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Stellation of Dodecahedron

Stellation of Dodecahedron
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2014-10-31 23:04
VectorCoordinate GeometrySolid GeometryGeometric Construction
In collections Polyeder, whose coordinates are golden numbers
This example demonstrates the stellation process of a dodecahedron, visually representing the extension of vertices, edges, and faces to form intricate star-like patterns. It involves core mathematical concepts such as polygon construction, spatial geometry, vector operations, and analytical geometry. The construction leverages GeoGebra commands like Polygon, Plane, IntersectPath, Intersect, and Line, as outlined in the summary. Suitable for high school and above, it serves as an educational tool for exploring solid geometry, spatial vectors, and geometric constructions.

GeoGebra Commands & Parameters

  1. Polygon(I, A, M, O, E) → poly1, i, a_1, m, o, e
  2. Polygon(O, F, T, S, E) → poly2, o_{10}, f, t, s, e_1
  3. Polygon(S, G, K, I, E) → poly3, s_1, g, k, i_1, e_2
  4. Polygon(T, H, P, G, S) → poly4, t_1, h, p, g_1, s_2
  5. Polygon(K, G, P, N, C) → poly5, k_1, g_2, p_1, n, c
  6. Polygon(C, Q, A, I, K) → poly6, c_1, q, a_2, i_2, k_2
  7. Polygon(R, B, M, A, Q) → poly7, r, b_1, m_1, a_3, q_1
  8. Polygon(N, D, R, Q, C) → poly8, n_1, d, r_1, q_2, c_2
  9. Polygon(H, L, D, N, P) → poly9, h_1, l, d_1, n_2, p_2
  10. Polygon(L, J, B, R, D) → poly10, l_1, j, b_2, r_2, d_2
  11. Polygon(F, J, L, H, T) → poly11, f_1, j_1, l_2, h_2, t_2
  12. Polygon(M, B, J, F, O) → poly12, m_2, b_3, j_2, f_2, o_2
  13. Plane(poly1) → u
  14. Plane(poly2) → v
  15. Plane(poly3) → w
  16. Plane(poly4) → u_1
  17. Plane(poly5) → v_1
  18. Plane(poly6) → w_1
  19. Plane(poly7) → u_2
  20. Plane(poly8) → v_2
  21. Plane(poly9) → w_2
  22. Plane(poly10) → c_3
  23. Plane(poly11) → d_3
  24. Plane(poly12) → e_3
  25. IntersectPath(c_3, d_3) → f_3
  26. IntersectPath(c_3, e_3) → g_3
  27. IntersectPath(c_3, u) → h_3
  28. IntersectPath(c_3, u_1) → i_3
  29. IntersectPath(c_3, u_2) → j_3
  30. IntersectPath(c_3, v) → k_3
  31. IntersectPath(c_3, v_1) → l_3
  32. IntersectPath(c_3, v_2) → m_3
  33. IntersectPath(c_3, w) → n_3
  34. IntersectPath(c_3, w_1) → p_3
  35. IntersectPath(c_3, w_2) → q_3
  36. IntersectPath(d_3, e_3) → r_3
  37. IntersectPath(d_3, u) → s_3
  38. IntersectPath(d_3, u_1) → t_3
  39. IntersectPath(d_3, u_2) → a_4
  40. IntersectPath(d_3, v) → b_4
  41. IntersectPath(d_3, v_1) → c_4
  42. IntersectPath(d_3, v_2) → d_4
  43. IntersectPath(d_3, w) → e_4
  44. IntersectPath(d_3, w_1) → f_4
  45. IntersectPath(d_3, w_2) → g_4
  46. IntersectPath(e_3, u) → h_4
  47. IntersectPath(e_3, u_1) → i_4
  48. IntersectPath(d_3, u_2) → j_4
  49. IntersectPath(d_3, v) → k_4
  50. IntersectPath(d_3, e_3) → l_4
  51. IntersectPath(e_3, v) → m_4
  52. IntersectPath(e_3, u_2) → n_4
  53. IntersectPath(e_3, u_1) → p_4
  54. IntersectPath(Plane[poly1], e_3) → q_4
  55. IntersectPath(v_1, e_3) → r_4
  56. IntersectPath(e_3, v_2) → s_4
  57. IntersectPath(e_3, w) → t_4
  58. IntersectPath(e_3, w_1) → a_5
  59. IntersectPath(e_3, w_2) → b_5
  60. IntersectPath(u, u_1) → c_5
  61. IntersectPath(u, u_2) → d_5
  62. IntersectPath(u, v) → e_5
  63. IntersectPath(u, v_1) → f_5
  64. IntersectPath(u, v_2) → g_5
  65. IntersectPath(u, v_2) → h_5
  66. IntersectPath(u, w) → i_5
  67. IntersectPath(u, w_1) → j_5
  68. IntersectPath(u, w_2) → k_5
  69. IntersectPath(u_1, u_2) → l_5
  70. IntersectPath(u_1, v) → m_5
  71. IntersectPath(u_1, v_1) → n_5
  72. IntersectPath(u_1, v_2) → p_5
  73. IntersectPath(u_1, w) → q_5
  74. IntersectPath(u_1, w_1) → r_5
  75. IntersectPath(u_1, w_2) → s_5
  76. IntersectPath(u_2, v) → t_5
  77. IntersectPath(u_2, v_1) → a_6
  78. IntersectPath(u_2, v_2) → b_6
  79. IntersectPath(u_2, w) → c_6
  80. IntersectPath(u_2, w_1) → d_6
  81. IntersectPath(u_2, w_2) → e_6
  82. IntersectPath(v, v_1) → f_6
  83. IntersectPath(v, v_2) → g_6
  84. IntersectPath(v, w) → h_6
  85. IntersectPath(v, w_1) → i_6
  86. IntersectPath(v, w_2) → j_6
  87. IntersectPath(v_1, v_2) → k_6
  88. IntersectPath(v_1, w) → l_6
  89. IntersectPath(v_1, w_1) → m_6
  90. IntersectPath(v_1, w_2) → n_6
  91. IntersectPath(v_2, w) → p_6
  92. IntersectPath(v_2, w_1) → q_6
  93. IntersectPath(v_2, w_2) → r_6
  94. IntersectPath(w, w_1) → s_6
  95. IntersectPath(w, w_2) → t_6
  96. IntersectPath(w_1, w_2) → a_7
  97. Intersect(s_3, t_3) → U
  98. Intersect(k_3, t_5) → V
  99. Intersect(t_4, h_6) → W
  100. Intersect(s_4, g_5) → Z
  101. Intersect(s_4, n_4) → A_1
  102. Intersect(l_3, k_6) → B_1
  103. Intersect(p_3, j_3) → C_1
  104. Intersect(e_5, j_5) → D_1
  105. Intersect(j_6, b_4) → E_1
  106. Intersect(q_6, s_6) → F_1
  107. Intersect(d_5, g_5) → G_1
  108. Intersect(b_5, q_3) → H_1
  109. Polygon(V, B, J) → poly13, j_7, v_3, b_7
  110. Polygon(V, J, F) → poly14, f_7, v_4, j_8
  111. Polygon(V, O, F) → poly15, f_8, v_5, o_3
  112. Polygon(V, M, O) → poly16, o_4, v_6, m_7
  113. Polygon(V, M, B) → poly17, b_8, v_7, m_8
  114. Polygon(U, F, T) → poly18, t_7, u_3, f_9
  115. Polygon(U, T, S) → poly19, s_7, u_4, t_8
  116. Polygon(U, S, E) → poly20, e_7, u_5, s_8
  117. Polygon(U, E, O) → poly21, o_5, u_6, e_8
  118. Polygon(U, O, F) → poly22, f_{10}, u_7, o_6
  119. Polygon(E_1, T, H) → poly23, h_7, e_9, t_9
  120. Polygon(E_1, H, P) → poly24, p_7, e_{10}, h_8
  121. Polygon(E_1, P, G) → poly25, g_7, e_{11}, p_8
  122. Polygon(E_1, G, S) → poly26, s_9, e_{12}, g_8
  123. Polygon(E_1, S, T) → poly27, t_{10}, e_{13}, s_{10}
  124. Polygon(H_1, J, L) → poly28, l_7, h_9, j_9
  125. Polygon(H_1, L, H) → poly29, h_{10}, h_{11}, l_8
  126. Polygon(H_1, H, T) → poly30, t_{11}, h_{12}, h_{13}
  127. Polygon(H_1, T, F) → poly31, f_{11}, h_{14}, t_{12}
  128. Polygon(H_1, F, J) → poly32, j_{10}, h_{15}, f_{12}
  129. Polygon(B_1, L, D) → poly33, d_7, b_9, l_9
  130. Polygon(B_1, D, N) → poly34, n_7, b_{10}, d_8
  131. Polygon(B_1, N, P) → poly35, p_9, b_{11}, n_8
  132. Polygon(B_1, P, H) → poly36, h_{16}, b_{12}, p_{10}
  133. Polygon(B_1, H, L) → poly37, l_{10}, b_{13}, h_{17}
  134. Polygon(L, A_1, J) → poly38, j_{11}, l_{11}, a_8
  135. Polygon(A_1, J, B) → poly39, b_{14}, a_9, j_{12}
  136. Polygon(A_1, B, R) → poly40, r_7, a_{10}, b_{15}
  137. Polygon(A_1, R, D) → poly41, d_9, a_{11}, r_8
  138. Polygon(A_1, D, L) → poly42, l_{12}, a_{12}, d_{10}
  139. Polygon(Z, R, B) → poly43, b_{16}, z_1, r_9
  140. Polygon(Z, B, M) → poly44, m_9, z_2, b_{17}
  141. Polygon(Z, M, A) → poly45, a_{13}, z_3, m_{10}
  142. Polygon(Z, A, Q) → poly46, q_7, z_4, a_{14}
  143. Polygon(Z, Q, R) → poly47, r_{10}, z_5, q_8
  144. Polygon(W, M, O) → poly48, o_7, w_3, m_{11}
  145. Polygon(W, O, E) → poly49, e_{14}, w_4, o_8
  146. Polygon(W, E, I) → poly50, i_7, w_5, e_{15}
  147. Polygon(W, I, A) → poly51, a_{15}, w_6, i_8
  148. Polygon(W, A, M) → poly52, m_{12}, w_7, a_{16}
  149. Polygon(D_1, E, S) → poly53, s_{11}, d_{11}, e_{16}
  150. Polygon(D_1, S, G) → poly54, g_9, d_{12}, s_{12}
  151. Polygon(D_1, G, K) → poly55, k_7, d_{13}, g_{10}
  152. Polygon(D_1, K, I) → poly56, i_9, d_{14}, k_8
  153. Polygon(D_1, I, E) → poly57, e_{17}, d_{15}, i_{10}
  154. Polygon(G_1, Q, A) → poly58, a_{17}, g_{11}, q_9
  155. Polygon(G_1, A, I) → poly59, i_{11}, g_{12}, a_{18}
  156. Polygon(G_1, I, K) → poly60, k_9, g_{13}, i_{12}
  157. Polygon(G_1, K, C) → poly61, c_7, g_{14}, k_{10}
  158. Polygon(G_1, C, Q) → poly62, q_{10}, g_{15}, c_8
  159. Polygon(N, F_1, P) → poly63, p_{11}, n_9, f_{13}
  160. Polygon(N, C, F_1) → poly64, f_{14}, n_{10}, c_9
  161. Polygon(C, K, F_1) → poly65, f_{15}, c_{10}, k_{11}
  162. Polygon(F_1, K, G) → poly66, g_{16}, f_{16}, k_{12}
  163. Polygon(F_1, G, P) → poly67, p_{12}, f_{17}, g_{17}
  164. Polygon(C_1, D, R) → poly68, r_{11}, c_{11}, d_{16}
  165. Polygon(C_1, R, Q) → poly69, q_{11}, c_{12}, r_{12}
  166. Polygon(C_1, Q, C) → poly70, c_{13}, c_{14}, q_{12}
  167. Polygon(C_1, C, N) → poly71, n_{11}, c_{15}, c_{16}
  168. Polygon(C_1, N, D) → poly72, d_{17}, c_{17}, n_{12}
  169. Intersect(c_4, f_5) → I_1
  170. Intersect(t_4, b_5) → J_1
  171. Intersect(a_7, b_5) → K_1
  172. Intersect(d_4, g_5) → N_1
  173. Intersect(k_3, i_6) → P_1
  174. Intersect(t_4, s_4) → M_1
  175. Intersect(a_7, i_6) → O_1
  176. Intersect(l_3, f_5) → Q_1
  177. Intersect(d_4, p_6) → R_1
  178. Intersect(k_3, l_3) → L_1
  179. Intersect(a_4, c_4) → S_1
  180. Intersect(h_6, d_4) → T_1
  181. Intersect(i_6, m_3) → U_1
  182. Intersect(j_4, i_6) → V_1
  183. Intersect(i_6, m_3) → W_1
  184. Intersect(p_5, i_4) → Z_1
  185. Intersect(j_6, e_6) → A_2
  186. Intersect(i_3, p_3) → B_2
  187. Line(E_1, D_1) → i_{13}
  188. Line(V, W) → j_{13}
  189. Intersect(i_{13}, j_{13}) → C_2
  190. Line(F_1, D_1) → k_{13}
  191. Line(H_1, U) → l_{13}
  192. Intersect(k_{13}, l_{13}) → D_2
  193. Line(D_1, U) → m_{13}
  194. Line(B_1, H_1) → n_{13}
  195. Intersect(n_{13}, m_{13}) → E_2
  196. Line(G_1, W) → p_{13}
  197. Line(E_1, U) → q_{13}
  198. Intersect(p_{13}, q_{13}) → F_2
  199. Polygon(F_1, D_1, W, Z, C_1) → poly73, f_{18}, d_{18}, w_8, z_6, c_{18}
  200. Polygon(G_1, D_1, U, V, Z) → poly74, g_{18}, d_{19}, u_8, v_8, z_7
  201. Polygon(D_1, F_1, B_1, H_1, U) → poly75, d_{20}, f_{19}, b_{18}, h_{18}, u_9
  202. Polygon(G_1, Z, A_1, B_1, F_1) → poly76, g_{19}, z_8, a_{19}, b_{19}, f_{20}
  203. Polygon(W, D_1, E_1, H_1, V) → poly77, w_9, d_{21}, e_{18}, h_{19}, v_9
  204. Polygon(D_1, G_1, C_1, B_1, E_1) → poly78, d_{22}, g_{20}, c_{19}, b_{20}, e_{19}
  205. Polygon(U, E_1, B_1, A_1, V) → poly79, u_{10}, e_{20}, b_{21}, a_{20}, v_{10}
  206. Polygon(E_1, F_1, C_1, A_1, H_1) → poly80, e_{21}, f_{21}, c_{20}, a_{21}, h_{20}
  207. Polygon(C_1, Z, V, H_1, B_1) → poly81, c_{21}, z_9, v_{11}, h_{21}, b_{22}
  208. Polygon(W, U, H_1, A_1, Z) → poly82, w_{10}, u_{11}, h_{22}, a_{22}, z_{10}
  209. Polygon(G_1, W, V, A_1, C_1) → poly83, g_{21}, w_{11}, v_{12}, a_{23}, c_{22}
  210. Polygon(G_1, F_1, E_1, U, W) → poly84, g_{22}, f_{22}, e_{22}, u_{12}, w_{12}
  211. Line(F_1, G_1) → r_{13}
  212. Line(A_1, Z) → s_{13}
  213. Line(A_1, B_1) → t_{13}
  214. Line(B_1, F_1) → i_{14}
  215. Line(Z, G_1) → j_{14}
  216. Intersect(j_{14}, i_{14}) → G_2
  217. Polygon(F_1, G_2, G_1, M_1, Z, N_1, A_1, Z_1, B_1, R_1) → poly85, f_{23}, r_{14}, g_{23}, g_{24}, m_{14}, z_{11}, n_{14}, a_{24}, z_{12}, b_{23}
  218. Line(G_1, W) → k_{14}
  219. Line(W, V) → l_{14}
  220. Line(V, A_1) → p_{14}
  221. Line(A_1, C_1) → q_{14}
  222. Line(C_1, G_1) → s_{14}
  223. Intersect(q_{14}, k_{14}) → H_2
  224. Polygon(C_1, H_2, G_1, C_2, W, F_2, V, A_2, A_1, S_1) → poly86, c_{23}, s_{15}, h_{23}, g_{25}, c_{24}, w_{13}, f_{24}, v_{13}, a_{25}, a_{26}
  225. Line(Z, W) → t_{14}
  226. Line(W, U) → i_{15}
  227. Line(U, H_1) → j_{15}
  228. Line(H_1, A_1) → k_{15}
  229. Polygon(Z, M_1, W, D_2, U, J_1, H_1, Z_1, A_1, K_1) → poly87, z_{13}, k_{16}, m_{15}, w_{14}, d_{23}, u_{13}, j_{16}, h_{24}, z_{14}, a_{27}
  230. Line(U, E_1) → l_{15}
  231. Line(E_1, B_1) → n_{15}
  232. Polygon(V, F_2, U, I_1, E_1, R_1, B_1, S_1, A_1, N_1) → poly88, v_{14}, n_{16}, f_{25}, u_{14}, i_{16}, e_{23}, r_{15}, b_{24}, s_{16}, a_{28}
  233. Line(E_1, F_1) → p_{15}
  234. Line(F_1, C_1) → q_{15}
  235. Line(H_1, E_1) → t_{15}
  236. Polygon(E_1, O_1, F_1, H_2, C_1, K_1, A_1, A_2, H_1, J_1) → poly89, e_{24}, j_{17}, o_{13}, f_{26}, h_{25}, c_{25}, k_{17}, a_{29}, a_{30}, h_{26}
  237. Line(B_1, H_1) → l_{16}
  238. Polygon(H_1, L_1, B_1, B_2, C_1, Q_1, Z, P_1, V, E_2) → poly90, h_{27}, e_{25}, l_{17}, b_{25}, b_{26}, c_{26}, q_{16}, z_{15}, p_{16}, v_{15}
  239. Polygon(U, D_2, D_1, G_2, F_1, B_2, B_1, Z_1, H_1, E_2) → poly91, u_{15}, e_{26}, d_{24}, d_{25}, g_{26}, f_{27}, b_{27}, b_{28}, z_{16}, h_{28}
  240. Polygon(C_2, D_1, O_1, E_1, L_1, H_1, A_2, V, P_1, W) → poly92, c_{27}, w_{15}, d_{26}, o_{14}, e_{27}, l_{18}, h_{29}, a_{31}, v_{16}, p_{17}
  241. Polygon(W, M_1, G_1, H_2, F_1, R_1, E_1, J_1, U, F_2) → poly93, w_{16}, f_{29}, m_{16}, g_{27}, h_{30}, f_{28}, r_{16}, e_{28}, j_{18}, u_{16}
  242. Polygon(D_1, C_2, G_1, Q_1, C_1, S_1, B_1, L_1, E_1, I_1) → poly94, d_{27}, i_{17}, c_{28}, g_{28}, q_{17}, c_{29}, s_{17}, b_{29}, l_{19}, e_{29}
  243. Polygon(F_1, O_1, D_1, D_2, W, P_1, Z, K_1, C_1, B_2) → poly95, f_{30}, b_{30}, o_{15}, d_{28}, d_{29}, w_{17}, p_{18}, z_{17}, k_{18}, c_{30}
  244. Polygon(Z, Q_1, G_1, G_2, D_1, I_1, U, E_2, V, N_1) → poly96, z_{18}, n_{17}, q_{18}, g_{29}, g_{30}, d_{30}, i_{18}, u_{17}, e_{30}, v_{17}
  245. Polygon(C_2, G_1, Q_1, C_1, S_1, B_1, L_1, E_1, I_1, D_1) → poly97, c_{31}, d_{31}, g_{31}, q_{19}, c_{32}, s_{18}, b_{31}, l_{20}, e_{31}, i_{19}
  246. Polygon(D_1, G_1, C_1, B_1, E_1) → poly98, d_{32}, g_{32}, c_{33}, b_{32}, e_{32}
  247. Polygon(G, D_1, K, G_1, C, C_1, N, B_1, P, E_1) → poly99, g_{33}, e_{33}, d_{33}, k_{19}, g_{34}, c_{34}, c_{35}, n_{19}, b_{33}, p_{19}

AI Drawing Prompt Reference

Draw the stellation of a dodecahedron by first constructing twelve pentagons as the faces of the dodecahedron, then creating planes from these faces and finding their intersections to generate new vertices and form a star-like pattern. The steps include constructing pentagons, creating planes, and finding plane intersections.
The prompt above was compiled from the original drawing commands and is for reference only.