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4V-Triacon Octahedron

4V-Triacon Octahedron
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2016-05-06 09:00
PolygonSolid GeometryGeometric Construction
This example demonstrates the construction and properties of a complex octahedron named 4V-Triacon. It involves core mathematical concepts such as polygon construction, spatial geometry, symmetry, rotation, and reflection. The construction utilizes GeoGebra commands like Mirror, Polygon, Pyramid, Segment, Midpoint, Intersect, Sphere, Ray, and Rotate. It is suitable for high school and above students studying solid geometry, spatial visualization, and GeoGebra software application.

GeoGebra Commands & Parameters

  1. Mirror(B, yAxis) → B'
  2. Mirror(A, yAxis) → A'
  3. Mirror(C, xAxis) → C'
  4. Polygon(A', A, B, B') → poly1, a', a_1, b, b'
  5. Pyramid(poly1, C) → c, faceAA'C, faceABC, faceBB'C, faceA'B'C, edgeA'C, edgeAC, edgeBC, edgeB'C
  6. Pyramid(poly1, C') → d, faceAA'C', faceABC', faceBB'C', faceA'B'C', edgeA'C', edgeAC', edgeBC', edgeB'C'
  7. Segment(A, C) → f
  8. Segment(C, B) → g
  9. Segment(B, A) → h
  10. Midpoint(h) → E
  11. Midpoint(f) → L
  12. Midpoint(g) → M
  13. Segment(A, E) → i
  14. Segment(E, B) → j
  15. Segment(B, M) → k
  16. Segment(M, C) → l
  17. Segment(C, L) → m
  18. Segment(L, A) → n
  19. Midpoint(n) → I
  20. Midpoint(m) → O
  21. Midpoint(l) → P
  22. Midpoint(k) → J
  23. Midpoint(j) → F
  24. Midpoint(i) → D
  25. Segment(C, E) → p
  26. Segment(L, B) → q
  27. Segment(M, A) → r
  28. Segment(M, O) → s
  29. Segment(M, F) → t
  30. Segment(L, P) → f_1
  31. Segment(L, D) → g_1
  32. Segment(E, I) → h_1
  33. Segment(E, J) → i_1
  34. Intersect(p, f_1) → N
  35. Intersect(p, q) → K
  36. Intersect(g_1, r) → G
  37. Intersect(t, i_1) → H
  38. Sphere(Z, (a * sqrt(2) / 2)) → u
  39. Ray(Z, O) → j_1
  40. Ray(Z, P) → k_1
  41. Ray(Z, N) → l_1
  42. Ray(Z, L) → m_1
  43. Ray(Z, M) → n_1
  44. Ray(Z, K) → p_1
  45. Ray(Z, I) → q_1
  46. Ray(Z, J) → r_1
  47. Ray(Z, G) → s_1
  48. Ray(Z, H) → t_1
  49. Ray(Z, D) → c_1
  50. Ray(Z, E) → d_1
  51. Ray(Z, F) → e_1
  52. Intersect(k_1, u) → Q, P'
  53. Intersect(j_1, u) → S, O'
  54. Intersect(l_1, u) → U, N'
  55. Intersect(n_1, u) → W, M'
  56. Intersect(e_1, u) → B_1, F'
  57. Intersect(t_1, u) → D_1, H'
  58. Intersect(p_1, u) → F_1, K'
  59. Intersect(m_1, u) → H_1, L'
  60. Intersect(c_1, u) → J_1, D'
  61. Intersect(d_1, u) → L_1, E'
  62. Intersect(s_1, u) → N_1, G'
  63. Intersect(q_1, u) → P_1, I'
  64. Intersect(r_1, u) → R_1, J'
  65. Intersect(edgeBC, edgeAC) → T_1
  66. Intersect(a_1, edgeAC) → U_1
  67. Intersect(edgeBC, b) → V_1
  68. Segment(T_1, O') → f_2
  69. Segment(O', L') → g_2
  70. Segment(L', I') → h_2
  71. Segment(I', U_1) → i_2
  72. Segment(U_1, D') → j_2
  73. Segment(D', E') → k_2
  74. Segment(E', F') → l_2
  75. Segment(F', V_1) → m_2
  76. Segment(V_1, J') → n_2
  77. Segment(J', M') → p_2
  78. Segment(M', P') → q_2
  79. Segment(P', T_1) → r_2
  80. Segment(P', N') → s_2
  81. Segment(N', O') → t_2
  82. Segment(N', T_1) → a_2
  83. Segment(L', K') → b_2
  84. Segment(K', H') → c_2
  85. Segment(H', V_1) → d_2
  86. Segment(N', K') → e_2
  87. Segment(K', E') → f_3
  88. Segment(U_1, G') → g_3
  89. Segment(G', K') → h_3
  90. Segment(K', M') → i_3
  91. Segment(M', H') → j_3
  92. Segment(H', F') → k_3
  93. Segment(L', G') → l_3
  94. Segment(G', D') → m_3
  95. Segment(E', G') → n_3
  96. Segment(G', I') → p_3
  97. Segment(E', H') → q_3
  98. Segment(H', J') → r_3
  99. Segment(L', N') → s_3
  100. Segment(N', M') → t_3
  101. Rotate(N', 90°, zAxis) → N''
  102. Rotate(P', 90°, zAxis) → P''
  103. Rotate(K', 90°, zAxis) → K''
  104. Rotate(M', 90°, zAxis) → M''
  105. Rotate(J', 90°, zAxis) → J''
  106. Rotate(V_1, 90°, zAxis) → R
  107. Rotate(F', 90°, zAxis) → F''
  108. Rotate(H', 90°, zAxis) → H''
  109. Rotate(E', 90°, zAxis) → E''
  110. Rotate(D', 90°, zAxis) → D''
  111. Rotate(G', 90°, zAxis) → G''
  112. Rotate(N'', 90°, zAxis) → N'''
  113. Rotate(P'', 90°, zAxis) → P'''
  114. Rotate(M'', 90°, zAxis) → M'''
  115. Rotate(J'', 90°, zAxis) → J'''
  116. Rotate(R, 90°, zAxis) → R'
  117. Rotate(H'', 90°, zAxis) → H'''
  118. Rotate(F'', 90°, zAxis) → F'''
  119. Rotate(K'', 90°, zAxis) → K'''
  120. Rotate(E'', 90°, zAxis) → E'''
  121. Rotate(D'', 90°, zAxis) → D'''
  122. Rotate(G'', 90°, zAxis) → G'''
  123. Segment(T_1, P'') → a_3
  124. Segment(P'', M'') → b_3
  125. Segment(M'', J'') → c_3
  126. Segment(J'', R) → d_3
  127. Segment(T_1, P''') → e_3
  128. Segment(P''', M''') → f_4
  129. Segment(M''', J''') → g_4
  130. Segment(J''', R') → h_4
  131. Rotate(N''', 90°, zAxis) → T
  132. Rotate(K''', 90°, zAxis) → V
  133. Rotate(G''', 90°, zAxis) → A_1
  134. Rotate(D''', 90°, zAxis) → C_1
  135. Rotate(E''', 90°, zAxis) → E_1
  136. Rotate(H''', 90°, zAxis) → G_1
  137. Rotate(F''', 90°, zAxis) → I_1
  138. Segment(P', N'') → i_4
  139. Segment(N'', T_1) → j_4
  140. Segment(N'', P'') → k_4
  141. Segment(P'', N''') → l_4
  142. Segment(N''', P''') → m_4
  143. Segment(N''', T_1) → n_4
  144. Segment(P''', T) → p_4
  145. Segment(T, O') → q_4
  146. Segment(T, T_1) → r_4
  147. Segment(M', K'') → s_4
  148. Segment(K'', M'') → t_4
  149. Segment(M'', N'') → a_4
  150. Segment(N'', M') → b_4
  151. Segment(N'', K'') → c_4
  152. Segment(K'', E'') → d_4
  153. Segment(J', G'') → e_4
  154. Segment(G'', V_1) → f_5
  155. Segment(V_1, D'') → g_5
  156. Segment(D'', E'') → h_5
  157. Segment(E'', F'') → i_5
  158. Segment(F'', R) → j_5
  159. Segment(G'', D'') → k_5
  160. Segment(G'', M') → l_5
  161. Segment(G'', K'') → m_5
  162. Segment(G'', E'') → n_5
  163. Segment(K'', H'') → p_5
  164. Segment(H'', E'') → q_5
  165. Segment(H'', F'') → r_5
  166. Segment(H'', R) → s_5
  167. Segment(J'', H'') → t_5
  168. Segment(H'', M'') → a_5
  169. Segment(N''', K''') → b_5
  170. Segment(K''', E''') → c_5
  171. Segment(E''', D''') → d_5
  172. Segment(R, D''') → e_5
  173. Segment(E''', F''') → f_6
  174. Segment(F''', R') → g_6
  175. Segment(J'', G''') → h_6
  176. Segment(G''', M'') → i_6
  177. Segment(G''', R) → j_6
  178. Segment(G''', D''') → k_6
  179. Segment(G''', E''') → l_6
  180. Segment(G''', K''') → m_6
  181. Segment(K''', M'') → n_6
  182. Segment(M'', N''') → p_6
  183. Segment(N''', M''') → q_6
  184. Segment(M''', K''') → r_6
  185. Segment(K''', H''') → s_6
  186. Segment(H''', R') → t_6
  187. Segment(J''', H''') → a_6
  188. Segment(H''', F''') → b_6
  189. Segment(E''', H''') → c_6
  190. Segment(H''', M''') → d_6
  191. Segment(R', C_1) → e_6
  192. Segment(C_1, E_1) → f_7
  193. Segment(E_1, I_1) → g_7
  194. Segment(I_1, U_1) → h_7
  195. Segment(I', G_1) → i_7
  196. Segment(G_1, U_1) → j_7
  197. Segment(G_1, I_1) → k_7
  198. Segment(R', A_1) → l_7
  199. Segment(A_1, J''') → m_7
  200. Segment(A_1, C_1) → n_7
  201. Segment(A_1, E_1) → p_7
  202. Segment(E_1, G_1) → q_7
  203. Segment(V, E_1) → r_7
  204. Segment(V, T) → s_7
  205. Segment(M''', T) → t_7
  206. Segment(T, L') → a_7
  207. Segment(L', V) → b_7
  208. Segment(M''', V) → c_7
  209. Segment(V, A_1) → d_7
  210. Segment(M''', A_1) → e_7
  211. Segment(V, G_1) → f_8
  212. Segment(G_1, L') → g_8
  213. Mirror(T_1, xOyPlane) → K_1
  214. Mirror(O', xOyPlane) → O''
  215. Mirror(N', xOyPlane) → N''_1
  216. Mirror(P', xOyPlane) → P''_1
  217. Mirror(L', xOyPlane) → L''
  218. Mirror(K', xOyPlane) → K''_1
  219. Mirror(M', xOyPlane) → M''_1
  220. Mirror(I', xOyPlane) → I''
  221. Mirror(G', xOyPlane) → G''_1
  222. Mirror(H', xOyPlane) → H''_1
  223. Mirror(J', xOyPlane) → J''_1
  224. Rotate(I'', (-(90°)), zAxis) → I'''
  225. Rotate(G''_1, (-(90°)), zAxis) → M_1
  226. Rotate(L'', (-(90°)), zAxis) → L'''
  227. Rotate(K''_1, (-(90°)), zAxis) → O_1
  228. Rotate(O'', (-(90°)), zAxis) → O'''
  229. Rotate(N''_1, (-(90°)), zAxis) → Q_1
  230. Rotate(M''_1, (-(90°)), zAxis) → W_1
  231. Rotate(J''_1, (-(90°)), zAxis) → Z_1
  232. Rotate(H''_1, (-(90°)), zAxis) → A_2
  233. Segment(I'', L'') → h_8
  234. Segment(L'', O'') → i_8
  235. Segment(O'', K_1) → j_8
  236. Segment(J''_1, M''_1) → k_8
  237. Segment(M''_1, P''_1) → l_8
  238. Segment(P''_1, K_1) → m_8
  239. Segment(I''', L''') → n_8
  240. Segment(L''', O''') → p_8
  241. Segment(O''', K_1) → q_8
  242. Rotate(A_2, (-(90°)), zAxis) → B_2
  243. Rotate(O_1, (-(90°)), zAxis) → C_2
  244. Rotate(Q_1, (-(90°)), zAxis) → D_2
  245. Rotate(O''', (-(90°)), zAxis) → E_2
  246. Rotate(L''', (-(90°)), zAxis) → F_2
  247. Rotate(I''', (-(90°)), zAxis) → G_2
  248. Rotate(M_1, (-(90°)), zAxis) → H_2
  249. Rotate(I''', (-(90°)), zAxis) → I_2
  250. Segment(G_2, F_2) → r_8
  251. Segment(F_2, E_2) → s_8
  252. Segment(E_2, K_1) → t_8
  253. Rotate(H_2, (-(90°)), zAxis) → K_2
  254. Rotate(C_2, (-(90°)), zAxis) → L_2
  255. Rotate(B_2, (-(90°)), zAxis) → M_2
  256. Rotate(D_2, (-(90°)), zAxis) → N_2
  257. Segment(N''_1, P''_1) → a_8
  258. Segment(P''_1, N_2) → b_8
  259. Segment(N_2, E_2) → c_8
  260. Segment(E_2, D_2) → d_8
  261. Segment(D_2, O''') → e_8
  262. Segment(O''', Q_1) → f_9
  263. Segment(Q_1, O'') → g_9
  264. Segment(O'', N''_1) → h_9
  265. Segment(N''_1, K_1) → i_9
  266. Segment(K_1, N_2) → j_9
  267. Segment(K_1, D_2) → k_9
  268. Segment(K_1, Q_1) → l_9
  269. Segment(K''_1, N''_1) → m_9
  270. Segment(D_2, C_2) → n_9
  271. Segment(Q_1, O_1) → p_9
  272. Segment(N_2, L_2) → q_9
  273. Segment(L'', Q_1) → r_9
  274. Segment(Q_1, L''') → s_9
  275. Segment(L''', D_2) → t_9
  276. Segment(F_2, D_2) → a_9
  277. Segment(F_2, N_2) → b_9
  278. Segment(L'', N''_1) → c_9
  279. Segment(N''_1, M''_1) → d_9
  280. Segment(M''_1, N_2) → e_9
  281. Segment(L'', K''_1) → f_{10}
  282. Segment(K''_1, M''_1) → g_{10}
  283. Segment(G''_1, K''_1) → h_{10}
  284. Segment(G''_1, I'') → j_{10}
  285. Segment(H''_1, J''_1) → k_{10}
  286. Segment(I'', A_2) → n_{10}
  287. Segment(J''_1, K_2) → p_{10}
  288. Segment(E', K''_1) → q_{10}
  289. Segment(E', H''_1) → r_{10}
  290. Segment(E', G''_1) → s_{10}
  291. Segment(F', H''_1) → t_{10}
  292. Segment(D', G''_1) → a_{10}
  293. Segment(U_1, I'') → b_{10}
  294. Segment(G''_1, U_1) → c_{10}
  295. Segment(U_1, A_2) → d_{10}
  296. Segment(I_1, A_2) → e_{10}
  297. Segment(V_1, J''_1) → f_{11}
  298. Segment(V_1, H''_1) → g_{11}
  299. Segment(V_1, K_2) → h_{11}
  300. Segment(K_2, D'') → i_{11}
  301. Segment(R, G_2) → j_{11}
  302. Segment(G_2, H_2) → k_{11}
  303. Segment(H_2, D''') → l_{11}
  304. Segment(G_2, M_2) → m_{11}
  305. Segment(F'', M_2) → n_{11}
  306. Segment(M_2, R) → p_{11}
  307. Segment(R, H_2) → q_{11}
  308. Segment(F''', B_2) → r_{11}
  309. Segment(B_2, I''') → s_{11}
  310. Segment(I''', M_1) → t_{11}
  311. Segment(M_1, C_1) → a_{11}
  312. Segment(R', I''') → b_{11}
  313. Segment(M_1, R') → c_{11}
  314. Segment(R', B_2) → d_{11}
  315. Segment(H''_1, K''_1) → i_{10}
  316. Segment(G''_1, L'') → l_{10}
  317. Segment(H''_1, M''_1) → m_{10}
  318. Segment(A_2, L'') → e_{11}
  319. Segment(O_1, L'') → f_{12}
  320. Segment(A_2, O_1) → g_{12}
  321. Segment(M_1, O_1) → h_{12}
  322. Segment(O_1, L''') → i_{12}
  323. Segment(H_2, C_2) → j_{12}
  324. Segment(C_2, L''') → k_{12}
  325. Segment(L''', M_1) → l_{12}
  326. Segment(B_2, L''') → m_{12}
  327. Segment(B_2, C_2) → n_{12}
  328. Segment(F_2, L_2) → p_{12}
  329. Segment(L_2, M''_1) → q_{12}
  330. Segment(L_2, K_2) → r_{12}
  331. Segment(L_2, M_2) → s_{12}
  332. Segment(F_2, C_2) → t_{12}
  333. Segment(H_2, F_2) → a_{12}
  334. Segment(F_2, M_2) → b_{12}
  335. Segment(K_2, M''_1) → c_{12}
  336. Segment(E'', L_2) → d_{12}
  337. Segment(K_2, E'') → e_{12}
  338. Segment(E'', M_2) → f_{13}
  339. Segment(E''', C_2) → g_{13}
  340. Segment(E''', B_2) → h_{13}
  341. Segment(E''', H_2) → i_{13}
  342. Segment(E_1, O_1) → j_{13}
  343. Segment(E_1, M_1) → k_{13}
  344. Segment(E_1, A_2) → l_{13}

AI Drawing Prompt Reference

Draw a complex octahedron by first creating a basic structure through mirroring and polygon construction, then forming symmetrical geometry using rotations and reflections. Next, build multiple pyramids from the base structure and connect key points to form a mesh. Finally, draw several spheres around the center point and radiate rays from the center to various key points to complete the overall shape.
The prompt above was compiled from the original drawing commands and is for reference only.