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Exploring Checkerboard Patterns Through Reflection

Exploring Checkerboard Patterns Through Reflection
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2015-01-31 23:04
PolygonCoordinate GeometryTransformation
This example demonstrates the creation of a checkerboard pattern by repeatedly reflecting an initial quadrilateral around its edges. It illustrates the principles of reflection and geometric construction. Commands used include Polygon and Mirror, suitable for middle school and above students learning geometric transformations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Mirror(A, b) → A'
  3. Mirror(B, b) → B'
  4. Mirror(C, b) → C'
  5. Mirror(D, b) → D'
  6. Polygon(A', B', C', D') → poly1', a', b', c', d'
  7. Mirror(A', d') → A''
  8. Mirror(B', d') → B''
  9. Mirror(C', d') → C''
  10. Mirror(D', d') → D''
  11. Polygon(A'', B'', C'', D'') → poly1'', a'', b'', c'', d''
  12. Mirror(A'', b'') → A'''
  13. Mirror(B'', b'') → B'''
  14. Mirror(C'', b'') → C'''
  15. Mirror(D'', b'') → D'''
  16. Polygon(A''', B''', C''', D''') → poly1''', a''', b''', c''', d'''
  17. Mirror(A''', d''') → E
  18. Mirror(B''', d''') → F
  19. Mirror(C''', d''') → G
  20. Mirror(D''', d''') → H
  21. Polygon(E, F, G, H) → poly2, e, f, g, h
  22. Mirror(A, c) → A'_1
  23. Mirror(B, c) → B'_1
  24. Mirror(C, c) → C'_1
  25. Mirror(D, c) → D'_1
  26. Polygon(A'_1, B'_1, C'_1, D'_1) → poly1'_1, a'_1, b'_1, c'_1, d'_1
  27. Mirror(A'_1, a'_1) → I
  28. Mirror(B'_1, a'_1) → J
  29. Mirror(C'_1, a'_1) → K
  30. Mirror(D'_1, a'_1) → L
  31. Polygon(I, J, K, L) → poly3, i, j, k, l
  32. Mirror(I, k) → I'
  33. Mirror(J, k) → J'
  34. Mirror(K, k) → K'
  35. Mirror(L, k) → L'
  36. Polygon(I', J', K', L') → poly3', i', j', k', l'
  37. Mirror(I', i') → I''
  38. Mirror(J', i') → J''
  39. Mirror(K', i') → K''
  40. Mirror(L', i') → L''
  41. Polygon(I'', J'', K'', L'') → poly3'', i'', j'', k'', l''
  42. Mirror(A, d) → A'_2
  43. Mirror(B, d) → B'_2
  44. Mirror(C, d) → C'_2
  45. Mirror(D, d) → D'_2
  46. Polygon(A'_2, B'_2, C'_2, D'_2) → poly1'_2, a'_2, b'_2, c'_2, d'_2
  47. Mirror(A'_2, b'_2) → M
  48. Mirror(B'_2, b'_2) → N
  49. Mirror(C'_2, b'_2) → O
  50. Mirror(D'_2, b'_2) → P
  51. Polygon(M, N, O, P) → poly4, m, n, o, p
  52. Mirror(M, p) → M'
  53. Mirror(N, p) → N'
  54. Mirror(O, p) → O'
  55. Mirror(P, p) → P'
  56. Polygon(M', N', O', P') → poly4', m', n', o', p'
  57. Mirror(M', n') → M''
  58. Mirror(N', n') → N''
  59. Mirror(O', n') → O''
  60. Mirror(P', n') → P''
  61. Polygon(M'', N'', O'', P'') → poly4'', m'', n'', o'', p''
  62. Mirror(A, a) → A'_3
  63. Mirror(B, a) → B'_3
  64. Mirror(C, a) → C'_3
  65. Mirror(D, a) → D'_3
  66. Polygon(A'_3, B'_3, C'_3, D'_3) → poly1'_3, a'_3, b'_3, c'_3, d'_3
  67. Mirror(A'_3, c'_3) → Q
  68. Mirror(B'_3, c'_3) → R
  69. Mirror(C'_3, c'_3) → S
  70. Mirror(D'_3, c'_3) → T
  71. Polygon(Q, R, S, T) → poly5, q, r, s, t
  72. Mirror(Q, q) → Q'
  73. Mirror(R, q) → R'
  74. Mirror(S, q) → S'
  75. Mirror(T, q) → T'
  76. Polygon(Q', R', S', T') → poly5', q', r', s', t'
  77. Mirror(Q', s') → Q''
  78. Mirror(R', s') → R''
  79. Mirror(S', s') → S''
  80. Mirror(T', s') → T''
  81. Polygon(Q'', R'', S'', T'') → poly5'', q'', r'', s'', t''
  82. Mirror(I'', j'') → I'''
  83. Mirror(J'', j'') → J'''
  84. Mirror(K'', j'') → K'''
  85. Mirror(L'', j'') → L'''
  86. Polygon(I''', J''', K''', L''') → poly3''', i''', j''', k''', l'''
  87. Mirror(I''', l''') → U
  88. Mirror(J''', l''') → V
  89. Mirror(K''', l''') → W
  90. Mirror(L''', l''') → Z
  91. Polygon(U, V, W, Z) → poly6, u, v, w, z
  92. Mirror(U, v) → U'
  93. Mirror(V, v) → V'
  94. Mirror(W, v) → W'
  95. Mirror(Z, v) → Z'
  96. Polygon(U', V', W', Z') → poly6', u', v', w', z'
  97. Mirror(U', z') → U''
  98. Mirror(V', z') → V''
  99. Mirror(W', z') → W''
  100. Mirror(Z', z') → Z''
  101. Polygon(U'', V'', W'', Z'') → poly6'', u'', v'', w'', z''
  102. Mirror(I'', l'') → I'''_1
  103. Mirror(J'', l'') → J'''_1
  104. Mirror(K'', l'') → K'''_1
  105. Mirror(L'', l'') → L'''_1
  106. Polygon(I'''_1, J'''_1, K'''_1, L'''_1) → poly3'''_1, i'''_1, j'''_1, k'''_1, l'''_1
  107. Mirror(I'''_1, j'''_1) → A_1
  108. Mirror(J'''_1, j'''_1) → B_1
  109. Mirror(K'''_1, j'''_1) → C_1
  110. Mirror(L'''_1, j'''_1) → D_1
  111. Polygon(A_1, B_1, C_1, D_1) → poly7, a_1, b_1, c_1, d_1
  112. Mirror(A_1, d_1) → E_1
  113. Mirror(B_1, d_1) → F_1
  114. Mirror(C_1, d_1) → G_1
  115. Mirror(D_1, d_1) → H_1
  116. Polygon(E_1, F_1, G_1, H_1) → poly7', e_1, f_1, g_1, h_1
  117. Mirror(E_1, f_1) → I_1
  118. Mirror(F_1, f_1) → J_1
  119. Mirror(G_1, f_1) → K_1
  120. Mirror(H_1, f_1) → L_1
  121. Polygon(I_1, J_1, K_1, L_1) → poly7'', i_1, j_1, k_1, l_1
  122. Mirror(E, g) → E'
  123. Mirror(F, g) → F'
  124. Mirror(G, g) → G'
  125. Mirror(H, g) → H'
  126. Polygon(E', F', G', H') → poly2', e', f', g', h'
  127. Mirror(E', e') → E''
  128. Mirror(F', e') → F''
  129. Mirror(G', e') → G''
  130. Mirror(H', e') → H''
  131. Polygon(E'', F'', G'', H'') → poly2'', e'', f'', g'', h''
  132. Mirror(E'', g'') → E'''
  133. Mirror(F'', g'') → F'''
  134. Mirror(G'', g'') → G'''
  135. Mirror(H'', g'') → H'''
  136. Polygon(E''', F''', G''', H''') → poly2''', e''', f''', g''', h'''
  137. Mirror(E, e) → E'_1
  138. Mirror(F, e) → F'_1
  139. Mirror(G, e) → G'_1
  140. Mirror(H, e) → H'_1
  141. Polygon(E'_1, F'_1, G'_1, H'_1) → poly2'_1, e'_1, f'_1, g'_1, h'_1
  142. Mirror(E'_1, g'_1) → Q_1
  143. Mirror(F'_1, g'_1) → R_1
  144. Mirror(G'_1, g'_1) → S_1
  145. Mirror(H'_1, g'_1) → T_1
  146. Polygon(Q_1, R_1, S_1, T_1) → poly9, q_1, r_1, s_1, t_1
  147. Mirror(Q_1, q_1) → U_1
  148. Mirror(R_1, q_1) → V_1
  149. Mirror(S_1, q_1) → W_1
  150. Mirror(T_1, q_1) → Z_1
  151. Polygon(U_1, V_1, W_1, Z_1) → poly9', u_1, v_1, w_1, z_1
  152. Mirror(U_1, w_1) → A_2
  153. Mirror(V_1, w_1) → B_2
  154. Mirror(W_1, w_1) → C_2
  155. Mirror(Z_1, w_1) → D_2
  156. Polygon(A_2, B_2, C_2, D_2) → poly9'', a_2, b_2, c_2, d_2
  157. Mirror(Q'', r'') → Q'''
  158. Mirror(R'', r'') → R'''
  159. Mirror(S'', r'') → S'''
  160. Mirror(T'', r'') → T'''
  161. Polygon(Q''', R''', S''', T''') → poly5''', q''', r''', s''', t'''
  162. Mirror(Q''', t''') → E_2
  163. Mirror(R''', t''') → F_2
  164. Mirror(S''', t''') → G_2
  165. Mirror(T''', t''') → H_2
  166. Polygon(E_2, F_2, G_2, H_2) → poly10, e_2, f_2, g_2, h_2
  167. Mirror(E_2, f_2) → I_2
  168. Mirror(F_2, f_2) → J_2
  169. Mirror(G_2, f_2) → K_2
  170. Mirror(H_2, f_2) → L_2
  171. Polygon(I_2, J_2, K_2, L_2) → poly10', i_2, j_2, k_2, l_2
  172. Mirror(Q'', t'') → Q'''_1
  173. Mirror(R'', t'') → R'''_1
  174. Mirror(S'', t'') → S'''_1
  175. Mirror(T'', t'') → T'''_1
  176. Polygon(Q'''_1, R'''_1, S'''_1, T'''_1) → poly5'''_1, q'''_1, r'''_1, s'''_1, t'''_1
  177. Mirror(Q'''_1, r'''_1) → Q_2
  178. Mirror(R'''_1, r'''_1) → R_2
  179. Mirror(S'''_1, r'''_1) → S_2
  180. Mirror(T'''_1, r'''_1) → T_2
  181. Polygon(Q_2, R_2, S_2, T_2) → poly11, q_2, r_2, s_2, t_2
  182. Mirror(Q_2, t_2) → U_2
  183. Mirror(R_2, t_2) → V_2
  184. Mirror(S_2, t_2) → W_2
  185. Mirror(T_2, t_2) → Z_2
  186. Polygon(U_2, V_2, W_2, Z_2) → poly11', u_2, v_2, w_2, z_2
  187. Mirror(U_2, v_2) → A_3
  188. Mirror(V_2, v_2) → B_3
  189. Mirror(W_2, v_2) → C_3
  190. Mirror(Z_2, v_2) → D_3
  191. Polygon(A_3, B_3, C_3, D_3) → poly11'', a_3, b_3, c_3, d_3
  192. Mirror(M'', m'') → M'''
  193. Mirror(N'', m'') → N'''
  194. Mirror(O'', m'') → O'''
  195. Mirror(P'', m'') → P'''
  196. Polygon(M''', N''', O''', P''') → poly4''', m''', n''', o''', p'''
  197. Mirror(M''', o''') → E_3
  198. Mirror(N''', o''') → F_3
  199. Mirror(O''', o''') → G_3
  200. Mirror(P''', o''') → H_3
  201. Polygon(E_3, F_3, G_3, H_3) → poly12, e_3, f_3, g_3, h_3
  202. Mirror(E_3, e_3) → I_3
  203. Mirror(F_3, e_3) → J_3
  204. Mirror(G_3, e_3) → K_3
  205. Mirror(H_3, e_3) → L_3
  206. Polygon(I_3, J_3, K_3, L_3) → poly12', i_3, j_3, k_3, l_3
  207. Mirror(M'', o'') → M'''_1
  208. Mirror(N'', o'') → N'''_1
  209. Mirror(O'', o'') → O'''_1
  210. Mirror(P'', o'') → P'''_1
  211. Polygon(M'''_1, N'''_1, O'''_1, P'''_1) → poly4'''_1, m'''_1, n'''_1, o'''_1, p'''_1
  212. Mirror(M'''_1, m'''_1) → Q_3
  213. Mirror(N'''_1, m'''_1) → R_3
  214. Mirror(O'''_1, m'''_1) → S_3
  215. Mirror(P'''_1, m'''_1) → T_3
  216. Polygon(Q_3, R_3, S_3, T_3) → poly13, q_3, r_3, s_3, t_3
  217. Mirror(Q_3, s_3) → U_3
  218. Mirror(R_3, s_3) → V_3
  219. Mirror(S_3, s_3) → W_3
  220. Mirror(T_3, s_3) → Z_3
  221. Polygon(U_3, V_3, W_3, Z_3) → poly13', u_3, v_3, w_3, z_3
  222. Mirror(U_3, u_3) → M_1
  223. Mirror(V_3, u_3) → N_1
  224. Mirror(W_3, u_3) → O_1
  225. Mirror(Z_3, u_3) → P_1
  226. Polygon(M_1, N_1, O_1, P_1) → poly13'', m_1, n_1, o_1, p_1
  227. Mirror(E''', e''') → M_2
  228. Mirror(F''', e''') → N_2
  229. Mirror(G''', e''') → O_2
  230. Mirror(H''', e''') → P_2
  231. Polygon(M_2, N_2, O_2, P_2) → poly8, m_2, n_2, o_2, p_2
  232. Mirror(I_2, l_2) → M_3
  233. Mirror(J_2, l_2) → N_3
  234. Mirror(K_2, l_2) → O_3
  235. Mirror(L_2, l_2) → P_3
  236. Polygon(M_3, N_3, O_3, P_3) → poly10'', m_3, n_3, o_3, p_3
  237. Mirror(I_3, k_3) → A_4
  238. Mirror(J_3, k_3) → B_4
  239. Mirror(K_3, k_3) → C_4
  240. Mirror(L_3, k_3) → D_4
  241. Polygon(A_4, B_4, C_4, D_4) → poly12'', a_4, b_4, c_4, d_4
  242. Mirror(Q, r) → Q'_1
  243. Mirror(R, r) → R'_1
  244. Mirror(S, r) → S'_1
  245. Mirror(T, r) → T'_1
  246. Polygon(Q'_1, R'_1, S'_1, T'_1) → poly5'_1, q'_1, r'_1, s'_1, t'_1
  247. Mirror(E_2, g_2) → E_4
  248. Mirror(F_2, g_2) → F_4
  249. Mirror(G_2, g_2) → G_4
  250. Mirror(H_2, g_2) → H_4
  251. Polygon(E_4, F_4, G_4, H_4) → poly10'_1, e_4, f_4, g_4, h_4
  252. Mirror(Q_1, t_1) → I_4
  253. Mirror(R_1, t_1) → J_4
  254. Mirror(S_1, t_1) → K_4
  255. Mirror(T_1, t_1) → L_4
  256. Polygon(I_4, J_4, K_4, L_4) → poly9'_1, i_4, j_4, k_4, l_4
  257. Mirror(A'', a'') → A'''_1
  258. Mirror(B'', a'') → B'''_1
  259. Mirror(C'', a'') → C'''_1
  260. Mirror(D'', a'') → D'''_1
  261. Polygon(A'''_1, B'''_1, C'''_1, D'''_1) → poly1'''_1, a'''_1, b'''_1, c'''_1, d'''_1
  262. Mirror(A'', c'') → A'''_2
  263. Mirror(B'', c'') → B'''_2
  264. Mirror(C'', c'') → C'''_2
  265. Mirror(D'', c'') → D'''_2
  266. Polygon(A'''_2, B'''_2, C'''_2, D'''_2) → poly1'''_2, a'''_2, b'''_2, c'''_2, d'''_2
  267. Mirror(U, u) → U'_1
  268. Mirror(V, u) → V'_1
  269. Mirror(W, u) → W'_1
  270. Mirror(Z, u) → Z'_1
  271. Polygon(U'_1, V'_1, W'_1, Z'_1) → poly6'_1, u'_1, v'_1, w'_1, z'_1
  272. Mirror(E'', h'') → E'''_1
  273. Mirror(F'', h'') → F'''_1
  274. Mirror(G'', h'') → G'''_1
  275. Mirror(H'', h'') → H'''_1
  276. Polygon(E'''_1, F'''_1, G'''_1, H'''_1) → poly2'''_1, e'''_1, f'''_1, g'''_1, h'''_1
  277. Mirror(I, j) → I'_1
  278. Mirror(J, j) → J'_1
  279. Mirror(K, j) → K'_1
  280. Mirror(L, j) → L'_1
  281. Polygon(I'_1, J'_1, K'_1, L'_1) → poly3'_1, i'_1, j'_1, k'_1, l'_1
  282. Mirror(A_1, a_1) → M_4
  283. Mirror(B_1, a_1) → N_4
  284. Mirror(C_1, a_1) → O_4
  285. Mirror(D_1, a_1) → P_4
  286. Polygon(M_4, N_4, O_4, P_4) → poly7'_1, m_4, n_4, o_4, p_4
  287. Mirror(Q_3, r_3) → Q_4
  288. Mirror(R_3, r_3) → R_4
  289. Mirror(S_3, r_3) → S_4
  290. Mirror(T_3, r_3) → T_4
  291. Polygon(Q_4, R_4, S_4, T_4) → poly13'_1, q_4, r_4, s_4, t_4
  292. Mirror(M, o) → M'_1
  293. Mirror(N, o) → N'_1
  294. Mirror(O, o) → O'_1
  295. Mirror(P, o) → P'_1
  296. Polygon(M'_1, N'_1, O'_1, P'_1) → poly4'_1, m'_1, n'_1, o'_1, p'_1
  297. Mirror(I, l) → I'_2
  298. Mirror(J, l) → J'_2
  299. Mirror(K, l) → K'_2
  300. Mirror(L, l) → L'_2
  301. Polygon(I'_2, J'_2, K'_2, L'_2) → poly3'_2, i'_2, j'_2, k'_2, l'_2
  302. Mirror(M, m) → M'_2
  303. Mirror(N, m) → N'_2
  304. Mirror(O, m) → O'_2
  305. Mirror(P, m) → P'_2
  306. Polygon(M'_2, N'_2, O'_2, P'_2) → poly4'_2, m'_2, n'_2, o'_2, p'_2
  307. Mirror(Q, t) → Q'_2
  308. Mirror(R, t) → R'_2
  309. Mirror(S, t) → S'_2
  310. Mirror(T, t) → T'_2
  311. Polygon(Q'_2, R'_2, S'_2, T'_2) → poly5'_2, q'_2, r'_2, s'_2, t'_2
  312. Mirror(Q_2, s_2) → U_4
  313. Mirror(R_2, s_2) → V_4
  314. Mirror(S_2, s_2) → W_4
  315. Mirror(T_2, s_2) → Z_4
  316. Polygon(U_4, V_4, W_4, Z_4) → poly11'_1, u_4, v_4, w_4, z_4
  317. Mirror(E_3, f_3) → A_5
  318. Mirror(F_3, f_3) → B_5
  319. Mirror(G_3, f_3) → C_5
  320. Mirror(H_3, f_3) → D_5
  321. Polygon(A_5, B_5, C_5, D_5) → poly12'_1, a_5, b_5, c_5, d_5

AI Drawing Prompt Reference

Draw a checkerboard pattern starting from a quadrilateral by repeatedly reflecting it around its edges. Begin with a quadrilateral, then reflect it along each edge to create new quadrilaterals, continuing this process with each new quadrilateral until a complex checkerboard pattern emerges.
The prompt above was compiled from the original drawing commands and is for reference only.