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Rosace Cube: Fusion of Rotating Petals and Geometric Construction

Rosace Cube: Fusion of Rotating Petals and Geometric Construction
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2020-02-24 15:12
CircleSequenceSolid GeometryTransformationGeometric Construction
① This example demonstrates a composite geometric figure called the 'Rosace Cube', which combines a cube with a series of rotating arcs, ellipses, and petal-like curves to create a three-dimensional decorative effect resembling rose petals. ② Core mathematical concepts include solid geometry (cube), planar curves (circles, ellipses, arcs), rotation transformations, mirror symmetry, angle bisectors, tangents, and sequence-generated point sets, highlighting the dynamic combination of geometric elements in space. ③ Construction highlights: From the construction summary, the example uses GeoGebra commands such as Segment, Rotate, Intersect, Line, Circle, Point, Tangent, OrthogonalLine, AngularBisector, Mirror, CircleArc, Sequence, Midpoint, Arc, Semicircle, and Cube, along with numerous auxiliary points (e.g., I_2, G_2, F_2) and parameters (e.g., n, a_3, c) to generate symmetric petal structures via sequences and rotations. ④ Target audience: Suitable for high school and above, this example is ideal for exploring geometric transformations, the integration of solid and planar figures, and parametric construction in dynamic geometry software, as well as for creative applications in mathematical art or modeling.

GeoGebra Commands & Parameters

  1. Segment(G_2, F_2) → m_1
  2. Segment(F_2, I_2) → i_1
  3. Rotate((7, 0), 90° + (180°) / n) → B
  4. Intersect(xAxis, yAxis) → A
  5. Line(A, F_2) → a_3
  6. Line(F_2, A) → n_1
  7. Circle(A, B) → c
  8. Point(c) → S_4
  9. Tangent(S_4, c) → j_2,
  10. Intersect(j_2, m_1) → U_4
  11. Intersect(c, n_1, 1) → J_2
  12. OrthogonalLine(J_2, n_1) → a_2
  13. Intersect(a_2, i_1) → K_2
  14. AngularBisector(J_2, K_2, F_2) → b_2
  15. Intersect(n_1, b_2) → L_2
  16. Circle(L_2, J_2) → k_3
  17. Point(k_3) → R_4
  18. Tangent(R_4, k_3) → i_2,
  19. Intersect(i_2, j_2) → V_4
  20. Intersect(i_2, m_1) → T_4
  21. AngularBisector(T_4, V_4, U_4) → m_2
  22. AngularBisector(T_4, U_4, V_4) → l_2
  23. Intersect(l_2, m_2) → W_4
  24. OrthogonalLine(W_4, m_1) → n_2
  25. Intersect(m_1, n_2) → Z_4
  26. Circle(W_4, Z_4) → t_5
  27. Line(A, B) → b
  28. Rotate(b, (360°) / n) → b'
  29. Point(b) → C
  30. Rotate(C, (360°) / n) → C'
  31. Rotate(B, (180°) / n) → B'
  32. Line(A, B') → a
  33. Rotate(a, 30°, B') → a'
  34. Intersect(b, a') → E
  35. Mirror(E, a) → E'_1
  36. CircleArc(E'_1, E, B') → e
  37. CircleArc(E, B', E'_1) → f
  38. Sequence(Arc[Rotate[Circle[E'_1, E], (i * (360°) / n)], Rotate[E, (i * (360°) / n)], Rotate[B', (i * (360°) / n)]], i, 1, n) → list2
  39. Sequence(Arc[Rotate[Circle[E, B'], (i * (360°) / n)], Rotate[B', (i * (360°) / n)], Rotate[E'_1, (i * (360°) / n)]], i, 1, n) → list3
  40. Midpoint(B', B) → G
  41. Line(A, G) → g
  42. Intersect(e, g, 1) → H
  43. CircleArc(E'_1, E, H) → h
  44. Mirror(E'_1, g) → I
  45. Mirror(E, g) → E'
  46. Mirror(H, g) → H'
  47. Arc(Mirror[Circle[E'_1, E], g], E', H') → h'
  48. Semicircle(C', C) → d
  49. Sequence(Semicircle[Rotate[C', (i * (360°) / n)], Rotate[C, (i * (360°) / n)]], i, 1, n) → list1
  50. Intersect(d, a) → D,
  51. Circle(A, F) → k
  52. Intersect(k, b, 2) → J
  53. Mirror(E', a) → E''
  54. Mirror(H', a) → H''_1
  55. Arc(Mirror[Mirror[Circle[E'_1, E], g], a], E'', H''_1) → h''
  56. Sequence(Arc[Rotate[Mirror[Circle[E'_1, E], g], (i * (360°) / n)], Rotate[E', (i * (360°) / n)], Rotate[H', (i * (360°) / n)]], i, 1, n) → list4
  57. Sequence(Arc[Rotate[Mirror[Mirror[Circle[E'_1, E], g], a], (i * (360°) / n)], Rotate[E'', (i * (360°) / n)], Rotate[H''_1, (i * (360°) / n)]], i, 1, n) → list5
  58. Sequence(Segment[Rotate[J, (i * (360°) / n)], Rotate[E, (i * (360°) / n)]], i, 1, n) → list6
  59. Sequence(Segment[Rotate[F, (i * (360°) / n)], Rotate[E', (i * (360°) / n)]], i, 1, n) → list7
  60. Midpoint(B', H') → K
  61. Line(E'_1, K) → i
  62. Intersect(h'', i, 1) → M
  63. Intersect(a, i) → L
  64. Circle(L, M) → p
  65. Intersect(p, a, 2) → N
  66. Sequence(Rotate[p, (i * (360°) / n)], i, 1, n) → list8
  67. Sequence(Segment[Rotate[N, (i * (360°) / n)], Rotate[B', (i * (360°) / n)]], i, 1, n) → list9
  68. OrthogonalLine(L, a) → j
  69. Midpoint(M, L) → O
  70. Rotate(O, 90°, L) → O'
  71. Rotate(O', 90°, L) → O''
  72. Rotate(O'', 90°, L) → O'''
  73. Semicircle(O', O) → q
  74. Semicircle(O'', O') → r
  75. Semicircle(O''', O'') → s
  76. Semicircle(O, O''') → t
  77. Sequence(Semicircle[Rotate[O', (i * (360°) / n)], Rotate[O, (i * (360°) / n)]], i, 1, n) → list10
  78. Sequence(Semicircle[Rotate[O'', (i * (360°) / n)], Rotate[O', (i * (360°) / n)]], i, 1, n) → list11
  79. Sequence(Semicircle[Rotate[O''', (i * (360°) / n)], Rotate[O'', (i * (360°) / n)]], i, 1, n) → list12
  80. Sequence(Semicircle[Rotate[O, (i * (360°) / n)], Rotate[O''', (i * (360°) / n)]], i, 1, n) → list13
  81. Midpoint(E'', E'_1) → P
  82. CircleArc(Q, E'', E'_1) → c_1
  83. Line(A, Q) → l
  84. Intersect(c_1, l, 1) → R
  85. Line(E'_1, R) → m
  86. Intersect(h'', m) → S,
  87. Midpoint(R, S) → T
  88. Circle(R, T) → d_1
  89. Intersect(c_1, d_1, 2) → U
  90. Intersect(c_1, d_1, 1) → V
  91. CircleArc(Q, E'', U) → e_1
  92. CircleArc(R, U, V) → f_1
  93. CircleArc(Q, V, E'_1) → g_1
  94. Sequence(Arc[Rotate[Circle[Q, E''], (i * (360°) / n)], Rotate[E'', (i * (360°) / n)], Rotate[U, (i * (360°) / n)]], i, 1, n) → list14
  95. Sequence(Arc[Rotate[Circle[R, U], (i * (360°) / n)], Rotate[U, (i * (360°) / n)], Rotate[V, (i * (360°) / n)]], i, 1, n) → list15
  96. Sequence(Arc[Rotate[Circle[Q, V], (i * (360°) / n)], Rotate[V, (i * (360°) / n)], Rotate[E'_1, (i * (360°) / n)]], i, 1, n) → list16
  97. Mirror(Q, a) → Q'_1
  98. Mirror(E'', a) → E'''
  99. Mirror(U, a) → U'_1
  100. Arc(Mirror[Circle[Q, E''], a], E''', U'_1) → h_1
  101. Mirror(R, a) → R'
  102. Mirror(U, a) → U'
  103. Mirror(V, a) → V'_1
  104. Arc(Mirror[Circle[R, U], a], U', V'_1) → k_1
  105. Mirror(Q, a) → Q'
  106. Mirror(V, a) → V'
  107. Mirror(E'_1, a) → W
  108. Arc(Mirror[Circle[Q, V], a], V', W) → p_1
  109. Sequence(Arc[Rotate[Mirror[Circle[Q, E''], a], (i * (360°) / n)], Rotate[E''', (i * (360°) / n)], Rotate[U'_1, (i * (360°) / n)]], i, 1, n) → list17
  110. Sequence(Arc[Rotate[Mirror[Circle[R, U], a], (i * (360°) / n)], Rotate[U', (i * (360°) / n)], Rotate[V'_1, (i * (360°) / n)]], i, 1, n) → list18
  111. Sequence(Arc[Rotate[Mirror[Circle[Q, V], a], (i * (360°) / n)], Rotate[V', (i * (360°) / n)], Rotate[W, (i * (360°) / n)]], i, 1, n) → list19
  112. Intersect(k, b, 1) → Z
  113. Midpoint(W, Z) → A_1
  114. Intersect(k, a, 1) → B_1
  115. Midpoint(E''', B_1) → C_1
  116. Circle(A_1, E''') → q_1
  117. Circle(E''', A_1) → r_1
  118. Intersect(r_1, q_1) → D_1, E_1, ,
  119. CircleArc(E_1, A_1, E''') → s_1
  120. Line(H', A) → a_1
  121. Intersect(s_1, a_1, 1) → F_1
  122. Mirror(E_1, a_1) → G_1
  123. CircleArc(E_1, A_1, F_1) → t_1
  124. CircleArc(G_1, F_1, C_1) → c_2
  125. Sequence(Arc[Rotate[Circle[E_1, A_1], (i * (360°) / n)], Rotate[A_1, (i * (360°) / n)], Rotate[F_1, (i * (360°) / n)]], i, 1, n) → list20
  126. Sequence(Arc[Rotate[Circle[G_1, F_1], (i * (360°) / n)], Rotate[F_1, (i * (360°) / n)], Rotate[C_1, (i * (360°) / n)]], i, 1, n) → list21
  127. Mirror(E_1, a) → H_1
  128. Mirror(A_1, a) → I_1
  129. Mirror(F_1, a) → J_1
  130. Arc(Mirror[Circle[E_1, A_1], a], I_1, J_1) → d_2
  131. Mirror(G_1, a) → K_1
  132. Mirror(F_1, a) → L_1
  133. Mirror(C_1, a) → M_1
  134. Arc(Mirror[Circle[G_1, F_1], a], L_1, M_1) → e_2
  135. Sequence(Arc[Rotate[Mirror[Circle[E_1, A_1], a], (i * (360°) / n)], Rotate[I_1, (i * (360°) / n)], Rotate[J_1, (i * (360°) / n)]], i, 1, n) → list22
  136. Sequence(Arc[Rotate[Mirror[Circle[G_1, F_1], a], (i * (360°) / n)], Rotate[L_1, (i * (360°) / n)], Rotate[M_1, (i * (360°) / n)]], i, 1, n) → list23
  137. Mirror(E'_1, b) → N_1
  138. Line(E'_1, B) → b_1
  139. Intersect(e, b_1, 1) → O_1
  140. Circle(B, O_1) → f_2
  141. Intersect(f_2, b, 1) → P_1
  142. Circle(P_1, B) → g_2
  143. Intersect(f_2, g_2) → Q_1, R_1, ,
  144. Circle(R_1, P_1) → h_2
  145. Circle(Q_1, P_1) → k_2
  146. Intersect(f_2, h_2, 2) → S_1
  147. Intersect(f_2, k_2, 2) → T_1
  148. Intersect(f_2, b, 2) → U_1
  149. Intersect(c, f_2, 1) → V_1
  150. Intersect(c, f_2, 2) → W_1
  151. Circle(W_1, P_1) → p_2
  152. Circle(V_1, P_1) → q_2
  153. Intersect(f_2, q_2, 2) → Z_1
  154. Intersect(f_2, p_2, 2) → A_2
  155. Circle(A_2, W_1) → r_2
  156. Circle(Z_1, V_1) → s_2
  157. Intersect(p_2, s_2, 1) → B_2
  158. Intersect(q_2, r_2, 2) → C_2
  159. CircleArc(W_1, B_2, P_1) → t_2
  160. Sequence(Arc[Rotate[Circle[W_1, B_2], (i * (360°) / n)], Rotate[B_2, (i * (360°) / n)], Rotate[P_1, (i * (360°) / n)]], i, 1, n) → list24
  161. CircleArc(B, V_1, W_1) → c_3
  162. Sequence(Arc[Rotate[Circle[B, V_1], (i * (360°) / n)], Rotate[V_1, (i * (360°) / n)], Rotate[W_1, (i * (360°) / n)]], i, 1, n) → list25
  163. CircleArc(V_1, P_1, C_2) → d_3
  164. CircleArc(A_2, C_2, W_1) → e_3
  165. Sequence(Arc[Rotate[Circle[A_2, C_2], (i * (360°) / n)], Rotate[C_2, (i * (360°) / n)], Rotate[W_1, (i * (360°) / n)]], i, 1, n) → list27
  166. CircleArc(Z_1, V_1, B_2) → f_3
  167. Sequence(Arc[Rotate[Circle[Z_1, V_1], (i * (360°) / n)], Rotate[V_1, (i * (360°) / n)], Rotate[B_2, (i * (360°) / n)]], i, 1, n) → list28
  168. Circle(B, B_2) → g_3
  169. Intersect(c, g_3, 1) → D_2
  170. Intersect(c, g_3, 2) → E_2
  171. CircleArc(B, D_2, E_2) → h_3
  172. Sequence(Arc[Rotate[Circle[B, D_2], (i * (360°) / n)], Rotate[D_2, (i * (360°) / n)], Rotate[E_2, (i * (360°) / n)]], i, 1, n) → list29
  173. Segment(H_2, I_2) → j_1
  174. Segment(H_2, G_2) → l_1
  175. Rotate(J_2, 90°, L_2) → O_2
  176. Rotate(J_2, 270°, L_2) → U_2
  177. Circle(U_2, J_2) → p_3
  178. Intersect(k_3, p_3, 2) → M_2
  179. CircleArc(U_2, J_2, M_2) → q_3
  180. CircleArc(O_2, M_2, J_2) → r_3
  181. Rotate(O_2, 60°, L_2) → N_2
  182. Rotate(M_2, 60°, L_2) → P_2
  183. Rotate(J_2, 60°, L_2) → Q_2
  184. Arc(Rotate[Circle[O_2, M_2], 60°, L_2], P_2, Q_2) → s_3
  185. Rotate(O_2, 120°, L_2) → R_2
  186. Rotate(M_2, 120°, L_2) → S_2
  187. Rotate(J_2, 120°, L_2) → T_2
  188. Arc(Rotate[Circle[O_2, M_2], 120°, L_2], S_2, T_2) → t_3
  189. Rotate(U_2, 60°, L_2) → V_2
  190. Rotate(J_2, 60°, L_2) → W_2
  191. Rotate(M_2, 60°, L_2) → Z_2
  192. Arc(Rotate[Circle[U_2, J_2], 60°, L_2], W_2, Z_2) → c_4
  193. Rotate(U_2, 120°, L_2) → A_3
  194. Rotate(J_2, 120°, L_2) → B_3
  195. Rotate(M_2, 120°, L_2) → C_3
  196. Arc(Rotate[Circle[U_2, J_2], 120°, L_2], B_3, C_3) → d_4
  197. Rotate(k_3, 90°, A) → e_4
  198. Rotate(e_4, 90°, A) → f_4
  199. Rotate(f_4, 90°, A) → g_4
  200. Rotate(S_2, 90°, A) → D_3
  201. Rotate(T_2, 90°, A) → E_3
  202. Arc(Rotate[Rotate[Circle[O_2, M_2], 120°, L_2], 90°, A], D_3, E_3) → h_4
  203. Rotate(B_3, 90°, A) → F_3
  204. Rotate(C_3, 90°, A) → G_3
  205. Arc(Rotate[Rotate[Circle[U_2, J_2], 120°, L_2], 90°, A], F_3, G_3) → k_4
  206. Rotate(U_2, 90°, A) → H_3
  207. Rotate(J_2, 90°, A) → I_3
  208. Rotate(M_2, 90°, A) → J_3
  209. Arc(Rotate[Circle[U_2, J_2], 90°, A], I_3, J_3) → p_4
  210. Rotate(O_2, 90°, A) → K_3
  211. Rotate(M_2, 90°, A) → L_3
  212. Rotate(J_2, 90°, A) → M_3
  213. Arc(Rotate[Circle[O_2, M_2], 90°, A], L_3, M_3) → q_4
  214. Rotate(P_2, 90°, A) → N_3
  215. Rotate(Q_2, 90°, A) → O_3
  216. Arc(Rotate[Rotate[Circle[O_2, M_2], 60°, L_2], 90°, A], N_3, O_3) → r_4
  217. Rotate(W_2, 90°, A) → P_3
  218. Rotate(Z_2, 90°, A) → Q_3
  219. Arc(Rotate[Rotate[Circle[U_2, J_2], 60°, L_2], 90°, A], P_3, Q_3) → s_4
  220. Rotate(P_3, 90°, A) → R_3
  221. Rotate(Q_3, 90°, A) → S_3
  222. Arc(Rotate[Rotate[Rotate[Circle[U_2, J_2], 60°, L_2], 90°, A], 90°, A], R_3, S_3) → t_4
  223. Rotate(N_3, 90°, A) → T_3
  224. Rotate(O_3, 90°, A) → U_3
  225. Arc(Rotate[Rotate[Rotate[Circle[O_2, M_2], 60°, L_2], 90°, A], 90°, A], T_3, U_3) → c_5
  226. Rotate(D_3, 90°, A) → V_3
  227. Rotate(E_3, 90°, A) → W_3
  228. Arc(Rotate[Rotate[Rotate[Circle[O_2, M_2], 120°, L_2], 90°, A], 90°, A], V_3, W_3) → d_5
  229. Rotate(F_3, 90°, A) → Z_3
  230. Rotate(G_3, 90°, A) → A_4
  231. Arc(Rotate[Rotate[Rotate[Circle[U_2, J_2], 120°, L_2], 90°, A], 90°, A], Z_3, A_4) → e_5
  232. Rotate(I_3, 90°, A) → B_4
  233. Rotate(J_3, 90°, A) → C_4
  234. Arc(Rotate[Rotate[Circle[U_2, J_2], 90°, A], 90°, A], B_4, C_4) → f_5
  235. Rotate(L_3, 90°, A) → D_4
  236. Rotate(M_3, 90°, A) → E_4
  237. Arc(Rotate[Rotate[Circle[O_2, M_2], 90°, A], 90°, A], D_4, E_4) → g_5
  238. Rotate(Z_3, 90°, A) → F_4
  239. Rotate(A_4, 90°, A) → G_4
  240. Arc(Rotate[Rotate[Rotate[Rotate[Circle[U_2, J_2], 120°, L_2], 90°, A], 90°, A], 90°, A], F_4, G_4) → h_5
  241. Rotate(V_3, 90°, A) → H_4
  242. Rotate(W_3, 90°, A) → I_4
  243. Arc(Rotate[Rotate[Rotate[Rotate[Circle[O_2, M_2], 120°, L_2], 90°, A], 90°, A], 90°, A], H_4, I_4) → k_5
  244. Rotate(D_4, 90°, A) → J_4
  245. Rotate(E_4, 90°, A) → K_4
  246. Arc(Rotate[Rotate[Rotate[Circle[O_2, M_2], 90°, A], 90°, A], 90°, A], J_4, K_4) → p_5
  247. Rotate(B_4, 90°, A) → L_4
  248. Rotate(C_4, 90°, A) → M_4
  249. Arc(Rotate[Rotate[Rotate[Circle[U_2, J_2], 90°, A], 90°, A], 90°, A], L_4, M_4) → q_5
  250. Rotate(T_3, 90°, A) → N_4
  251. Rotate(U_3, 90°, A) → O_4
  252. Arc(Rotate[Rotate[Rotate[Rotate[Circle[O_2, M_2], 60°, L_2], 90°, A], 90°, A], 90°, A], N_4, O_4) → r_5
  253. Rotate(R_3, 90°, A) → P_4
  254. Rotate(S_3, 90°, A) → Q_4
  255. Arc(Rotate[Rotate[Rotate[Rotate[Circle[U_2, J_2], 60°, L_2], 90°, A], 90°, A], 90°, A], P_4, Q_4) → s_5
  256. Rotate(t_5, 90°) → d_6
  257. Rotate(t_5, 180°) → e_6
  258. Rotate(t_5, 270°) → f_6
  259. Mirror(t_5, a_3) → c_6
  260. Rotate(c_6, 90°) → g_6
  261. Rotate(c_6, 180°) → h_6
  262. Rotate(c_6, 270°) → k_6
  263. Mirror(Z_4, W_4) → A_5
  264. Rotate(A_5, 90°, W_4) → B_5
  265. Mirror(B_5, W_4) → C_5
  266. Semicircle(B_5, Z_4) → p_6
  267. Semicircle(Z_4, C_5) → q_6
  268. Semicircle(C_5, A_5) → r_6
  269. Semicircle(A_5, B_5) → s_6
  270. Mirror(B_5, a_3) → D_5
  271. Mirror(Z_4, a_3) → E_5
  272. Semicircle(E_5, D_5) → t_6
  273. Mirror(C_5, a_3) → F_5
  274. Mirror(A_5, a_3) → G_5
  275. Semicircle(G_5, F_5) → c_7
  276. Mirror(Z_4, a_3) → H_5
  277. Mirror(C_5, a_3) → I_5
  278. Semicircle(I_5, H_5) → d_7
  279. Mirror(A_5, a_3) → J_5
  280. Mirror(B_5, a_3) → K_5
  281. Semicircle(K_5, J_5) → e_7
  282. Rotate(B_5, 90°) → L_5
  283. Rotate(Z_4, 90°) → M_5
  284. Semicircle(L_5, M_5) → f_7
  285. Rotate(Z_4, 90°) → N_5
  286. Rotate(C_5, 90°) → O_5
  287. Semicircle(N_5, O_5) → g_7
  288. Rotate(C_5, 90°) → P_5
  289. Rotate(A_5, 90°) → Q_5
  290. Semicircle(P_5, Q_5) → h_7
  291. Rotate(A_5, 90°) → R_5
  292. Rotate(B_5, 90°) → S_5
  293. Semicircle(R_5, S_5) → k_7
  294. Rotate(B_5, 180°) → T_5
  295. Rotate(Z_4, 180°) → U_5
  296. Semicircle(T_5, U_5) → p_7
  297. Rotate(Z_4, 180°) → V_5
  298. Rotate(C_5, 180°) → W_5
  299. Semicircle(V_5, W_5) → q_7
  300. Rotate(C_5, 180°) → Z_5
  301. Rotate(A_5, 180°) → A_6
  302. Semicircle(Z_5, A_6) → r_7
  303. Rotate(A_5, 180°) → B_6
  304. Rotate(B_5, 180°) → C_6
  305. Semicircle(B_6, C_6) → s_7
  306. Rotate(B_5, 270°) → D_6
  307. Rotate(Z_4, 270°) → E_6
  308. Semicircle(D_6, E_6) → t_7
  309. Rotate(Z_4, 270°) → F_6
  310. Rotate(C_5, 270°) → G_6
  311. Semicircle(F_6, G_6) → c_8
  312. Rotate(C_5, 270°) → H_6
  313. Rotate(A_5, 270°) → I_6
  314. Semicircle(H_6, I_6) → d_8
  315. Rotate(A_5, 270°) → J_6
  316. Rotate(B_5, 270°) → K_6
  317. Semicircle(J_6, K_6) → e_8
  318. Rotate(E_5, 90°) → L_6
  319. Rotate(D_5, 90°) → M_6
  320. Semicircle(L_6, M_6) → f_8
  321. Rotate(E_5, 180°) → N_6
  322. Rotate(D_5, 180°) → O_6
  323. Semicircle(N_6, O_6) → g_8
  324. Rotate(E_5, 270°) → P_6
  325. Rotate(D_5, 270°) → Q_6
  326. Semicircle(P_6, Q_6) → h_8
  327. Rotate(G_5, 90°) → R_6
  328. Rotate(F_5, 90°) → S_6
  329. Semicircle(R_6, S_6) → k_8
  330. Rotate(I_5, 90°) → T_6
  331. Rotate(H_5, 90°) → U_6
  332. Semicircle(T_6, U_6) → p_8
  333. Rotate(K_5, 90°) → V_6
  334. Rotate(J_5, 90°) → W_6
  335. Semicircle(V_6, W_6) → q_8
  336. Rotate(G_5, 180°) → Z_6
  337. Rotate(F_5, 180°) → A_7
  338. Semicircle(Z_6, A_7) → r_8
  339. Rotate(I_5, 180°) → B_7
  340. Rotate(H_5, 180°) → C_7
  341. Semicircle(B_7, C_7) → s_8
  342. Rotate(K_5, 180°) → D_7
  343. Rotate(J_5, 180°) → E_7
  344. Semicircle(D_7, E_7) → t_8
  345. Rotate(G_5, 270°) → F_7
  346. Rotate(F_5, 270°) → G_7
  347. Semicircle(F_7, G_7) → c_9
  348. Rotate(I_5, 270°) → H_7
  349. Rotate(H_5, 270°) → I_7
  350. Semicircle(H_7, I_7) → d_9
  351. Rotate(K_5, 270°) → J_7
  352. Rotate(J_5, 270°) → K_7
  353. Semicircle(J_7, K_7) → e_9
  354. Point(t_5) → L_7
  355. Point(c) → M_7
  356. Tangent(M_7, c) → b_3,
  357. Tangent(L_7, t_5) → i_3,
  358. Intersect(b_3, i_3) → N_7
  359. Intersect(b_3, m_1) → O_7
  360. Intersect(m_1, i_3) → P_7
  361. AngularBisector(P_7, O_7, N_7) → j_3
  362. AngularBisector(N_7, P_7, O_7) → l_3
  363. Intersect(j_3, l_3) → Q_7
  364. OrthogonalLine(Q_7, m_1) → m_3
  365. Intersect(m_3, m_1) → R_7
  366. Circle(Q_7, R_7) → f_9
  367. Point(f_9) → S_7
  368. Point(c) → T_7
  369. Tangent(T_7, c) → n_3,
  370. Tangent(S_7, f_9) → a_4,
  371. Intersect(a_4, n_3) → U_7
  372. Intersect(m_1, a_4) → V_7
  373. Intersect(m_1, n_3) → W_7
  374. AngularBisector(V_7, W_7, U_7) → b_4
  375. AngularBisector(U_7, V_7, W_7) → i_4
  376. Intersect(b_4, i_4) → Z_7
  377. OrthogonalLine(Z_7, m_1) → j_4
  378. Intersect(j_4, m_1) → A_8
  379. Circle(Z_7, A_8) → g_9
  380. Intersect(f_9, m_1, 1) → B_8
  381. Rotate(B_8, 120°, Q_7) → C_8
  382. Rotate(C_8, 120°, Q_7) → D_8
  383. CircleArc(D_8, B_8, C_8) → h_9
  384. CircleArc(B_8, C_8, D_8) → k_9
  385. CircleArc(C_8, D_8, B_8) → p_9
  386. Line(C_8, B_8) → l_4
  387. Line(B_8, D_8) → m_4
  388. Line(D_8, C_8) → n_4
  389. Mirror(B_8, n_4) → E_8
  390. Mirror(C_8, n_4) → F_8
  391. Mirror(D_8, n_4) → G_8
  392. Arc(Mirror[Circle[B_8, C_8], n_4], F_8, G_8) → q_9
  393. Mirror(C_8, m_4) → H_8
  394. Mirror(D_8, m_4) → I_8
  395. Mirror(B_8, m_4) → J_8
  396. Arc(Mirror[Circle[C_8, D_8], m_4], I_8, J_8) → r_9
  397. Mirror(D_8, l_4) → K_8
  398. Mirror(B_8, l_4) → L_8
  399. Mirror(C_8, l_4) → M_8
  400. Arc(Mirror[Circle[D_8, B_8], l_4], L_8, M_8) → s_9
  401. Mirror(f_9, a_3) → t_9
  402. Mirror(g_9, a_3) → c_{10}
  403. Mirror(F_8, a_3) → N_8
  404. Mirror(G_8, a_3) → O_8
  405. Arc(Mirror[Mirror[Circle[B_8, C_8], n_4], a_3], N_8, O_8) → d_{10}
  406. Mirror(L_8, a_3) → P_8
  407. Mirror(M_8, a_3) → Q_8
  408. Arc(Mirror[Mirror[Circle[D_8, B_8], l_4], a_3], P_8, Q_8) → e_{10}
  409. Mirror(I_8, a_3) → R_8
  410. Mirror(J_8, a_3) → S_8
  411. Arc(Mirror[Mirror[Circle[C_8, D_8], m_4], a_3], R_8, S_8) → f_{10}
  412. Rotate(f_9, 90°) → g_{10}
  413. Rotate(f_9, 180°) → h_{10}
  414. Rotate(f_9, 270°) → k_{10}
  415. Rotate(t_9, 90°) → p_{10}
  416. Rotate(t_9, 180°) → q_{10}
  417. Rotate(t_9, 270°) → r_{10}
  418. Rotate(g_9, 270°) → s_{10}
  419. Rotate(g_9, 180°) → t_{10}
  420. Rotate(g_9, 90°) → c_{11}
  421. Rotate(c_{10}, 90°) → d_{11}
  422. Rotate(c_{10}, 180°) → e_{11}
  423. Rotate(c_{10}, 270°) → f_{11}
  424. Rotate(F_8, 90°) → T_8
  425. Rotate(G_8, 90°) → U_8
  426. Arc(Rotate[Mirror[Circle[B_8, C_8], n_4], 90°], T_8, U_8) → g_{11}
  427. Rotate(I_8, 90°) → V_8
  428. Rotate(J_8, 90°) → W_8
  429. Arc(Rotate[Mirror[Circle[C_8, D_8], m_4], 90°], V_8, W_8) → h_{11}
  430. Rotate(L_8, 90°) → Z_8
  431. Rotate(M_8, 90°) → A_9
  432. Arc(Rotate[Mirror[Circle[D_8, B_8], l_4], 90°], Z_8, A_9) → k_{11}
  433. Rotate(L_8, 180°) → B_9
  434. Rotate(M_8, 180°) → C_9
  435. Arc(Rotate[Mirror[Circle[D_8, B_8], l_4], 180°], B_9, C_9) → p_{11}
  436. Rotate(F_8, 180°) → D_9
  437. Rotate(G_8, 180°) → E_9
  438. Arc(Rotate[Mirror[Circle[B_8, C_8], n_4], 180°], D_9, E_9) → q_{11}
  439. Rotate(I_8, 180°) → F_9
  440. Rotate(J_8, 180°) → G_9
  441. Arc(Rotate[Mirror[Circle[C_8, D_8], m_4], 180°], F_9, G_9) → r_{11}
  442. Rotate(I_8, 270°) → H_9
  443. Rotate(J_8, 270°) → I_9
  444. Arc(Rotate[Mirror[Circle[C_8, D_8], m_4], 270°], H_9, I_9) → s_{11}
  445. Rotate(L_8, 270°) → J_9
  446. Rotate(M_8, 270°) → K_9
  447. Arc(Rotate[Mirror[Circle[D_8, B_8], l_4], 270°], J_9, K_9) → t_{11}
  448. Rotate(F_8, 270°) → L_9
  449. Rotate(G_8, 270°) → M_9
  450. Arc(Rotate[Mirror[Circle[B_8, C_8], n_4], 270°], L_9, M_9) → c_{12}
  451. Rotate(N_8, 90°) → N_9
  452. Rotate(O_8, 90°) → O_9
  453. Arc(Rotate[Mirror[Mirror[Circle[B_8, C_8], n_4], a_3], 90°], N_9, O_9) → d_{12}
  454. Rotate(P_8, 90°) → P_9
  455. Rotate(Q_8, 90°) → Q_9
  456. Arc(Rotate[Mirror[Mirror[Circle[D_8, B_8], l_4], a_3], 90°], P_9, Q_9) → e_{12}
  457. Rotate(R_8, 90°) → R_9
  458. Rotate(S_8, 90°) → S_9
  459. Arc(Rotate[Mirror[Mirror[Circle[C_8, D_8], m_4], a_3], 90°], R_9, S_9) → f_{12}
  460. Rotate(R_8, 180°) → T_9
  461. Rotate(S_8, 180°) → U_9
  462. Arc(Rotate[Mirror[Mirror[Circle[C_8, D_8], m_4], a_3], 180°], T_9, U_9) → g_{12}
  463. Rotate(P_8, 180°) → V_9
  464. Rotate(Q_8, 180°) → W_9
  465. Arc(Rotate[Mirror[Mirror[Circle[D_8, B_8], l_4], a_3], 180°], V_9, W_9) → h_{12}
  466. Rotate(N_8, 180°) → Z_9
  467. Rotate(O_8, 180°) → A_{10}
  468. Arc(Rotate[Mirror[Mirror[Circle[B_8, C_8], n_4], a_3], 180°], Z_9, A_{10}) → k_{12}
  469. Rotate(N_8, 270°) → B_{10}
  470. Rotate(O_8, 270°) → C_{10}
  471. Arc(Rotate[Mirror[Mirror[Circle[B_8, C_8], n_4], a_3], 270°], B_{10}, C_{10}) → p_{12}
  472. Rotate(P_8, 270°) → D_{10}
  473. Rotate(Q_8, 270°) → E_{10}
  474. Arc(Rotate[Mirror[Mirror[Circle[D_8, B_8], l_4], a_3], 270°], D_{10}, E_{10}) → q_{12}
  475. Rotate(R_8, 270°) → F_{10}
  476. Rotate(S_8, 270°) → G_{10}
  477. Arc(Rotate[Mirror[Mirror[Circle[C_8, D_8], m_4], a_3], 270°], F_{10}, G_{10}) → r_{12}
  478. Line(G_2, F_2) → i_5
  479. Rotate(l1, 90°, i_5) → l1'
  480. Line(F_2, I_2) → j_5
  481. Intersect(j_1, l_1) → H_{10}
  482. Line(I_2, xAxis) → l_5
  483. Line(G_2, yAxis) → m_5
  484. Rotate(l1, (-(90°)), l_5) → l1'_1
  485. Rotate(l1, 90°, j_5) → l1'_2
  486. Rotate(l1, (-(90°)), m_5) → l1'_3
  487. Point(Circle[F_2, Distance[G_2, F_2], Segment[G_2, F_2]]) → I_{10}
  488. Cube(G_2, F_2, I_{10}) → o, J_{10}, face5, face6, arête2, arête1, arête4, arête6, arête3, arête8, arête10, arête9, K_{10}, arête5, arête11, arête7, arête12, L_{10}, M_{10}, N_{10}, face1, face3, face2, face4
  489. Rotate(l2, 90°, i_5) → l2'
  490. Rotate(l2, (-(90°)), l_5) → l2'_1
  491. Rotate(l2, 90°, j_5) → l2'_2
  492. Rotate(l2, (-(90°)), m_5) → l2'_3
  493. Line(L_{10}, M_{10}) → n_5
  494. Rotate(l1'_2, 90°, n_5) → l3
  495. Rotate(l2'_2, 90°, n_5) → l4

AI Drawing Prompt Reference

Create a three-dimensional rose cube: a cube framework combined with rotating petal-like curves and arcs. Start from a central point, construct auxiliary lines and circles, then use rotations and mirror symmetry to generate elliptical arcs and petal shapes. Arrange these elements around a central axis in equal angular steps, creating a layered, symmetric floral pattern. Include tangent lines, angle bisectors, and multiple intersecting circles to enhance the geometric complexity and visual depth.
The prompt above was compiled from the original drawing commands and is for reference only.