Tessellation by Translation: Constructing a Parallelogram Grid with Vectors

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2014-03-30 05:26

In collections Tesselation by Translation Tesselation by Translation

This example demonstrates how to create a plane tessellation by translating a base polygon (a parallelogram) along two directions, producing a seamless grid that covers the plane. The core mathematical concepts involved are translation transformations, vectors, parallelograms, and the logic of tessellation. Construction highlights: using the Polygon command to create the base polygon poly1, defining vectors u and v from points, applying the Translate command to shift poly1 along u and v to generate translated copies (e.g., stroke1', stroke2'), and using AttachCopyToView to fix the view. Points and vectors are used to control the translation directions and distances. This example is suitable for middle or high school geometry classes or enrichment activities, helping students visualize how translation and vectors work together to create repeating patterns and understand the geometric principles behind tiling.

GeoGebra Commands & Parameters

Polygon(A, B, 4) → poly1, a, b, c, d, C, D
AttachCopyToView(PolyLine[(3.7, -3.7600000000000002), (3.62, -3.7600000000000002), (3.54, -3.7800000000000002), (3.48, -3.8200000000000003), (3.42, -3.86), (3.36, -3.9), (3.3000000000000003, -3.94), (3.24, -3.98), (3.18, -4.04), (3.1, -4.1), (3.02, -4.16), (2.96, -4.18), (2.84, -4.24), (2.7800000000000002, -4.26), (2.7, -4.32), (2.64, -4.34), (2.56, -4.4), (2.48, -4.42), (2.4, -4.46), (2.32, -4.48), (2.2600000000000002, -4.5200000000000005), (2.2, -4.58), (2.14, -4.62), (2.08, -4.68), (2.02, -4.72), (2.02, -4.64), (2.02, -4.5600000000000005), (2.02, -4.48), (2.02, -4.38), (2.02, -4.3), (2.02, -4.22), (2.08, -4.18), (2.14, -4.16), (2.2, -4.14), (2.2600000000000002, -4.12), (2.32, -4.08), (2.38, -4.0600000000000005), (2.36, -4), (2.3000000000000003, -3.98), (2.24, -3.94), (2.16, -3.9), (2.1, -3.88), (2.04, -3.8200000000000003), (1.98, -3.7800000000000002), (1.92, -3.7600000000000002), (1.86, -3.74), (1.82, -3.68), (1.76, -3.64), (1.74, -3.58), (1.8, -3.54), (1.86, -3.48), (1.92, -3.46), (1.98, -3.42), (2.06, -3.38), (2.12, -3.3200000000000003), (2.18, -3.3000000000000003), (2.24, -3.2600000000000002), (2.32, -3.2600000000000002), (2.38, -3.22), (2.44, -3.18), (2.5, -3.16), (2.56, -3.1), (2.62, -3.08), (2.68, -3.06), (2.6, -3.06), (2.5, -3.06), (2.4, -3.06), (2.3000000000000003, -3.06), (2.22, -3.06), (2.14, -3.06), (2.06, -3.06), (1.98, -3.06), (1.9000000000000001, -3.06), (1.82, -3.02), (1.74, -3), (1.6400000000000001, -3), (1.58, -2.98), (1.5, -2.98), (1.42, -2.98), (1.34, -2.98), (1.26, -2.98), (1.18, -3), (1.1, -3), (1.02, -3), (0.9400000000000001, -3), (0.86, -3), (0.78, -3), (0.7000000000000001, -3), (0.64, -2.98), (0.56, -2.98), (0.48, -2.98), (0.42, -2.96), (0.36, -3), (0.3, -3.02), (0.22, -3.02), (0.16, -3.04), (0.1, -3.06), (0.02, -3.08), (-0.06, -3.08), (-0.14, -3.08), (-0.22, -3.08), (-0.22, -3.16), (-0.22, -3.24), (-0.24, -3.3000000000000003), (-0.26, -3.36), (-0.32, -3.4), (-0.38, -3.42), (-0.44, -3.46), (-0.5, -3.5), (-0.56, -3.52), (-0.62, -3.54), (-0.68, -3.56), (-0.74, -3.58), (-0.8, -3.6), (-0.8, -3.68), (-0.76, -3.74), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke1
Point(a) → E
Point(c) → F
Vector(E, F) → u
Translate(stroke1, u) → stroke1'
AttachCopyToView(PolyLine[(-0.78, -3.7600000000000002), (-0.86, -3.7600000000000002), (-0.92, -3.7800000000000002), (-0.98, -3.8000000000000003), (-1.06, -3.8200000000000003), (-1.12, -3.84), (-1.12, -3.74), (-1.12, -3.62), (-1.1, -3.54), (-1.06, -3.48), (-1, -3.42), (-0.9400000000000001, -3.38), (-0.88, -3.36), (-0.8200000000000001, -3.3200000000000003), (-0.76, -3.2800000000000002), (-0.74, -3.22), (-0.74, -3.14), (-0.74, -3.02), (-0.74, -2.94), (-0.74, -2.84), (-0.74, -2.7600000000000002), (-0.7000000000000001, -2.7), (-0.64, -2.68), (-0.56, -2.68), (-0.48, -2.64), (-0.4, -2.62), (-0.34, -2.6), (-0.28, -2.56), (-0.22, -2.52), (-0.16, -2.5), (-0.1, -2.46), (-0.04, -2.42), (0.02, -2.38), (0.06, -2.3000000000000003), (0.06, -2.22), (0.06, -2.14), (0.04, -2.08), (0.02, -2.02), (0.02, -1.94), (0.02, -1.86), (-0.06, -1.86), (-0.14, -1.86), (-0.2, -1.82), (-0.28, -1.82), (-0.36, -1.82), (-0.44, -1.82), (-0.5, -1.8), (-0.58, -1.78), (-0.64, -1.76), (-0.62, -1.82), (-0.58, -1.9000000000000001), (-0.52, -1.96), (-0.46, -2), (-0.42, -2.06), (-0.36, -2.1), (-0.34, -2.16), (-0.28, -2.18), (-0.36, -2.18), (-0.44, -2.18), (-0.52, -2.2), (-0.6, -2.22), (-0.68, -2.22), (-0.74, -2.2), (-0.8, -2.14), (-0.8200000000000001, -2.08), (-0.84, -2.02), (-0.88, -1.96), (-0.9, -1.9000000000000001), (-0.9400000000000001, -1.84), (-0.96, -1.76), (-0.98, -1.7), (-1, -1.6400000000000001), (-1.04, -1.58), (-1.06, -1.52), (-1.08, -1.46), (-1.02, -1.44), (-0.9400000000000001, -1.42), (-0.88, -1.4000000000000001), (-0.8200000000000001, -1.3800000000000001), (-0.74, -1.36), (-0.66, -1.34), (-0.6, -1.32), (-0.54, -1.3), (-0.48, -1.28), (-0.4, -1.28), (-0.32, -1.28), (-0.24, -1.28), (-0.16, -1.28), (-0.1, -1.26), (-0.04, -1.24), (0.04, -1.24), (0.1, -1.22), (0.12, -1.16), (0.14, -1.08), (0.2, -1), (0.24, -0.92), (0.28, -0.86), (0.3, -0.8), (0.34, -0.74), (0.38, -0.68), (0.42, -0.62), (0.48, -0.56), (0.54, -0.5), (0.6, -0.46), (0.66, -0.4), (0.72, -0.36), (0.78, -0.32), (0.8, -0.26), (0.8200000000000001, -0.2), (0.86, -0.14), (0.9, -0.08), (0.92, -0.02), (0.92, 0.06), (0.9, 0.14), (0.86, 0.2), (0.8200000000000001, 0.26), (0.8, 0.32), (0.84, 0.4), (0.88, 0.46), (0.9400000000000001, 0.54), (0.96, 0.6), (1.02, 0.66), (1.02, 0.74), (1.06, 0.8), (1, 0.84), (0.9, 0.8200000000000001), (0.8200000000000001, 0.8200000000000001), (0.74, 0.8200000000000001), (0.66, 0.8200000000000001), (0.58, 0.8200000000000001), (0.5, 0.8200000000000001), (0.42, 0.8200000000000001), (0.34, 0.8200000000000001), (0.26, 0.8200000000000001), (0.2, 0.8), (0.14, 0.74), (0.08, 0.7000000000000001), (0.04, 0.64), (-0.02, 0.62), (-0.08, 0.58), (-0.14, 0.54), (-0.2, 0.48), (-0.26, 0.44), (-0.34, 0.44), (-0.42, 0.44), (-0.48, 0.46), (-0.54, 0.5), (-0.62, 0.5), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke2
Point(d) → G
Point(b) → H
Vector(G, H) → v
Translate(stroke2, v) → stroke2'
Point(stroke1') → I
Point(stroke1') → J
Point(stroke1') → K
Point(stroke1') → L
Point(stroke1') → M
Point(stroke1') → N
Point(stroke1') → O
Point(stroke1') → P
Point(stroke1') → Q
Point(stroke1') → R
Point(stroke1') → S
Point(stroke1') → T
Point(stroke2') → U
Point(stroke2') → V
Point(stroke2') → W
Point(stroke2') → Z
Point(stroke2') → A_1
Point(stroke2') → B_1
Point(stroke2') → C_1
Point(stroke2') → D_1
Point(stroke2') → E_1
Point(stroke2') → F_1
Point(stroke2') → G_1
Point(stroke2') → I_1
Point(stroke2') → J_1
Point(stroke2') → K_1
Point(stroke2') → L_1
Point(stroke1) → M_1
Point(stroke1) → N_1
Point(stroke1) → O_1
Point(stroke1) → P_1
Point(stroke1) → Q_1
Point(stroke1) → R_1
Point(stroke1) → S_1
Point(stroke1) → T_1
Point(stroke1) → U_1
Point(stroke1) → V_1
Point(stroke2) → W_1
Point(stroke2) → Z_1
Point(stroke2) → A_2
Point(stroke2) → B_2
Point(stroke2) → C_2
Point(stroke2) → D_2
Point(stroke2) → E_2
Point(stroke2) → F_2
Point(stroke2) → G_2
Point(stroke2) → H_2
Point(stroke2) → I_2
Point(stroke2) → J_2
Point(stroke2) → K_2
Point(stroke2) → L_2
Point(stroke2) → M_2
Point(stroke2) → N_2
Polygon(D, I, R, S, T, C, U, V, W, Z, A_1, B_1, J, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, K_1, L_1, K, B, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, U_1, L, V_1, A, W_1, Z_1, A_2, B_2, C_2, D_2, E_2, F_2, M, G_2, H_2, I_2, J_2, K_2, F, L_2, M_2, N_2, N, O, P, Q) → poly2, d_1, q, r, s, t, c_1, u_1, v_1, w, z, a_1, i, b_1, c_2, d_2, e_1, f_1, g_1, h_1, i_1, j_1, k_1, j, l_1, b_2, m_1, n_1, o_1, p_1, q_1, r_1, s_1, t_1, k, u_2, v_2, a_2, w_1, z_1, a_3, b_3, c_3, d_3, e_2, l, f_2, g_2, h_2, i_2, j_2, k_2, f, l_2, m_2, n_2, m, n, o, p
Vector(W_1, I) → e
Translate(stroke1, e) → stroke1'_1
Translate(stroke1', e) → stroke1''
Translate(stroke2, e) → stroke2'_1
Translate(stroke2', e) → stroke2''
Translate(D, e) → D'
Translate(I, e) → I'
Translate(J, e) → J'
Translate(K, e) → K'
Translate(L, e) → L'
Translate(M, e) → M'
Translate(N, e) → N'
Translate(O, e) → O'
Translate(P, e) → P'
Translate(Q, e) → Q'
Translate(R, e) → R'
Translate(S, e) → S'
Translate(T, e) → T'
Translate(C, e) → C'
Translate(U, e) → U'
Translate(V, e) → V'
Translate(W, e) → W'
Translate(Z, e) → Z'
Translate(A_1, e) → O_2
Translate(B_1, e) → P_2
Translate(C_1, e) → Q_2
Translate(D_1, e) → R_2
Translate(E_1, e) → S_2
Translate(F_1, e) → T_2
Translate(G_1, e) → U_2
Translate(H_1, e) → V_2
Translate(I_1, e) → W_2
Translate(J_1, e) → Z_2
Translate(K_1, e) → A_3
Translate(L_1, e) → B_3
Translate(B, e) → B'
Translate(M_1, e) → C_3
Translate(N_1, e) → D_3
Translate(O_1, e) → E_3
Translate(P_1, e) → F_3
Translate(Q_1, e) → G_3
Translate(R_1, e) → H_3
Translate(S_1, e) → I_3
Translate(T_1, e) → J_3
Translate(U_1, e) → K_3
Translate(V_1, e) → L_3
Translate(A, e) → A'
Translate(W_1, e) → M_3
Translate(Z_1, e) → N_3
Translate(A_2, e) → O_3
Translate(B_2, e) → P_3
Translate(C_2, e) → Q_3
Translate(D_2, e) → R_3
Translate(E_2, e) → S_3
Translate(F_2, e) → T_3
Translate(G_2, e) → U_3
Translate(H_2, e) → V_3
Translate(I_2, e) → W_3
Translate(J_2, e) → Z_3
Translate(K_2, e) → A_4
Translate(F, e) → F'
Translate(L_2, e) → B_4
Translate(M_2, e) → C_4
Translate(N_2, e) → D_4
Polygon(D', I', R', S', T', C', U', V', W', Z', O_2, P_2, J', Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, K', B', C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, K_3, L', L_3, A', M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, M', U_3, V_3, W_3, Z_3, A_4, F', B_4, C_4, D_4, N', O', P', Q') → poly2', d', q', r', s', t', c', u', v', w', z', o_2, i', p_2, q_2, r_2, s_2, t_2, u_3, v_3, w_2, z_2, a_4, j', b_4, b', c_4, d_4, e_3, f_3, g_3, h_3, i_3, j_3, k', k_3, l_3, a', m_3, n_3, o_3, p_3, q_3, r_3, s_3, l', t_3, u_4, v_4, w_3, z_3, a_5, f', b_5, c_5, d_5, m', n', o', p'
Translate(E, e) → E'
Translate(G, e) → G'
Translate(H, e) → H'
AttachCopyToView(PolyLine[(0, 5.5600000000000005), (0.08, 5.5600000000000005), (0.14, 5.6000000000000005), (0.2, 5.62), (0.26, 5.66), (0.32, 5.7), (0.38, 5.72), (0.46, 5.72), (NaN, NaN), (0.04, 5.58), (0.12, 5.6000000000000005), (0.18, 5.64), (0.22, 5.7), (0.28, 5.72), (0.34, 5.74), (0.42, 5.74), (0.5, 5.74), (0.58, 5.74), (NaN, NaN), (0.2, 5.5600000000000005), (0.26, 5.54), (0.32, 5.5600000000000005), (0.34, 5.62), (0.34, 5.7), (NaN, NaN), (-0.32, 5.3), (-0.24, 5.32), (-0.18, 5.34), (-0.16, 5.28), (-0.22, 5.24), (NaN, NaN), (-0.12, 5.1000000000000005), (-0.1, 5.16), (-0.08, 5.24), (-0.04, 5.3), (0.02, 5.32), (0.08, 5.36), (0.16, 5.36), (NaN, NaN), (2.4, 3.9), (2.42, 3.96), (2.46, 4.0200000000000005), (2.52, 4.0600000000000005), (2.58, 4.12), (2.66, 4.16), (2.72, 4.22), (2.7800000000000002, 4.26), (2.84, 4.28), (2.9, 4.32), (2.96, 4.34), (3.02, 4.36), (3.1, 4.38), (3.16, 4.42), (3.24, 4.46), (3.2600000000000002, 4.5200000000000005), (3.3000000000000003, 4.58), (3.36, 4.62), (3.44, 4.7), (3.5, 4.74), (3.56, 4.78), (3.62, 4.8), (3.68, 4.82), (3.74, 4.86), (3.8200000000000003, 4.86), (3.92, 4.86), (4, 4.86), (4.08, 4.86), (4.16, 4.86), (4.24, 4.86), (4.32, 4.86), (4.4, 4.86), (4.48, 4.86), (4.54, 4.9), (4.58, 4.96), (4.64, 5.04), (4.68, 5.1000000000000005), (4.72, 5.16), (4.78, 5.22), (4.82, 5.28), (4.88, 5.3), (NaN, NaN), (3.66, 4.78), (3.74, 4.78), (3.8200000000000003, 4.78), (3.92, 4.78), (3.98, 4.76), (4.04, 4.74), (4.1, 4.72), (4.16, 4.7), (4.22, 4.68), (4.3, 4.68), (4.4, 4.68), (4.48, 4.66), (4.54, 4.64), (4.62, 4.62), (4.68, 4.6000000000000005), (4.74, 4.58), (4.82, 4.5600000000000005), (4.88, 4.5), (4.9, 4.44), (4.94, 4.38), (4.96, 4.32), (5.0200000000000005, 4.26), (5.08, 4.22), (5.14, 4.18), (NaN, NaN), (3.6, 4.78), (3.64, 4.68), (3.66, 4.58), (3.68, 4.5200000000000005), (3.7, 4.46), (3.72, 4.4), (3.7600000000000002, 4.34), (3.8000000000000003, 4.28), (3.84, 4.2), (3.9, 4.12), (3.94, 4.04), (4, 3.98), (4.0600000000000005, 3.92), (4.1, 3.84), (4.14, 3.7600000000000002), (4.2, 3.72), (4.26, 3.68), (4.3, 3.62), (4.34, 3.56), (4.4, 3.52), (4.44, 3.46), (4.5, 3.42), (4.5600000000000005, 3.4), (4.58, 3.34), (NaN, NaN), (3.52, 4.76), (3.5, 4.7), (3.5, 4.6000000000000005), (3.48, 4.5200000000000005), (3.46, 4.44), (3.46, 4.36), (3.46, 4.28), (3.46, 4.2), (3.46, 4.12), (3.46, 4.04), (3.46, 3.96), (3.46, 3.88), (3.46, 3.8000000000000003), (3.46, 3.72), (3.46, 3.64), (3.46, 3.56), (3.48, 3.5), (3.5, 3.44), (3.52, 3.38), (3.56, 3.3200000000000003), (3.58, 3.24), (3.58, 3.16), (NaN, NaN), (2.4, 3.96), (2.48, 3.96), (2.56, 3.96), (2.66, 3.96), (2.72, 3.92), (2.7800000000000002, 3.86), (2.84, 3.8200000000000003), (2.86, 3.7600000000000002), (2.92, 3.72), (2.98, 3.66), (3.02, 3.6), (3.04, 3.54), (3.1, 3.46), (3.12, 3.4), (3.16, 3.34), (3.22, 3.3000000000000003), (3.2800000000000002, 3.24), (3.34, 3.2), (3.4, 3.16), (3.46, 3.12), (3.52, 3.08), (3.56, 3.02), (3.62, 3), (NaN, NaN), (-0.1, 1.04), (-0.04, 1.06), (0.02, 1.08), (0.08, 1.1), (0.14, 1.1400000000000001), (0.2, 1.16), (0.28, 1.16), (NaN, NaN), (0.02, 1.02), (0.1, 1.04), (0.16, 1.08), (NaN, NaN), (-0.52, 0.62), (-0.46, 0.66), (NaN, NaN), (2.1, -0.66), (2.12, -0.56), (2.18, -0.48), (2.22, -0.4), (2.2800000000000002, -0.34), (2.36, -0.3), (2.4, -0.24), (2.48, -0.18), (2.56, -0.12), (2.62, -0.08), (2.68, -0.06), (2.7600000000000002, -0.06), (2.84, -0.06), (2.92, -0.06), (3, -0.06), (3.06, -0.04), (3.14, 0.02), (3.2, 0.04), (3.2600000000000002, 0.06), (3.3200000000000003, 0.1), (3.4, 0.14), (3.48, 0.14), (3.56, 0.14), (3.64, 0.16), (3.7, 0.22), (3.7800000000000002, 0.22), (3.84, 0.24), (3.92, 0.24), (4, 0.24), (4.0600000000000005, 0.26), (4.12, 0.3), (4.18, 0.32), (4.24, 0.34), (4.28, 0.4), (4.34, 0.46), (4.38, 0.52), (4.44, 0.58), (4.48, 0.64), (NaN, NaN), (3.7, 0.22), (3.7800000000000002, 0.22), (3.86, 0.22), (3.94, 0.22), (4.04, 0.22), (4.1, 0.2), (4.18, 0.2), (4.26, 0.18), (4.32, 0.16), (4.4, 0.14), (4.46, 0.12), (4.5200000000000005, 0.1), (4.58, 0.06), (4.64, 0), (4.7, -0.02), (4.76, -0.06), (4.82, -0.08), (4.9, -0.08), (4.96, -0.1), (5.0200000000000005, -0.12), (5.1000000000000005, -0.12), (5.18, -0.12), (NaN, NaN), (3.74, 0.22), (3.74, 0.14), (3.7800000000000002, 0.08), (3.8000000000000003, 0.02), (3.84, -0.04), (3.86, -0.1), (3.9, -0.16), (3.92, -0.22), (3.94, -0.28), (3.96, -0.34), (4, -0.4), (4.0600000000000005, -0.42), (4.08, -0.5), (4.12, -0.56), (4.14, -0.62), (4.16, -0.68), (4.18, -0.74), (4.22, -0.8), (4.24, -0.86), (4.26, -0.9400000000000001), (4.28, -1), (4.28, -1.08), (4.3, -1.1400000000000001), (4.32, -1.2), (4.34, -1.26), (NaN, NaN), (3.68, 0.24), (3.68, 0.12), (3.66, 0.02), (3.64, -0.06), (3.62, -0.16), (3.56, -0.28), (3.54, -0.34), (3.52, -0.4), (3.5, -0.46), (3.48, -0.56), (3.44, -0.64), (3.42, -0.7000000000000001), (3.42, -0.8), (3.4, -0.88), (3.4, -0.96), (3.4, -1.04), (3.38, -1.12), (3.36, -1.22), (3.36, -1.32), (3.36, -1.4000000000000001), (3.34, -1.48), (NaN, NaN), (2.12, -0.58), (2.22, -0.58), (2.2800000000000002, -0.6), (2.36, -0.62), (2.42, -0.66), (2.5, -0.7000000000000001), (2.56, -0.74), (2.62, -0.78), (2.68, -0.8200000000000001), (2.74, -0.86), (2.8000000000000003, -0.92), (2.84, -0.98), (2.9, -1.02), (2.96, -1.08), (3.04, -1.08), (3.1, -1.1400000000000001), (3.12, -1.2), (3.16, -1.26), (3.18, -1.32), (3.24, -1.36), (3.2600000000000002, -1.42), (3.3000000000000003, -1.48), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke3

AI Drawing Prompt Reference

Create a plane tessellation using a parallelogram as the base tile. Define two translation vectors along the sides of the parallelogram. Translate the parallelogram repeatedly along these vectors to generate adjacent copies, continuing the process to cover the entire plane seamlessly. Add points on the translated shapes to indicate corresponding positions under translation. The final image shows a regular, continuous grid pattern demonstrating how translation and vectors create a repeating tiling.

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