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HEX Geometric Logic via Vector Translations

HEX Geometric Logic via Vector Translations
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2021-02-18 10:11
PolygonVectorTransformationGeometric Construction
In collections GeoGebra and Laser Cutting
This example demonstrates a HEX-style planar construction, showing how a repeated geometric pattern can be generated by chaining vector-based translations. It focuses on key ideas: building polygonal (and polyline) structures, representing direction and magnitude using vectors, and using Translate as a rigid transformation to copy and relocate geometric components systematically. The construction uses core GeoGebra commands and objects: Point to define anchor vertices (e.g., A, B, C…N), Polygon/PolyLine to form the main HEX shapes and intermediate outlines (poly1–poly4), Segment for side connections, Vector to specify the translation directions, and Translate to replicate the polygons and related objects (including transformed variants like f_1, g_1, h_1) across the plane. Suitable for middle/high school geometry practice, interactive math exploration, and lessons on the logic of vector transformations and geometric construction workflows.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Polygon(A, B, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  3. Polygon(D, C, 6) → poly2, l, J, m, n, p, q, r, G, H, I
  4. Polygon(I, H, 6) → poly3, s, N, t, a, b, c, d, K, L, M
  5. Polygon(M, L, 6) → poly4, e, R, f_1, g_1, h_1, i_1, j_1, O, P, Q
  6. Polygon(Q, P, 6) → poly5, k_1, V, l_1, m_1, n_1, p_1, q_1, S, T, U
  7. Polygon(U, T, 6) → poly6, r_1, B_1, s_1, t_1, a_1, b_1, c_1, W, Z, A_1
  8. Polygon(A_1, Z, 6) → poly7, d_1, F_1, e_1, f_2, g_2, h_2, i_2, C_1, D_1, E_1
  9. Polygon(E, D, 6) → poly8, j_2, J_1, k_2, l_2, m_2, n_2, p_2, G_1, H_1, I_1
  10. Polygon(J, I, 6) → poly9, q_2, N_1, r_2, s_2, t_2, a_2, b_2, K_1, L_1, M_1
  11. Polygon(N, M, 6) → poly10, c_2, R_1, d_2, e_2, f_3, g_3, h_3, O_1, P_1, Q_1
  12. Polygon(R, Q, 6) → poly11, i_3, V_1, j_3, k_3, l_3, m_3, n_3, S_1, T_1, U_1
  13. Polygon(V, U, 6) → poly12, p_3, B_2, q_3, r_3, s_3, t_3, a_3, W_1, Z_1, A_2
  14. Polygon(B_1, A_1, 6) → poly13, b_3, F_2, c_3, d_3, e_3, f_4, g_4, C_2, D_2, E_2
  15. Polygon(A_2, Z_1, 6) → poly14, h_4, J_2, i_4, j_4, k_4, l_4, m_4, G_2, H_2, I_2
  16. Polygon(U_1, T_1, 6) → poly15, n_4, N_2, p_4, q_4, r_4, s_4, t_4, K_2, L_2, M_2
  17. Polygon(Q_1, P_1, 6) → poly16, a_4, R_2, b_4, c_4, d_4, e_4, f_5, O_2, P_2, Q_2
  18. Polygon(M_1, L_1, 6) → poly17, g_5, V_2, h_5, i_5, j_5, k_5, l_5, S_2, T_2, U_2
  19. Polygon(I_1, H_1, 6) → poly18, m_5, B_3, n_5, p_5, q_5, r_5, s_5, W_2, Z_2, A_3
  20. Polygon(F, E, 6) → poly19, t_5, F_3, a_5, b_5, c_5, d_5, e_5, C_3, D_3, E_3
  21. Polygon(D_3, J_1, 6) → poly20, f_6, J_3, g_6, h_6, i_6, j_6, k_6, G_3, H_3, I_3
  22. Polygon(E_3, D_3, 6) → poly21, l_6, N_3, m_6, n_6, p_6, q_6, r_6, K_3, L_3, M_3
  23. Polygon(M_3, L_3, 6) → poly22, s_6, R_3, t_6, a_6, b_6, c_6, d_6, O_3, P_3, Q_3
  24. Polygon(L_3, J_3, 6) → poly23, e_6, V_3, f_7, g_7, h_7, i_7, j_7, S_3, T_3, U_3
  25. Polygon(I_3, B_3, 6) → poly24, k_7, B_4, l_7, m_7, n_7, p_7, q_7, W_3, Z_3, A_4
  26. Polygon(A_3, V_2, 6) → poly25, r_7, F_4, s_7, t_7, a_7, b_7, c_7, C_4, D_4, E_4
  27. Polygon(U_2, R_2, 6) → poly26, d_7, J_4, e_7, f_8, g_8, h_8, i_8, G_4, H_4, I_4
  28. Polygon(Q_2, N_2, 6) → poly27, j_8, N_4, k_8, l_8, m_8, n_8, p_8, K_4, L_4, M_4
  29. Polygon(M_2, J_2, 6) → poly28, q_8, R_4, r_8, s_8, t_8, a_8, b_8, O_4, P_4, Q_4
  30. Polygon(M_4, L_4, 6) → poly29, c_8, V_4, d_8, e_8, f_9, g_9, h_9, S_4, T_4, U_4
  31. Polygon(I_4, H_4, 6) → poly30, i_9, B_5, j_9, k_9, l_9, m_9, n_9, W_4, Z_4, A_5
  32. Polygon(E_4, D_4, 6) → poly31, p_9, F_5, q_9, r_9, s_9, t_9, a_9, C_5, D_5, E_5
  33. Polygon(A_4, Z_3, 6) → poly32, b_9, J_5, c_9, d_9, e_9, f_{10}, g_{10}, G_5, H_5, I_5
  34. Polygon(U_3, T_3, 6) → poly33, h_{10}, N_5, i_{10}, j_{10}, k_{10}, l_{10}, m_{10}, K_5, L_5, M_5
  35. Polygon(P_3, O_3, 6) → poly34, n_{10}, R_5, p_{10}, q_{10}, r_{10}, s_{10}, t_{10}, O_5, P_5, Q_5
  36. Polygon(Q_3, P_3, 6) → poly35, a_{10}, V_5, b_{10}, c_{10}, d_{10}, e_{10}, f_{11}, S_5, T_5, U_5
  37. Polygon(T_5, R_5, 6) → poly36, g_{11}, B_6, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, W_5, Z_5, A_6
  38. Polygon(Q_5, N_5, 6) → poly37, m_{11}, F_6, n_{11}, p_{11}, q_{11}, r_{11}, s_{11}, C_6, D_6, E_6
  39. Polygon(M_5, J_5, 6) → poly38, t_{11}, J_6, a_{11}, b_{11}, c_{11}, d_{11}, e_{11}, G_6, H_6, I_6
  40. Polygon(I_5, F_5, 6) → poly39, f_{12}, N_6, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, K_6, L_6, M_6
  41. Polygon(E_5, B_5, 6) → poly40, l_{12}, R_6, m_{12}, n_{12}, p_{12}, q_{12}, r_{12}, O_6, P_6, Q_6
  42. Polygon(A_5, V_4, 6) → poly41, s_{12}, V_6, t_{12}, a_{12}, b_{12}, c_{12}, d_{12}, S_6, T_6, U_6
  43. Polygon(Q_6, P_6, 6) → poly42, e_{12}, B_7, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, W_6, Z_6, A_7
  44. Polygon(M_6, L_6, 6) → poly43, k_{13}, F_7, l_{13}, m_{13}, n_{13}, p_{13}, q_{13}, C_7, D_7, E_7
  45. Polygon(I_6, H_6, 6) → poly44, r_{13}, J_7, s_{13}, t_{13}, a_{13}, b_{13}, c_{13}, G_7, H_7, I_7
  46. Polygon(E_6, D_6, 6) → poly45, d_{13}, N_7, e_{13}, f_{14}, g_{14}, h_{14}, i_{14}, K_7, L_7, M_7
  47. Polygon(A_6, Z_5, 6) → poly46, j_{14}, R_7, k_{14}, l_{14}, m_{14}, n_{14}, p_{14}, O_7, P_7, Q_7
  48. Polygon(U_5, T_5, 6) → poly47, q_{14}, V_7, r_{14}, s_{14}, t_{14}, a_{14}, b_{14}, S_7, T_7, U_7
  49. Polygon(T_7, B_6, 6) → poly48, c_{14}, B_8, d_{14}, e_{14}, f_{15}, g_{15}, h_{15}, W_7, Z_7, A_8
  50. Polygon(U_7, T_7, 6) → poly49, i_{15}, F_8, j_{15}, k_{15}, l_{15}, m_{15}, n_{15}, C_8, D_8, E_8
  51. PolyLine(E_8, F_8, E_3, F_3, F, A, B, C, G, H, K, L, U_7, O, P, S, T, W, Z, C_1, D_1, E_1, F_1, V_7, D_2, E_2, H_2, I_2, P_4, Q_4, T_4, U_4, T_6, U_6, U_5, Z_6, A_7, B_7, E_7, F_7, I_7, J_7, M_7, N_7, Q_7, V_5, R_7, A_8, B_8, D_8, Q_3, R_3, M_3, N_3) → p_{15}
  52. Point(p_{15}) → G_8
  53. Point(p_{15}) → H_8
  54. Segment(G_8, H_8) → q_{15}
  55. PolyLine(U_7, T_7, L_6, Q_6, P_6, U_6, B_6, A_6, Z_5, E_6, D_6, I_6, H_6, M_6) → r_{15}
  56. Vector(U_7, U_5) → u
  57. Translate(U_7, u) → I_8
  58. Translate(T_7, u) → J_8
  59. Translate(B_6, u) → K_8
  60. Translate(A_6, u) → L_8
  61. Translate(Z_5, u) → M_8
  62. Translate(E_6, u) → N_8
  63. Translate(D_6, u) → O_8
  64. Translate(I_6, u) → P_8
  65. Translate(H_6, u) → Q_8
  66. Translate(M_6, u) → R_8
  67. Translate(L_6, u) → S_8
  68. Translate(Q_6, u) → T_8
  69. Translate(P_6, u) → U_8
  70. Translate(U_6, u) → V_8
  71. PolyLine(I_8, J_8, S_8, T_8, U_8, V_8, K_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8) → s_{15}
  72. Translate(I_8, u) → W_8
  73. Translate(J_8, u) → Z_8
  74. Translate(K_8, u) → A_9
  75. Translate(L_8, u) → B_9
  76. Translate(M_8, u) → C_9
  77. Translate(N_8, u) → D_9
  78. Translate(O_8, u) → E_9
  79. Translate(P_8, u) → F_9
  80. Translate(Q_8, u) → G_9
  81. Translate(R_8, u) → H_9
  82. Translate(S_8, u) → I_9
  83. Translate(T_8, u) → J_9
  84. Translate(U_8, u) → K_9
  85. Translate(V_8, u) → L_9
  86. PolyLine(W_8, Z_8, I_9, J_9, K_9, L_9, A_9, B_9, C_9, D_9, E_9, F_9, G_9, H_9) → t_{15}
  87. Translate(W_8, u) → M_9
  88. Translate(Z_8, u) → N_9
  89. Translate(A_9, u) → O_9
  90. Translate(B_9, u) → P_9
  91. Translate(C_9, u) → Q_9
  92. Translate(D_9, u) → R_9
  93. Translate(E_9, u) → S_9
  94. Translate(F_9, u) → T_9
  95. Translate(G_9, u) → U_9
  96. Translate(H_9, u) → V_9
  97. Translate(I_9, u) → W_9
  98. Translate(J_9, u) → Z_9
  99. Translate(K_9, u) → A_{10}
  100. Translate(L_9, u) → B_{10}
  101. PolyLine(M_9, N_9, W_9, Z_9, A_{10}, B_{10}, O_9, P_9, Q_9, R_9, S_9, T_9, U_9, V_9) → a_{15}
  102. Translate(M_9, u) → C_{10}
  103. Translate(N_9, u) → D_{10}
  104. Translate(O_9, u) → E_{10}
  105. Translate(P_9, u) → F_{10}
  106. Translate(Q_9, u) → G_{10}
  107. Translate(R_9, u) → H_{10}
  108. Translate(S_9, u) → I_{10}
  109. Translate(T_9, u) → J_{10}
  110. Translate(U_9, u) → K_{10}
  111. Translate(V_9, u) → L_{10}
  112. Translate(W_9, u) → M_{10}
  113. Translate(Z_9, u) → N_{10}
  114. Translate(A_{10}, u) → O_{10}
  115. Translate(B_{10}, u) → P_{10}
  116. PolyLine(C_{10}, D_{10}, M_{10}, N_{10}, O_{10}, P_{10}, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, J_{10}, K_{10}, L_{10}) → b_{15}
  117. Translate(C_{10}, u) → Q_{10}
  118. Translate(D_{10}, u) → R_{10}
  119. Translate(E_{10}, u) → S_{10}
  120. Translate(F_{10}, u) → T_{10}
  121. Translate(G_{10}, u) → U_{10}
  122. Translate(H_{10}, u) → V_{10}
  123. Translate(I_{10}, u) → W_{10}
  124. Translate(J_{10}, u) → Z_{10}
  125. Translate(K_{10}, u) → A_{11}
  126. Translate(L_{10}, u) → B_{11}
  127. Translate(M_{10}, u) → C_{11}
  128. Translate(N_{10}, u) → D_{11}
  129. Translate(O_{10}, u) → E_{11}
  130. Translate(P_{10}, u) → F_{11}
  131. PolyLine(Q_{10}, R_{10}, C_{11}, D_{11}, E_{11}, F_{11}, S_{10}, T_{10}, U_{10}, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}) → c_{15}
  132. PolyLine(B_8, T_7, E_{10}, R_{10}, S_{10}, C, B_6, J_8, K_8, Z_8, A_9, N_9, O_9, D_{10}) → d_{15}
  133. Vector(N_7, J_7) → v
  134. Translate(B_8, v) → G_{11}
  135. Translate(T_7, v) → H_{11}
  136. Translate(B_6, v) → I_{11}
  137. Translate(J_8, v) → J_{11}
  138. Translate(K_8, v) → K_{11}
  139. Translate(Z_8, v) → L_{11}
  140. Translate(A_9, v) → M_{11}
  141. Translate(N_9, v) → N_{11}
  142. Translate(O_9, v) → O_{11}
  143. Translate(D_{10}, v) → P_{11}
  144. Translate(E_{10}, v) → Q_{11}
  145. Translate(R_{10}, v) → R_{11}
  146. Translate(S_{10}, v) → S_{11}
  147. Translate(C, v) → C'
  148. PolyLine(G_{11}, H_{11}, Q_{11}, R_{11}, S_{11}, C', I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}) → e_{15}
  149. Translate(G_{11}, v) → T_{11}
  150. Translate(H_{11}, v) → U_{11}
  151. Translate(I_{11}, v) → V_{11}
  152. Translate(J_{11}, v) → W_{11}
  153. Translate(K_{11}, v) → Z_{11}
  154. Translate(L_{11}, v) → A_{12}
  155. Translate(M_{11}, v) → B_{12}
  156. Translate(N_{11}, v) → C_{12}
  157. Translate(O_{11}, v) → D_{12}
  158. Translate(P_{11}, v) → E_{12}
  159. Translate(Q_{11}, v) → F_{12}
  160. Translate(R_{11}, v) → G_{12}
  161. Translate(S_{11}, v) → H_{12}
  162. Translate(C', v) → C''
  163. PolyLine(T_{11}, U_{11}, F_{12}, G_{12}, H_{12}, C'', V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, D_{12}, E_{12}) → f_{16}
  164. Translate(T_{11}, v) → I_{12}
  165. Translate(U_{11}, v) → J_{12}
  166. Translate(V_{11}, v) → K_{12}
  167. Translate(W_{11}, v) → L_{12}
  168. Translate(Z_{11}, v) → M_{12}
  169. Translate(A_{12}, v) → N_{12}
  170. Translate(B_{12}, v) → O_{12}
  171. Translate(C_{12}, v) → P_{12}
  172. Translate(D_{12}, v) → Q_{12}
  173. Translate(E_{12}, v) → R_{12}
  174. Translate(F_{12}, v) → S_{12}
  175. Translate(G_{12}, v) → T_{12}
  176. Translate(H_{12}, v) → U_{12}
  177. Translate(C'', v) → C'''
  178. PolyLine(I_{12}, J_{12}, S_{12}, T_{12}, U_{12}, C''', K_{12}, L_{12}, M_{12}, N_{12}, O_{12}, P_{12}, Q_{12}, R_{12}) → g_{16}
  179. Translate(I_{12}, v) → V_{12}
  180. Translate(J_{12}, v) → W_{12}
  181. Translate(K_{12}, v) → Z_{12}
  182. Translate(L_{12}, v) → A_{13}
  183. Translate(M_{12}, v) → B_{13}
  184. Translate(N_{12}, v) → C_{13}
  185. Translate(O_{12}, v) → D_{13}
  186. Translate(P_{12}, v) → E_{13}
  187. Translate(Q_{12}, v) → F_{13}
  188. Translate(R_{12}, v) → G_{13}
  189. Translate(S_{12}, v) → H_{13}
  190. Translate(T_{12}, v) → I_{13}
  191. Translate(U_{12}, v) → J_{13}
  192. Translate(C''', v) → K_{13}
  193. PolyLine(V_{12}, W_{12}, H_{13}, I_{13}, J_{13}, K_{13}, Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}, G_{13}) → h_{16}
  194. Translate(V_{12}, v) → L_{13}
  195. Translate(W_{12}, v) → M_{13}
  196. Translate(Z_{12}, v) → N_{13}
  197. Translate(A_{13}, v) → O_{13}
  198. Translate(B_{13}, v) → P_{13}
  199. Translate(C_{13}, v) → Q_{13}
  200. Translate(D_{13}, v) → R_{13}
  201. Translate(E_{13}, v) → S_{13}
  202. Translate(F_{13}, v) → T_{13}
  203. Translate(G_{13}, v) → U_{13}
  204. Translate(H_{13}, v) → V_{13}
  205. Translate(I_{13}, v) → W_{13}
  206. Translate(J_{13}, v) → Z_{13}
  207. Translate(K_{13}, v) → A_{14}
  208. PolyLine(L_{13}, M_{13}, V_{13}, W_{13}, Z_{13}, A_{14}, N_{13}, O_{13}, P_{13}, Q_{13}, R_{13}, S_{13}, T_{13}, U_{13}) → i_{16}

AI Drawing Prompt Reference

Create a flat geometric pattern composed of many regular hexagons tightly packed together, forming a honeycomb-like grid. Start with one central hexagon at the bottom, then add adjacent hexagons layer by layer upward and outward, each sharing a full side with an existing one. All hexagons are equal in size and orientation, with edges aligned, covering a large rectangular area. Show only the outlines of the hexagons without fill, keeping lines clear for geometric observation.
The prompt above was compiled from the original drawing commands and is for reference only.