3.4.6 Museo de Onda: Azulejo 6 – Geometric Transformations in a Parallelepiped T

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2023-03-22 21:50

In collections geometriadinamica.es

This GeoGebra example demonstrates how to construct a decorative tile pattern inspired by the Museo de Onda (Spain) using geometric transformations within a parallelepiped framework. The core mathematical concepts include geometric transformations (rotation, reflection, translation), the structure of a parallelepiped, tangent lines, midpoints, perpendiculars, and perimeter. The construction uses GeoGebra commands such as Line, Point, OrthogonalLine, Intersect, Midpoint, Mirror, Rotate, and Vector, involving objects like A, B, b, G, e, J, h, f, H, g, E, I, B_1, a, vp, vg, dif, c, vs, vr, vt, v, vd, gr, D, F, K, L, M, N, O, P, Q, R, S, T, U, V, W, Z. By applying rotations and reflections, a symmetric tile motif is generated. This resource is suitable for middle and high school students learning geometric transformations, symmetry, and spatial reasoning, and can be used in interdisciplinary projects connecting mathematics and art.

GeoGebra Commands & Parameters

Line(A, B) → b
Line(G, b) → e
Point(e) → J
OrthogonalLine(J, e) → h
OrthogonalLine(G, e) → f
Point(f) → H
OrthogonalLine(H, f) → g
Intersect(g, h) → E
Midpoint(E, H) → I
Midpoint(I, E) → B_1
Midpoint(G, J) → D
Midpoint(J, E) → F
Midpoint(H, G) → K
Midpoint(D, I) → L
Midpoint(K, L) → M
Midpoint(L, F) → N
Midpoint(F, J) → O
Midpoint(J, D) → P
Midpoint(G, D) → Q
Midpoint(K, G) → R
Midpoint(Q, M) → S
Midpoint(L, D) → T
Midpoint(P, N) → U
Midpoint(F, E) → V
Midpoint(K, H) → W
Midpoint(W, V) → Z
Midpoint(H, I) → A_1
Midpoint(N, B_1) → C_1
Midpoint(Z, W) → D_1
Midpoint(D, U) → E_1
Midpoint(U, J) → F_1
Midpoint(T, Q) → G_1
Midpoint(S, G) → H_1
Midpoint(S, K) → I_1
Midpoint(M, T) → J_1
Midpoint(T, N) → K_1
Midpoint(N, O) → L_1
Midpoint(F, C_1) → M_1
Midpoint(C_1, L) → N_1
Midpoint(L, D_1) → O_1
Midpoint(D_1, K) → P_1
Midpoint(H, D_1) → Q_1
Midpoint(D_1, I) → R_1
Midpoint(I, C_1) → S_1
Midpoint(C_1, E) → T_1
Mirror(F_1, h) → U_1
Mirror(L_1, h) → V_1
Mirror(M_1, h) → W_1
Mirror(T_1, h) → Z_1
Mirror(B_1, h) → A_2
Mirror(L_1, h) → C_2
Line(S, A_1) → t
Line(I, T) → a_1
Line(U, B_1) → b_1
Line(E_1, N_1) → c_1
Line(F_1, M_1) → d_1
Line(G_1, R_1) → e_1
Line(W_1, P_1) → g_1
Line(Q_1, Z_1) → h_1
Point(a_1) → F_2
Point(b) → A1
Min(90, ((a - 1) * 90)) → r1
Rotate(F_2, ((-r1))°, T) → K_{11}
Rotate(T, ((-r1))°, T) → T'_7
Rotate(U, ((-r1))°, T) → U'_3
Rotate(T, ((-r1))°, T) → T'_8
Rotate(T, ((-r1))°, T'_8) → T'_9
Rotate(F_2, ((-r1))°, T) → J_{13}
Rotate(L, ((-r1))°, T) → L'_2
Mirror(U'_3, b_1) → A_{14}
Mirror(T'_8, b_1) → B_{14}
Mirror(T'_9, b_1) → O_{15}
Mirror(K_{11}, b_1) → G_{16}
Mirror(T'_7, b_1) → H_{16}
Mirror(J_{13}, b_1) → J_{17}
Mirror(L'_2, b_1) → K_{17}
Midpoint(C_1, E) → K_{20}
Line(G_1, O_1) → l_1
OrthogonalLine(G_1, l_1) → m_1
OrthogonalLine(J_1, l_1) → q_1
OrthogonalLine(O_1, l_1) → r_1
OrthogonalLine(R_1, l_1) → s_1
OrthogonalLine(E_1, m_1) → t_1
OrthogonalLine(F_1, m_1) → a_2
Point(a_1) → C
Min(180, ((a - 1) * 180)) → r2
Min(270, ((a - 1) * 270)) → r3
Rotate(T, r1°, T) → T'_4
Rotate(F_2, r1°, T) → G_2
Rotate(T, r2°, T) → T'_5
Rotate(F_2, r2°, T) → J_2
Rotate(T, r3°, T) → T'_6
Rotate(F_2, r3°, T) → N_2
Rotate(F_2, r1°, T) → Q_2
Rotate(C, r1°, T) → C'
Rotate(F_2, r2°, T) → Z_2
Rotate(C, r2°, T) → C'_1
Rotate(F_2, r3°, T) → H_3
Rotate(C, r3°, T) → C'_2
Rotate(C, r1°, T) → C'_3
Rotate(L, r1°, T) → L'_1
Rotate(K_1, r1°, T) → Q_3
Rotate(C, r2°, T) → C'_4
Rotate(L, r2°, T) → L'_3
Rotate(K_1, r2°, T) → Z_3
Rotate(C, r3°, T) → C'_5
Rotate(L, r3°, T) → L'_4
Rotate(K_1, r3°, T) → H_4
Mirror(C'_4, a_1) → F_5
Mirror(L'_3, a_1) → G_5
Mirror(Z_3, a_1) → J_5
Mirror(Z_2, a_1) → O_5
Mirror(C'_1, a_1) → P_5
Mirror(T'_5, a_1) → W_5
Mirror(J_2, a_1) → Z_5
Mirror(C, a_1) → C'_6
Mirror(L, a_1) → L'_5
Mirror(K_1, a_1) → R_6
Mirror(F_2, a_1) → T_6
Mirror(C, a_1) → C'_7
Mirror(T, a_1) → T'_{10}
Mirror(F_2, a_1) → W_6
Min(180, ((a - 5.5) * 180)) → r4
Vector(S, D) → u
Vector(S, L) → w
Line(S, O) → q
Line(F, M) → r
Line(D_1, V) → s
Intersect(t, m_1) → B_2
Intersect(a_1, m_1) → P_2
Intersect(b_1, m_1) → R_2
Intersect(h, m_1) → S_2
Intersect(a_2, q) → T_2
Intersect(t_1, q) → W_2
Intersect(l_1, q) → A_3
Intersect(t, q_1) → B_3
Intersect(q_1, a_1) → C_3
Intersect(b_1, q_1) → E_3
Intersect(h, q_1) → G_3
Intersect(a_2, r) → I_3
Intersect(r, t_1) → J_3
Intersect(r, l_1) → K_3
Intersect(r_1, a_1) → L_3
Intersect(b_1, r_1) → N_3
Intersect(h, r_1) → P_3
Intersect(t, r_1) → R_3
Intersect(l_1, s) → S_3
Intersect(s, t_1) → T_3
Intersect(s, a_2) → U_3
Vector(U, P) → z
Vector(U, O) → u_1
Rotate(pic1, 270°, D_1) → pic1'
Rotate(pic1', 270°, D_1) → pic1''
Rotate(pic1'', 270°, D_1) → pic1'''
Rotate(pic1', 270°, M) → pic1''_1
Rotate(pic1''_1, 270°, M) → pic2
Rotate(pic1''_1, 270°, S) → pic3
Rotate(pic3, 270°, S) → pic3'
Rotate(pic3, 270°, Q) → pic3'_1
Rotate(pic3'_1, 270°, Q) → pic4
Rotate(pic4, 270°, D) → pic4'
Rotate(pic4', 270°, D) → pic4''
Rotate(pic2, 270°, L) → pic2'
Rotate(pic1'', 270°, Z) → pic1'''_1
Rotate(pic1'''_1, 270°, Z) → pic5
Rotate(pic4', 270°, P) → pic4''_1
Rotate(pic4''_1, 270°, P) → pic6
Rotate(pic4''_1, 270°, J) → pic7
Rotate(pic6, 270°, O) → pic6'
Rotate(pic6', 270°, O) → pic6''
Rotate(pic6'', 270°, O) → pic6'''
Rotate(pic6''', 270°, N) → pic8
Rotate(pic8, 270°, C_1) → pic8'
Rotate(pic6'', 270°, F) → pic6'''_1
Rotate(pic6'''_1, 270°, V) → pic9

AI Drawing Prompt Reference

Create a decorative tile pattern using geometric transformations, starting by drawing a base line segment and marking points on it, then constructing perpendicular lines through those points to form a set of parallel lines; build a grid of numerous midpoints between these perpendicular lines and connect them to form complex geometric shapes; finally reflect part of the design across one of the perpendicular lines as an axis of symmetry to generate a symmetric tile motif within a parallelepiped framework.

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