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Space Exploration

Space Exploration
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2016-11-14 03:00
PolygonSolid GeometryTransformationGeometric Construction
This example demonstrates the interaction between spheres and polygons in spatial geometry. Core math concepts include spheres, polygons, symmetry, rotation, and intersection calculations. Construction highlights from the summary involve commands like Mirror, Polygon, Midpoint, Circle, Intersect, Line, Plane, Sphere, IntersectConic, and Rotate, along with objects such as A, B, C, D, and their transformed versions. It is suitable for high school and above students to learn and understand solid geometry, geometric transformations, and spatial relationships, ideal for teaching demonstrations and geometric exploration.

GeoGebra Commands & Parameters

  1. Mirror(A, B) → A'
  2. Mirror(B, A') → B'
  3. Mirror(C, D) → C'
  4. Mirror(D, C') → D'
  5. Mirror(D', B') → D''
  6. Mirror(C', A') → C''
  7. Mirror(D, B) → D'_1
  8. Mirror(C, A) → C'_1
  9. Polygon(D', C', A', B') → wielokąt1, d', c', a', b'
  10. Polygon(C', D, B, A') → wielokąt2, c'_1, d, b, a'_1
  11. Polygon(D, C, A, B) → wielokąt3, d_1, c, a, b_1
  12. Polygon(B', A', C'', D'') → wielokąt4, b'_1, a'_2, c'', d''
  13. Polygon(A', B, D'_1, C'') → wielokąt5, a'_3, b_2, d'_1, c''_1
  14. Polygon(B, A, C'_1, D'_1) → wielokąt6, b_3, a_1, c'_2, d'_2
  15. Mirror(D', G) → D''_1
  16. Mirror(B', H) → B''
  17. Mirror(H, B'') → H'
  18. Mirror(G, D''_1) → G'
  19. Mirror(D''_1, G') → I
  20. Mirror(B'', H') → B'''
  21. Polygon(G, D', B', H) → wielokąt7, g, d'_3, b'_2, h
  22. Polygon(D''_1, G, H, B'') → wielokąt8, d''_1, g_1, h_1, b''
  23. Polygon(G', D''_1, B'', H') → wielokąt9, g', d''_2, b''_1, h'
  24. Polygon(I, G', H', B''') → wielokąt10, i, g'_1, h'_1, b'''
  25. Mirror(C, F) → C'_2
  26. Mirror(A, E) → A'_1
  27. Mirror(E, A'_1) → E'
  28. Mirror(F, C'_2) → F'
  29. Mirror(C'_2, F') → J
  30. Mirror(A'_1, E') → K
  31. Polygon(J, F', E', K) → wielokąt11, j, f', e', k
  32. Polygon(F', C'_2, A'_1, E') → wielokąt12, f'_1, c'_3, a'_4, e'_1
  33. Polygon(C'_2, F, E, A'_1) → wielokąt13, c'_4, f, e, a'_5
  34. Polygon(F, C, A, E) → wielokąt14, f_1, c_1, a_2, e_1
  35. Mirror(J, L) → J'
  36. Mirror(K, M) → K'
  37. Polygon(I, J', K', B''') → wielokąt15, i_1, j', k', b'''_1
  38. Polygon(J', L, M, K') → wielokąt16, j'_1, l, m, k'_1
  39. Polygon(L, J, K, M) → wielokąt17, l_1, j_1, k_1, m_1
  40. Mirror(J, K) → J'_1
  41. Mirror(F', E') → F''
  42. Mirror(C'_2, A'_1) → N
  43. Polygon(K, E', F'', J'_1) → wielokąt18, k_2, e'_2, f'', j'_2
  44. Polygon(E', A'_1, N, F'') → wielokąt19, e'_3, a'_6, n, f''_1
  45. Mirror(D''_1, B'') → O
  46. Mirror(G', H') → G''
  47. Mirror(I, B''') → I'
  48. Polygon(B'', H', G'', O) → wielokąt20, b''_2, h'_2, g'', o
  49. Polygon(H', B''', I', G'') → wielokąt21, h'_3, b'''_2, i', g''_1
  50. Mirror(I', P) → I''
  51. Midpoint(P, C'') → Q
  52. Midpoint(I'', D'_1) → R
  53. Midpoint(P, Q) → S
  54. Midpoint(I'', R) → T
  55. Midpoint(Q, C'') → U
  56. Midpoint(R, D'_1) → V
  57. Midpoint(N, C'_1) → W
  58. Midpoint(O, D'') → Z
  59. Polygon(J'_1, F'', T, I'') → wielokąt22, j'_3, f''_2, t, i''
  60. Polygon(S, G'', I', P) → wielokąt24, s_1, g''_2, i'_1, p_1
  61. Polygon(F'', N, R, T) → wielokąt25, f''_3, n_1, r, t_2
  62. Polygon(T, R, Q, S) → wielokąt26, t_3, r_1, q, s_2
  63. Polygon(S, Q, O, G'') → wielokąt27, s_3, q_1, o_1, g''_3
  64. Polygon(N, W, V, R) → wielokąt28, n_2, w, v, r_2
  65. Polygon(R, V, U, Q) → wielokąt29, r_3, v_1, u, q_2
  66. Polygon(Q, U, Z, O) → wielokąt30, q_3, u_1, z_1, o_2
  67. Polygon(U, C'', D'', Z) → wielokąt23, u_2, c''_2, d''_3, z_2
  68. Polygon(U, V, D'_1, C'') → wielokąt31, u_3, v_2, d'_4, c''_3
  69. Polygon(V, W, C'_1, D'_1) → wielokąt32, v_3, w_1, c'_5, d'_5
  70. Polygon(P, I'', B_1, A_1) → wielokąt33, p, i''_1, b_4, a_3
  71. Polygon(I'', T, D_1, B_1) → wielokąt34, i''_2, t_1, d_2, b_5
  72. Polygon(S, P, A_1, C_1) → wielokąt35, s, p_2, a_4, c_2
  73. Polygon(C_1, A_1, B_1, D_1) → wielokąt36, c_3, a_5, b_6, d_3
  74. Polygon(T, S, C_1, D_1) → wielokąt37, t_4, s_4, c_4, d_4
  75. Circle(K, 1, k_1) → e_2
  76. Circle(J'_1, 1, i'') → f_2
  77. Intersect(f_2, e_2) → E_1, F_1, ,
  78. Polygon(I_1, H_1, P, I') → wielokąt38, i_2, h_2, p_3, i'_2
  79. Polygon(I_1, B''', K', H_1) → wielokąt39, i_3, b'''_3, k'_2, h_3
  80. Polygon(K', M, G_1, H_1) → wielokąt40, k'_3, m_2, g_2, h_4
  81. Polygon(H_1, G_1, I'', P) → wielokąt41, h_5, g_3, i''_3, p_4
  82. Polygon(M, K, E_1, G_1) → wielokąt42, m_3, k_3, e_3, g_4
  83. Polygon(G_1, E_1, J'_1, I'') → wielokąt43, g_5, e_4, j'_4, i''_4
  84. Polygon(B''', I_1, I') → wielokąt44, i'_3, b'''_4, i_4
  85. Polygon(E_1, K, J'_1) → wielokąt45, j'_5, e_5, k_4
  86. Line(F, G) → j_2
  87. Circle(C, 1, d_1) → f_3
  88. Circle(F, 1, j_2) → f_4
  89. Intersect(f_4, f_3) → J_1, K_1, ,
  90. Polygon(Q_1, J, F') → wielokąt46, f'_2, q_4, j_3
  91. Polygon(Q_1, O_1, F') → wielokąt47, f'_3, q_5, o_3
  92. Polygon(O_1, C'_2, F') → wielokąt48, f'_4, o_4, c'_6
  93. Polygon(O_1, M_1, C'_2) → wielokąt49, c'_7, o_5, m_4
  94. Polygon(M_1, F, C'_2) → wielokąt50, c'_8, m_5, f_5
  95. Polygon(M_1, K_1, F) → wielokąt51, f_6, m_6, k_5
  96. Polygon(K_1, C, F) → wielokąt52, f_7, k_6, c_5
  97. Polygon(L_1, G, D') → wielokąt53, d'_6, l_2, g_6
  98. Polygon(L_1, N_1, G) → wielokąt54, g_7, l_3, n_3
  99. Polygon(N_1, D''_1, G) → wielokąt55, g_8, n_4, d''_4
  100. Polygon(N_1, P_1, D''_1) → wielokąt56, d''_5, n_5, p_5
  101. Polygon(P_1, G', D''_1) → wielokąt57, d''_6, p_6, g'_2
  102. Polygon(P_1, R_1, G') → wielokąt58, g'_3, p_7, r_4
  103. Polygon(R_1, I, G') → wielokąt59, g'_4, r_5, i_5
  104. Polygon(L_1, T_1, V_1, N_1) → wielokąt60, l_4, t_5, v_4, n_6
  105. Polygon(S_1, K_1, M_1, U_1) → wielokąt62, s_6, k_7, m_7, u_5
  106. Polygon(V_1, Z_1, P_1, N_1) → wielokąt63, v_6, z_3, p_8, n_7
  107. Polygon(V_1, U_1, W_1, Z_1) → wielokąt64, v_7, u_6, w_2, z_4
  108. Polygon(U_1, M_1, O_1, W_1) → wielokąt65, u_7, m_8, o_6, w_3
  109. Polygon(W_1, O_1, Q_1, A_2) → wielokąt66, w_4, o_7, q_6, a_6
  110. Polygon(W_1, A_2, B_2, Z_1) → wielokąt67, w_5, a_7, b_7, z_5
  111. Polygon(Z_1, B_2, R_1, P_1) → wielokąt68, z_6, b_8, r_6, p_9
  112. Polygon(B_2, J', I, R_1) → wielokąt69, b_9, j'_6, i_6, r_7
  113. Polygon(B_2, A_2, L, J') → wielokąt70, b_{10}, a_8, l_5, j'_7
  114. Polygon(A_2, Q_1, J, L) → wielokąt71, a_9, q_7, j_4, l_6
  115. Polygon(S_1, D, C, K_1) → wielokąt61, s_5, d_5, c_6, k_8
  116. Polygon(S_1, T_1, C', D) → wielokąt72, s_7, t_6, c'_9, d_6
  117. Polygon(T_1, L_1, D', C') → wielokąt73, t_7, l_7, d'_7, c'_{10}
  118. Intersect(m_7, u_6) → E_2
  119. Polygon(C_2, S_1, U_1) → wielokąt74, u_4, c_7, s_8
  120. Polygon(V_1, D_2, C_2, U_1) → wielokąt75, v_5, d_7, c_8, u_8
  121. Polygon(D_2, V_1, T_1) → wielokąt76, t_8, d_8, v_8
  122. Polygon(G_2, T_1, D_2) → wielokąt78, d_{10}, g_{10}, t_9
  123. Polygon(F_2, C_2, D_2, G_2) → wielokąt77, f_8, c_9, d_9, g_9
  124. Polygon(F_2, S_1, T_1, G_2) → wielokąt79, f_9, s_9, t_{10}, g_{11}
  125. Polygon(C_2, F_2, S_1) → wielokąt80, s_{10}, c_{10}, f_{10}
  126. Circle(A, 1, a_2) → e_6
  127. Circle(C'_1, 1, w_1) → h_6
  128. Intersect(h_6, e_6) → H_2, I_2, ,
  129. Polygon(A'_1, K_2, N) → wielokąt81, n_8, a'_7, k_9
  130. Polygon(A'_1, E, J_2, K_2) → wielokąt82, a'_8, e_7, j_5, k_{10}
  131. Polygon(E, A, H_2, J_2) → wielokąt83, e_8, a_{10}, h_7, j_6
  132. Polygon(H_2, A, C'_1) → wielokąt84, c'_{11}, h_8, a_{11}
  133. Polygon(N, W, M_2, N_2) → wielokąt85, n_9, w_6, m_9, n_{10}
  134. Polygon(W, C'_1, L_2, M_2) → wielokąt86, w_7, c'_{12}, l_8, m_{10}
  135. Polygon(H_2, L_2, C'_1) → wielokąt87, c'_{13}, h_9, l_9
  136. Polygon(K_2, N, N_2) → wielokąt88, n_{11}, k_{11}, n_{12}
  137. Plane(zAxis, xAxis) → i_7
  138. Mirror(L_2, i_7) → O_2
  139. Mirror(M_2, i_7) → P_2
  140. Mirror(N_2, i_7) → Q_2
  141. Mirror(H_2, i_7) → R_2
  142. Mirror(J_2, i_7) → S_2
  143. Mirror(K_2, i_7) → T_2
  144. Polygon(B', H, S_2, R_2) → wielokąt89, b'_3, h_{10}, s_{11}, r_8
  145. Polygon(D'', R_2, B') → wielokąt90, b'_4, d''_7, r_9
  146. Polygon(D'', R_2, O_2) → wielokąt91, o_8, d''_8, r_{10}
  147. Polygon(H, B'', T_2, S_2) → wielokąt92, h_{11}, b''_3, t_{11}, s_{12}
  148. Polygon(T_2, B'', O) → wielokąt93, o_9, t_{12}, b''_4
  149. Polygon(T_2, O, Q_2) → wielokąt94, q_8, t_{13}, o_{10}
  150. Polygon(O, Z, P_2, Q_2) → wielokąt95, o_{11}, z_7, p_{10}, q_9
  151. Polygon(Z, D'', O_2, P_2) → wielokąt96, z_8, d''_9, o_{12}, p_{11}
  152. Mirror(B'', T_2) → B'''_1
  153. Mirror(B', R_2) → B''_1
  154. Mirror(R_2, B''_1) → U_2
  155. Mirror(O_2, B''_1) → V_2
  156. Mirror(T_2, B'''_1) → W_2
  157. Mirror(Q_2, B'''_1) → Z_2
  158. Mirror(A_3, B'''_1) → B_3
  159. Mirror(C_3, B''_1) → D_3
  160. Midpoint(C_3, A_3) → E_3
  161. Midpoint(D_3, B_3) → F_3
  162. Polygon(C_3, R_2, S_2, E_3) → wielokąt97, c_{11}, r_{11}, s_{13}, e_9
  163. Polygon(E_3, S_2, T_2, A_3) → wielokąt98, e_{10}, s_{14}, t_{14}, a_{12}
  164. Midpoint(V_2, Z_2) → G_3
  165. Midpoint(U_2, W_2) → H_3
  166. Polygon(V_2, C_3, E_3, G_3) → wielokąt99, v_9, c_{12}, e_{11}, g_{12}
  167. Polygon(G_3, E_3, A_3, Z_2) → wielokąt100, g_{13}, e_{12}, a_{13}, z_9
  168. Polygon(V_2, G_3, H_3, U_2) → wielokąt101, v_{10}, g_{14}, h_{12}, u_9
  169. Polygon(G_3, Z_2, W_2, H_3) → wielokąt102, g_{15}, z_{10}, w_8, h_{13}
  170. Polygon(H_3, F_3, B_3, W_2) → wielokąt103, h_{14}, f_{11}, b_{11}, w_9
  171. Polygon(U_2, D_3, F_3, H_3) → wielokąt104, u_{10}, d_{11}, f_{12}, h_{15}
  172. Polygon(D_3, O_2, P_2, F_3) → wielokąt105, d_{12}, o_{13}, p_{12}, f_{13}
  173. Polygon(P_2, Q_2, B_3, F_3) → wielokąt106, p_{13}, q_{10}, b_{12}, f_{14}
  174. Mirror(A_3, i_7) → I_3
  175. Mirror(E_3, i_7) → J_3
  176. Mirror(C_3, i_7) → K_3
  177. Mirror(Z_2, i_7) → L_3
  178. Mirror(G_3, i_7) → M_3
  179. Mirror(V_2, i_7) → N_3
  180. Mirror(U_2, i_7) → O_3
  181. Mirror(H_3, i_7) → P_3
  182. Mirror(W_2, i_7) → Q_3
  183. Mirror(D_3, i_7) → R_3
  184. Mirror(F_3, i_7) → S_3
  185. Mirror(B_3, i_7) → T_3
  186. Polygon(H_2, K_3, J_3, J_2) → wielokąt107, h_{16}, k_{12}, j_7, j_8
  187. Polygon(J_2, J_3, I_3, K_2) → wielokąt108, j_9, j_{10}, i_8, k_{13}
  188. Polygon(K_3, N_3, M_3, J_3) → wielokąt109, k_{14}, n_{13}, m_{11}, j_{11}
  189. Polygon(J_3, M_3, L_3, I_3) → wielokąt110, j_{12}, m_{12}, l_{10}, i_9
  190. Polygon(M_3, N_3, O_3, P_3) → wielokąt111, m_{13}, n_{14}, o_{14}, p_{14}
  191. Polygon(M_3, P_3, Q_3, L_3) → wielokąt112, m_{14}, p_{15}, q_{11}, l_{11}
  192. Polygon(R_3, L_2, M_2, S_3) → wielokąt113, r_{12}, l_{12}, m_{15}, s_{15}
  193. Polygon(S_3, M_2, N_2, T_3) → wielokąt114, s_{16}, m_{16}, n_{15}, t_{15}
  194. Polygon(O_3, R_3, S_3, P_3) → wielokąt115, o_{15}, r_{13}, s_{17}, p_{16}
  195. Polygon(P_3, S_3, T_3, Q_3) → wielokąt116, p_{17}, s_{18}, t_{16}, q_{12}
  196. Sphere(A_3, T_2) → i_{10}
  197. Sphere(T_2, A_3) → w_{10}
  198. IntersectConic(i_{10}, w_{10}) → c_{13}
  199. Sphere(Z_2, sqrt(2)) → i_{11}
  200. IntersectConic(i_{10}, i_{11}) → d_{13}
  201. Intersect(c_{13}, d_{13}) → U_3, V_3, ,
  202. Line(B''_1, B'''_1) → i_{12}
  203. Rotate(V_3, 120°, i_{12}) → W_3
  204. Rotate(W_3, 120°, i_{12}) → Z_3
  205. Polygon(A_3, T_2, V_3) → wielokąt118, v_{12}, a_{14}, t_{17}
  206. Polygon(A_3, V_3, W_3, Z_2) → wielokąt119, a_{15}, v_{13}, w_{12}, z_{12}
  207. Polygon(Z_2, W_3, W_2) → wielokąt120, w_{13}, z_{13}, w_{14}
  208. Polygon(W_3, Z_3, B_3, W_2) → wielokąt121, w_{15}, z_{14}, b_{13}, w_{16}
  209. Polygon(V_3, T_2, Q_2, Z_3) → wielokąt122, v_{14}, t_{18}, q_{13}, z_{15}
  210. Polygon(Z_3, Q_2, B_3) → wielokąt123, b_{14}, z_{16}, q_{14}
  211. Mirror(V_3, i_7) → A_4
  212. Mirror(Z_3, i_7) → B_4
  213. Mirror(W_3, i_7) → C_4
  214. Polygon(K_2, A_4, B_4, N_2) → wielokąt125, k_{15}, a_{17}, b_{16}, n_{16}
  215. Polygon(K_2, I_3, A_4) → wielokąt126, a_{18}, k_{16}, i_{13}
  216. Polygon(I_3, L_3, C_4, A_4) → wielokąt127, i_{14}, l_{13}, c_{15}, a_{19}
  217. Polygon(B_4, T_3, N_2) → wielokąt128, n_{17}, b_{17}, t_{19}
  218. Polygon(C_4, Q_3, T_3, B_4) → wielokąt129, c_{16}, q_{15}, t_{20}, b_{18}
  219. Polygon(C_4, L_3, Q_3) → wielokąt130, q_{16}, c_{17}, l_{14}
  220. Plane(M_3, J_3, P_3) → u_{11}
  221. Mirror(A_4, u_{11}) → D_4
  222. Mirror(C_4, u_{11}) → E_4
  223. Mirror(B_4, u_{11}) → F_4
  224. Mirror(V_3, u_{11}) → G_4
  225. Mirror(W_3, u_{11}) → H_4
  226. Mirror(Z_3, u_{11}) → I_4
  227. Polygon(K_3, H_2, D_4) → wielokąt131, d_{14}, k_{17}, h_{17}
  228. Polygon(F_4, D_4, H_2, L_2) → wielokąt132, f_{15}, d_{15}, h_{18}, l_{15}
  229. Polygon(N_3, K_3, D_4, E_4) → wielokąt133, n_{18}, k_{18}, d_{16}, e_{13}
  230. Polygon(N_3, E_4, O_3) → wielokąt134, o_{16}, n_{19}, e_{14}
  231. Polygon(E_4, F_4, R_3, O_3) → wielokąt135, e_{15}, f_{16}, r_{14}, o_{17}
  232. Polygon(F_4, L_2, R_3) → wielokąt136, r_{15}, f_{17}, l_{16}
  233. Polygon(R_2, G_4, I_4, O_2) → wielokąt137, r_{16}, g_{16}, i_{15}, o_{18}
  234. Polygon(R_2, C_3, G_4) → wielokąt138, g_{17}, r_{17}, c_{18}
  235. Polygon(G_4, C_3, V_2, H_4) → wielokąt139, g_{18}, c_{19}, v_{15}, h_{19}
  236. Polygon(H_4, V_2, U_2) → wielokąt140, u_{12}, h_{20}, v_{16}
  237. Polygon(I_4, H_4, U_2, D_3) → wielokąt141, i_{16}, h_{21}, u_{13}, d_{17}
  238. Polygon(O_2, I_4, D_3) → wielokąt142, d_{18}, o_{19}, i_{17}
  239. Sphere(E_4, D_4) → j_{13}
  240. Sphere(D_4, F_4) → u_{14}
  241. Polygon(D_4, F_4, E_4) → wielokąt117, e_{16}, d_{19}, f_{18}
  242. Polygon(G_4, H_4, I_4) → wielokąt124, i_{18}, g_{19}, h_{22}
  243. Sphere(V_3, W_3) → v_{11}
  244. Sphere(W_3, Z_3) → w_{11}
  245. Sphere(Z_3, V_3) → c_{14}
  246. IntersectConic(w_{11}, v_{11}) → e_{17}
  247. IntersectConic(c_{14}, w_{11}) → q_{17}
  248. Intersect(q_{17}, e_{17}) → J_4, K_4, ,
  249. Mirror(K_4, i_7) → L_4
  250. Polygon(W_3, V_3, K_4) → wielokąt143, k_{19}, w_{17}, v_{17}
  251. Polygon(W_3, K_4, Z_3) → wielokąt144, z_{11}, w_{18}, k_{20}
  252. Polygon(K_4, V_3, Z_3) → wielokąt145, z_{17}, k_{21}, v_{18}
  253. Polygon(A_4, L_4, B_4) → wielokąt146, b_{15}, a_{16}, l_{17}
  254. Polygon(A_4, C_4, L_4) → wielokąt147, l_{18}, a_{20}, c_{20}
  255. Polygon(C_4, B_4, L_4) → wielokąt148, l_{19}, c_{21}, b_{19}

AI Drawing Prompt Reference

Draw a spatial geometry figure with multiple polygons after mirroring points. Start by creating points A, B, C, and D, then mirror these points to generate A', B', C', D', and so on. Use these points to construct several quadrilaterals, such as wielokąt1 and wielokąt2. Finally, generate more points and polygons through mirroring, like wielokąt7 and wielokąt8, to form a complex spatial geometric structure.
The prompt above was compiled from the original drawing commands and is for reference only.