
Mirror(P_3, h') → P_6
Mirror(Q_3, h') → Q_6
Mirror(O', h') → O''
Mirror(R_3, h') → R_6
Mirror(S_3, h') → S_6
Mirror(T_3, h') → T_6
Mirror(P', h') → P''
Mirror(U_3, h') → U_6
Mirror(V_3, h') → V_6
Mirror(Q', h') → Q''
Mirror(R', h') → R''
Mirror(S', h') → S''
Mirror(H'', h') → H'''
Mirror(T', h') → T''
Mirror(U', h') → U''
Mirror(V', h') → V''
Mirror(I'', h') → I'''
Mirror(W', h') → W''
Mirror(Z', h') → Z''
Mirror(W_3, h') → W_6
Mirror(J'', h') → J'''
Mirror(Z_3, h') → Z_6
Mirror(A_4, h') → A_7
Mirror(B_4, h') → B_7
Mirror(K'', h') → K'''
Mirror(C_4, h') → C_7
Mirror(D_4, h') → D_7
Mirror(W_1, h'''_5) → E_7
Mirror(Z_1, h'''_5) → F_7
Mirror(A_2, h'''_5) → G_7
Mirror(B_2, h'''_5) → H_7
Mirror(C_2, h'''_5) → I_7
Mirror(D_2, h'''_5) → J_7
Mirror(E_2, h'''_5) → K_7
Mirror(F_2, h'''_5) → L_7
Mirror(G_2, h'''_5) → M_7
Mirror(H_2, h'''_5) → N_7
Mirror(I_2, h'''_5) → O_7
Mirror(J_2, h'''_5) → P_7
Mirror(K_2, h'''_5) → Q_7
Mirror(L_2, h'''_5) → R_7
Mirror(M_2, h'''_5) → S_7
Mirror(N_2, h'''_5) → T_7
Mirror(O_2, h'''_5) → U_7
Mirror(P_2, h'''_5) → V_7
Mirror(Q_2, h'''_5) → W_7
Mirror(R_2, h'''_5) → Z_7
Mirror(S_2, h'''_5) → A_8
Mirror(T_2, h'''_5) → B_8
Mirror(U_2, h'''_5) → C_8
Mirror(V_2, h'''_5) → D_8
Mirror(W_2, h'''_5) → E_8
Mirror(Z_2, h'''_5) → F_8
Mirror(A_3, h'''_5) → G_8
Mirror(B_3, h'''_5) → H_8
Mirror(C_3, h'''_5) → I_8
Mirror(D_3, h'''_5) → J_8
Mirror(E_3, h'''_5) → K_8
Mirror(F_3, h'''_5) → L_8
Mirror(G_3, h'''_5) → M_8
Mirror(H_3, h'''_5) → N_8
Mirror(I_3, h'''_5) → O_8
Mirror(E_7, h'') → P_8
Mirror(F_7, h'') → Q_8
Mirror(G_7, h'') → R_8
Mirror(H_7, h'') → S_8
Mirror(I_7, h'') → T_8
Mirror(J_7, h'') → U_8
Mirror(K_7, h'') → V_8
Mirror(L_7, h'') → W_8
Mirror(M_7, h'') → Z_8
Mirror(N_7, h'') → A_9
Mirror(O_7, h'') → B_9
Mirror(P_7, h'') → C_9
Mirror(Q_7, h'') → D_9
Mirror(R_7, h'') → E_9
Mirror(S_7, h'') → F_9
Mirror(T_7, h'') → G_9
Mirror(U_7, h'') → H_9
Mirror(V_7, h'') → I_9
Mirror(W_7, h'') → J_9
Mirror(Z_7, h'') → K_9
Mirror(A_8, h'') → L_9
Mirror(B_8, h'') → M_9
Mirror(C_8, h'') → N_9
Mirror(D_8, h'') → O_9
Mirror(E_8, h'') → P_9
Mirror(F_8, h'') → Q_9
Mirror(G_8, h'') → R_9
Mirror(H_8, h'') → S_9
Mirror(I_8, h'') → T_9
Mirror(J_8, h'') → U_9
Mirror(K_8, h'') → V_9
Mirror(L_8, h'') → W_9
Mirror(M_8, h'') → Z_9
Mirror(N_8, h'') → A_{10}
Mirror(O_8, h'') → B_{10}
Mirror(P_8, h) → C_{10}
Mirror(Q_8, h) → D_{10}
Mirror(R_8, h) → E_{10}
Mirror(S_8, h) → F_{10}
Mirror(T_8, h) → G_{10}
Mirror(U_8, h) → H_{10}
Mirror(V_8, h) → I_{10}
Mirror(W_8, h) → J_{10}
Mirror(Z_8, h) → K_{10}
Mirror(A_9, h) → L_{10}
Mirror(B_9, h) → M_{10}
Mirror(C_9, h) → N_{10}
Mirror(D_9, h) → O_{10}
Mirror(E_9, h) → P_{10}
Mirror(F_9, h) → Q_{10}
Mirror(G_9, h) → R_{10}
Mirror(H_9, h) → S_{10}
Mirror(I_9, h) → T_{10}
Mirror(J_9, h) → U_{10}
Mirror(K_9, h) → V_{10}
Mirror(L_9, h) → W_{10}
Mirror(M_9, h) → Z_{10}
Mirror(N_9, h) → A_{11}
Mirror(O_9, h) → B_{11}
Mirror(P_9, h) → C_{11}
Mirror(Q_9, h) → D_{11}
Mirror(R_9, h) → E_{11}
Mirror(S_9, h) → F_{11}
Mirror(T_9, h) → G_{11}
Mirror(U_9, h) → H_{11}
Mirror(V_9, h) → I_{11}
Mirror(W_9, h) → J_{11}
Mirror(Z_9, h) → K_{11}
Mirror(A_{10}, h) → L_{11}
Mirror(B_{10}, h) → M_{11}
Mirror(C''_5, a'_2) → N_{11}
Mirror(C'''_5, a'_2) → O_{11}
Mirror(Q_5, a'_2) → P_{11}
Mirror(D'_5, a'_2) → Q_{11}
Mirror(D''_5, a'_2) → R_{11}
Mirror(D'''_5, a'_2) → S_{11}
Mirror(R_5, a'_2) → T_{11}
Mirror(E'_5, a'_2) → U_{11}
Mirror(E''_5, a'_2) → V_{11}
Mirror(E'''_5, a'_2) → W_{11}
Mirror(S_5, a'_2) → Z_{11}
Mirror(F'_5, a'_2) → A_{12}
Mirror(F''_5, a'_2) → B_{12}
Mirror(F'''_5, a'_2) → C_{12}
Mirror(T_5, a'_2) → D_{12}
Mirror(G'_5, a'_2) → E_{12}
Mirror(G''_4, a'_2) → F_{12}
Mirror(U_5, a'_2) → G_{12}
Mirror(V_5, a'_2) → H_{12}
Mirror(W_5, a'_2) → I_{12}
Mirror(H'_3, a'_2) → J_{12}
Mirror(Z_5, a'_2) → K_{12}
Mirror(A_6, a'_2) → L_{12}
Mirror(B_6, a'_2) → M_{12}
Mirror(I'_3, a'_2) → N_{12}
Mirror(C_6, a'_2) → O_{12}
Mirror(D_6, a'_2) → P_{12}
Mirror(E_6, a'_2) → Q_{12}
Mirror(J'_3, a'_2) → R_{12}
Mirror(F_6, a'_2) → S_{12}
Mirror(G_6, a'_2) → T_{12}
Mirror(H_6, a'_2) → U_{12}
Mirror(K'_3, a'_2) → V_{12}
Mirror(I_6, a'_2) → W_{12}
Mirror(G'''_3, a'_2) → Z_{12}

AI Drawing Prompt Reference

Draw a symmetric wallpaper pattern using Dn symmetry. Start by drawing a line segment, then rotate this segment around one endpoint to get a second point. Next, draw a line connecting these two points and find the angle bisector of the angle formed. Use mirror symmetry operations on the line segments and angle bisectors successively to create a symmetric pattern. Repeat this process to expand the pattern, ultimately forming a complex symmetric wallpaper design.

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