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Duality of Cubes and Octahedra

Duality of Cubes and Octahedra
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2023-03-14 00:22
Solid GeometryParametric EquationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the duality between cubes and octahedra. It involves core mathematical concepts in geometry and topology. The construction uses GeoGebra commands such as Ellipse, Point, Midpoint, Segment to build models of cubes and octahedra. Suitable for middle to high school students and mathematics education.

GeoGebra Commands & Parameters

  1. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  2. Point(mira1) → A
  3. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  4. Point(mira2) → B
  5. punto3d(A, B, M, O_0, O_r, 1, -1, 1, U, L) → V3'
  6. punto3d(A, B, M, O_0, O_r, -1, -1, -1, U, L) → V8'
  7. punto3d(A, B, M, O_0, O_r, 1, 1, -1, U, L) → V1'
  8. Midpoint(V1', V8') → octV6'
  9. Midpoint(V1', V3') → octV1'
  10. punto3d(A, B, M, O_0, O_r, -1, 1, 1, U, L) → V5'
  11. Midpoint(V1', V5') → octV2'
  12. Midpoint(V8', V5') → octV3'
  13. Midpoint(V3', V8') → octV4'
  14. Midpoint(V3', V5') → octV5'
  15. punto3d(A, B, M, O_0, O_r, 1, -1, -1, U, L) → V4'
  16. punto3d(A, B, M, O_0, O_r, -1, -1, 1, U, L) → V7'
  17. punto3d(A, B, M, O_0, O_r, -1, 1, -1, U, L) → V6'
  18. punto3d(A, B, M, O_0, O_r, 1, 1, 1, U, L) → V2'
  19. Segment(octV1', octV2') → octaedroA1
  20. Segment(octV1', octV2') → octaedroA1'
  21. Segment(octV2', octV6') → octaedroA10
  22. Segment(octV2', octV6') → octaedroA10'
  23. Segment(octV3', octV6') → octaedroA11
  24. Segment(octV3', octV6') → octaedroA11'
  25. Segment(octV4', octV6') → octaedroA12
  26. Segment(octV4', octV6') → octaedroA12'
  27. Segment(octV3', octV2') → octaedroA2
  28. Segment(octV3', octV2') → octaedroA2'
  29. Segment(octV3', octV4') → octaedroA3
  30. Segment(octV3', octV4') → octaedroA3'
  31. Segment(octV1', octV4') → octaedroA4
  32. Segment(octV1', octV4') → octaedroA4'
  33. Segment(octV1', octV5') → octaedroA5
  34. Segment(octV1', octV5') → octaedroA5'
  35. Segment(octV2', octV5') → octaedroA6
  36. Segment(octV2', octV5') → octaedroA6'
  37. Segment(octV3', octV5') → octaedroA7
  38. Segment(octV3', octV5') → octaedroA7'
  39. Segment(octV4', octV5') → octaedroA8
  40. Segment(octV4', octV5') → octaedroA8'
  41. Segment(octV1', octV6') → octaedroA9
  42. Segment(octV1', octV6') → octaedroA9'
  43. Segment(V2', V1') → cuboA1'
  44. Segment(V3', V7') → cuboA10'
  45. Segment(V5', V6') → cuboA11'
  46. Segment(V7', V8') → cuboA12'
  47. Segment(V2', V3') → cuboA2'
  48. Segment(V3', V4') → cuboA3'
  49. Segment(V4', V1') → cuboA4'
  50. Segment(V1', V6') → cuboA5'
  51. Segment(V6', V8') → cuboA6'
  52. Segment(V8', V4') → cuboA7'
  53. Segment(V2', V5') → cuboA8'
  54. Segment(V5', V7') → cuboA9'
  55. Circle(C, 9.5) → d
  56. Point(d) → G
  57. OrthogonalLine(C, xAxis) → e
  58. Intersect(d, e, 2) → H
  59. Angle(G, C, H) → ang
  60. If(t < 0, t + 5, t) → t5
  61. If(t < 3, 1, t - 1) → t4
  62. If(t ≥ 1, If[t ≥ 3, 4 - t, 1], t) → t1
  63. If(t > 2, 2, 1) → t32
  64. If(t < 1, 0, If[t > 2, 3 - t, t - 1]) → t2
  65. If(t > 2, (t2 - 1) / 2, 0) → t31
  66. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  67. Angle(M, O_r, B) → θ
  68. Angle(U, O_r, A) → φ
  69. Vector(O_r, O_0) → u
  70. Translate(Xr, u) → X0
  71. Translate(Zr, u) → Z0
  72. Translate(Yr, u) → Y0
  73. Vector(O_0, Z0) → v
  74. Vector(O_0, Y0) → w
  75. Vector(O_0, X0) → z
  76. Circle(O_0, L) → c
  77. Line(O_0, Y0) → a
  78. Line(O_0, yAxis) → b
  79. Line(O_0, X0) → g
  80. Line(Z0, O_0) → j
  81. Translate(P1r, u) → P1
  82. Rotate(A, botonpaso, O_r) → botonA
  83. Rotate(B, botonpaso, O_r) → botonB
  84. Segment(Mira3, Mira2) → k
  85. Segment((-1, 0), U) → l
  86. Segment((0, -1), M) → m
  87. punto3d(A, B, M, O_0, O_r, 1 / t6, 1 / t6, -1 / t6, U, L) → V1
  88. punto3d(A, B, M, O_0, O_r, 1 / t6, 1 / t6, 1 / t6, U, L) → V2
  89. punto3d(A, B, M, O_0, O_r, 1 / t6, -1 / t6, -1 / t6, U, L) → V4
  90. punto3d(A, B, M, O_0, O_r, -1 / t6, 1 / t6, 1 / t6, U, L) → V5
  91. punto3d(A, B, M, O_0, O_r, -1 / t6, 1 / t6, -1 / t6, U, L) → V6
  92. punto3d(A, B, M, O_0, O_r, -1 / t6, -1 / t6, 1 / t6, U, L) → V7
  93. punto3d(A, B, M, O_0, O_r, -1 / t6, -1 / t6, -1 / t6, U, L) → V8
  94. punto3d(A, B, M, O_0, O_r, 1 / t6, -1 / t6, 1 / t6, U, L) → V3
  95. Midpoint(V1, V3) → octV1
  96. Midpoint(V1, V5) → octV2
  97. Midpoint(V8, V5) → octV3
  98. Midpoint(V3, V8) → octV4
  99. Midpoint(V3, V5) → octV5
  100. Midpoint(V1, V8) → octV6
  101. Midpoint(V1, V2) → cub1M
  102. Midpoint(V3, V2) → cub2M
  103. Midpoint(V1, V6) → cub4M
  104. Midpoint(V7, V8) → cub5M
  105. Angle(V3, cub1M, V4) → cub1Ang
  106. If(cub1Ang < 180°, V2, V3) → cubV2b
  107. If(cub1Ang < 180°, V1, V4) → cubV1b
  108. Polygon(V3, V4, cubV1b, cubV2b) → cub1, cub1v3, cub1v4, cub1v1, cub1v2
  109. Angle(V6, cub1M, V5) → cub3Ang
  110. If(cub3Ang < 180°, V2, V5) → cubV2c
  111. If(cub3Ang < 180°, V1, V6) → cubV1c
  112. Polygon(V6, V5, cubV2c, cubV1c) → cub3, cub3v6, cub3v5, cub3v2, cub3v1
  113. Angle(V5, cub5M, V6) → cub5Ang
  114. If(cub5Ang < 180°, V5, V7) → cubV5b
  115. If(cub5Ang < 180°, V6, V8) → cubV6b
  116. Polygon(V8, V7, cubV5b, cubV6b) → cub5, cub5v8, cub5v7, cub5v5, cub5v6
  117. Angle(V4, cub5M, V3) → cub6Ang
  118. If(cub6Ang < 180°, V7, V3) → cubV7b
  119. If(cub6Ang < 180°, V8, V4) → cubV8b
  120. Polygon(V4, V3, cubV7b, cubV8b) → cub6, cub6v4, cub6v3, cub6v7, cub6v8
  121. Angle(V5, cub2M, V7) → cub2Ang
  122. If(cub2Ang < 180°, V3, V7) → cubV3b
  123. If(cub2Ang < 180°, V2, V5) → cubV2d
  124. Polygon(V7, V5, cubV2d, cubV3b) → cub2, cub2v7, cub2v5, cub2v2, cub2v3
  125. Angle(V4, cub4M, V8) → cub4Ang
  126. Segment(V2, V1) → cubA1'
  127. Segment(V2, V3) → cubA2'
  128. Segment(V3, V4) → cubA3'
  129. Segment(V4, V1) → cubA4'
  130. Segment(V1, V6) → cubA5'
  131. Segment(V8, V6) → cubA6'
  132. Segment(V4, V8) → cubA7'
  133. Segment(V2, V5) → cubA8'
  134. Segment(V5, V7) → cubA9'
  135. Segment(V3, V7) → cubA10'
  136. Segment(V5, V6) → cubA11'
  137. Segment(V7, V8) → cubA12'
  138. Segment(octV1, octV2) → octA1'
  139. Segment(octV3, octV2) → octA2'
  140. Segment(octV3, octV4) → octA3'
  141. Segment(octV1, octV4) → octA4'
  142. Segment(octV1, octV5) → octA5'
  143. Segment(octV2, octV5) → octA6'
  144. Segment(octV3, octV5) → octA7'
  145. Segment(octV4, octV5) → octA8'
  146. Segment(octV1, octV6) → octA9'
  147. Segment(octV2, octV6) → octA10'
  148. Segment(octV3, octV6) → octA11'
  149. Segment(octV4, octV6) → octA12'
  150. Segment(octV1, octV2) → octA1
  151. Segment(octV3, octV2) → octA2
  152. Segment(octV3, octV4) → octA3
  153. Segment(octV1, octV4) → octA4
  154. Segment(octV1, octV5) → octA5
  155. Segment(octV2, octV5) → octA6
  156. Segment(octV3, octV5) → octA7
  157. Segment(octV4, octV5) → octA8
  158. Segment(octV1, octV6) → octA9
  159. Segment(octV2, octV6) → octA10
  160. Segment(octV3, octV6) → octA11
  161. Segment(octV4, octV6) → octA12
  162. Segment(V1, V2) → cubA1
  163. Segment(V2, V3) → cubA2
  164. Segment(V3, V4) → cubA3
  165. Segment(V4, V1) → cubA4
  166. Segment(V1, V6) → cubA5
  167. Segment(V6, V8) → cubA6
  168. Segment(V8, V4) → cubA7
  169. Segment(V2, V5) → cubA8
  170. Segment(V5, V7) → cubA9
  171. Segment(V7, V3) → cubA10
  172. Segment(V6, V5) → cubA11
  173. Segment(V7, V8) → cubA12
  174. If(cub4Ang < 180°, V1, V4) → cubV1d
  175. If(cub4Ang < 180°, V6, V8) → cubV6c
  176. Polygon(V8, V4, cubV1d, cubV6c) → cub4, cub4v8, cub4v4, cub4v1, cub4v6
  177. Angle(octV2, octV1, octV5) → oct1Ang
  178. If(oct1Ang < 180°, octV1, octV2) → octV1b
  179. Angle(octV3, octV2, octV5) → oct2Ang
  180. If(oct2Ang < 180°, octV2, octV3) → octV2b
  181. Angle(octV4, octV3, octV5) → oct3Ang
  182. If(oct3Ang < 180°, octV3, octV4) → octV3b
  183. Angle(octV1, octV4, octV5) → oct4Ang
  184. If(oct4Ang < 180°, octV4, octV1) → octV4b
  185. Angle(octV6, octV1, octV2) → oct5Ang
  186. If(oct5Ang < 180°, octV1, octV2) → octV1c
  187. Angle(octV3, octV6, octV2) → oct6Ang
  188. If(oct6Ang < 180°, octV6, octV3) → octV6b
  189. Angle(octV6, octV3, octV4) → oct7Ang
  190. If(oct7Ang < 180°, octV3, octV4) → octV3c
  191. Angle(octV6, octV4, octV1) → oct8Ang
  192. If(oct8Ang < 180°, octV4, octV1) → octV4c
  193. Polygon(octV1b, octV2, octV5) → oct1, oct1v5, oct1v1, oct1v2
  194. Polygon(octV2b, octV3, octV5) → oct2, oct2v5, oct2v2, oct2v3
  195. Polygon(octV3b, octV4, octV5) → oct3, oct3v5, oct3v3, oct3v4
  196. Polygon(octV4b, octV1, octV5) → oct4, oct4v5, oct4v4, oct4v1
  197. Polygon(octV1c, octV2, octV6) → oct5, oct5v6, oct5v1, oct5v2
  198. Polygon(octV6b, octV2, octV3) → oct6, oct6v3, oct6v6, oct6v2
  199. Polygon(octV3c, octV4, octV6) → oct7, oct7v6, oct7v3, oct7v4
  200. Polygon(octV4c, octV1, octV6) → oct8, oct8v6, oct8v4, oct8v1
  201. Midpoint(V3', V2') → Ma2
  202. Midpoint(V4', V1') → Ma4
  203. Midpoint(V1', V6') → Ma5
  204. Midpoint(V7', V5') → Ma9
  205. Midpoint(V4', V8') → Ma7
  206. Midpoint(V8', V6') → Ma6
  207. Midpoint(V2', V5') → Ma8
  208. punto3d(A, B, M, O_0, O_r, (1 - t1) / t4, -1 / t4, 1 / t4, U, L) → T01b
  209. punto3d(A, B, M, O_0, O_r, 1 / t4, -1 / t4, (1 - t1) / t4, U, L) → T01c
  210. punto3d(A, B, M, O_0, O_r, 1 / t4, (-1 + t1) / t4, 1 / t4, U, L) → T01a
  211. Polygon(T01c, T01a, T01b) → poly1, A0110', A0113', A0109'
  212. punto3d(A, B, M, O_0, O_r, (1 - t1) / t4, -1 / t4, -1 / t4, U, L) → T02a
  213. punto3d(A, B, M, O_0, O_r, 1 / t4, (-1 + t1) / t4, -1 / t4, U, L) → T02b
  214. punto3d(A, B, M, O_0, O_r, 1 / t4, -1 / t4, (-1 + t1) / t4, U, L) → T02c
  215. Polygon(T02c, T02b, T02a) → poly2, A0210', A0214', A0209'
  216. punto3d(A, B, M, O_0, O_r, 1 / t4, 1 / t4, (-1 + t1) / t4, U, L) → T03c
  217. punto3d(A, B, M, O_0, O_r, 1 / t4, (1 - t1) / t4, -1 / t4, U, L) → T03b
  218. punto3d(A, B, M, O_0, O_r, (1 - t1) / t4, 1 / t4, -1 / t4, U, L) → T03a
  219. Polygon(T03b, T03c, T03a) → poly3, A0310', A0312', A0314'
  220. punto3d(A, B, M, O_0, O_r, (-1 + t1) / t4, 1 / t4, -1 / t4, U, L) → T04a
  221. punto3d(A, B, M, O_0, O_r, -1 / t4, (1 - t1) / t4, -1 / t4, U, L) → T04b
  222. punto3d(A, B, M, O_0, O_r, -1 / t4, 1 / t4, (-1 + t1) / t4, U, L) → T04c
  223. Polygon(T04b, T04c, T04a) → poly4, A0411', A0412', A0414'
  224. punto3d(A, B, M, O_0, O_r, (-1 + t1) / t4, 1 / t4, 1 / t4, U, L) → T05a
  225. punto3d(A, B, M, O_0, O_r, -1 / t4, (1 - t1) / t4, 1 / t4, U, L) → T05b
  226. punto3d(A, B, M, O_0, O_r, -1 / t4, 1 / t4, (1 - t1) / t4, U, L) → T05c
  227. Polygon(T05b, T05a, T05c) → poly5, A0513', A0512', A0511'
  228. punto3d(A, B, M, O_0, O_r, 1 / t4, 1 / t4, (1 - t1) / t4, U, L) → T06c
  229. punto3d(A, B, M, O_0, O_r, 1 / t4, (1 - t1) / t4, 1 / t4, U, L) → T06b
  230. punto3d(A, B, M, O_0, O_r, (1 - t1) / t4, 1 / t4, 1 / t4, U, L) → T06a
  231. Polygon(T06b, T06c, T06a) → poly6, A0610', A0612', A0613'
  232. punto3d(A, B, M, O_0, O_r, (-1 + t1) / t4, -1 / t4, -1 / t4, U, L) → T07a
  233. punto3d(A, B, M, O_0, O_r, -1 / t4, (-1 + t1) / t4, -1 / t4, U, L) → T07b
  234. punto3d(A, B, M, O_0, O_r, -1 / t4, -1 / t4, (-1 + t1) / t4, U, L) → T07c
  235. Polygon(T07c, T07a, T07b) → poly7, A0709', A0714', A0711'
  236. punto3d(A, B, M, O_0, O_r, -1 / t4, -1 / t4, (1 - t1) / t4, U, L) → T08c
  237. punto3d(A, B, M, O_0, O_r, -1 / t4, (-1 + t1) / t4, 1 / t4, U, L) → T08b
  238. punto3d(A, B, M, O_0, O_r, (-1 + t1) / t4, -1 / t4, 1 / t4, U, L) → T08a
  239. Polygon(T08a, T08b, T08c) → poly8, A0813', A0811', A0809'
  240. Polygon(T07a, T02a, T02b, T03b, T03a, T04a, T04b, T07b) → poly14, A0914', A0214, A1014', A0314, A1214', A0414, A1114', A0714
  241. Angle(T06c, M10, T06b) → angM10
  242. Angle(T05c, M12, T05a) → angM12
  243. Angle(T05a, M13, T05b) → angM13
  244. Angle(T01a, T01c, T01b) → angM01
  245. Angle(T02a, T02c, T02b) → angM02
  246. Angle(T03a, T03b, T03c) → angM03
  247. Angle(T04b, T04a, T04c) → angM04
  248. Angle(T05c, T05a, T05b) → angM05
  249. Angle(T06a, T06c, T06b) → angM06
  250. Angle(T07c, T07a, T07b) → angM07
  251. Angle(T08b, T08a, T08c) → angM08
  252. Angle(T08a, M09, T08c) → angM09
  253. Angle(T05b, M11, T05c) → angM11
  254. Angle(T04b, M14, T04a) → angM14
  255. punto3d(A, B, M, O_0, O_r, 0, (-1 + t2) / t32, 1 + t31, U, L) → Q01a
  256. punto3d(A, B, M, O_0, O_r, 0, -1 - t31, (1 - t2) / t32, U, L) → Q01b
  257. punto3d(A, B, M, O_0, O_r, (-1 + t2) / t32, 0, 1 + t31, U, L) → Q02a
  258. punto3d(A, B, M, O_0, O_r, -1 - t31, 0, (1 - t2) / t32, U, L) → Q02b
  259. punto3d(A, B, M, O_0, O_r, 0, (1 - t2) / t32, 1 + t31, U, L) → Q03a
  260. punto3d(A, B, M, O_0, O_r, 0, 1 + t31, (1 - t2) / t32, U, L) → Q03b
  261. punto3d(A, B, M, O_0, O_r, (1 - t2) / t32, 0, 1 + t31, U, L) → Q04a
  262. punto3d(A, B, M, O_0, O_r, 1 + t31, 0, (1 - t2) / t32, U, L) → Q04b
  263. punto3d(A, B, M, O_0, O_r, (1 - t2) / t32, -1 - t31, 0, U, L) → Q05a
  264. punto3d(A, B, M, O_0, O_r, 1 + t31, (-1 + t2) / t32, 0, U, L) → Q05b
  265. punto3d(A, B, M, O_0, O_r, (-1 + t2) / t32, -1 - t31, 0, U, L) → Q06a
  266. punto3d(A, B, M, O_0, O_r, -1 - t31, (-1 + t2) / t32, 0, U, L) → Q06b
  267. punto3d(A, B, M, O_0, O_r, -1 - t31, (1 - t2) / t32, 0, U, L) → Q07a
  268. punto3d(A, B, M, O_0, O_r, (-1 + t2) / t32, 1 + t31, 0, U, L) → Q07b
  269. punto3d(A, B, M, O_0, O_r, (1 - t2) / t32, 1 + t31, 0, U, L) → Q08a
  270. punto3d(A, B, M, O_0, O_r, 1 + t31, (1 - t2) / t32, 0, U, L) → Q08b
  271. punto3d(A, B, M, O_0, O_r, 0, -1 - t31, (-1 + t2) / t32, U, L) → Q09a
  272. punto3d(A, B, M, O_0, O_r, 0, (-1 + t2) / t32, -1 - t31, U, L) → Q09b
  273. punto3d(A, B, M, O_0, O_r, -1 - t31, 0, (-1 + t2) / t32, U, L) → Q10a
  274. punto3d(A, B, M, O_0, O_r, 0, 1 + t31, (-1 + t2) / t32, U, L) → Q11a
  275. punto3d(A, B, M, O_0, O_r, 0, (1 - t2) / t32, -1 - t31, U, L) → Q11b
  276. punto3d(A, B, M, O_0, O_r, (-1 + t2) / t32, 0, -1 - t31, U, L) → Q10b
  277. punto3d(A, B, M, O_0, O_r, 1 + t31, 0, (-1 + t2) / t32, U, L) → Q12a
  278. punto3d(A, B, M, O_0, O_r, (1 - t2) / t32, 0, -1 - t31, U, L) → Q12b
  279. Polygon(T08a, T08c, T07c, T07a, T02a, T02c, T01c, T01b) → poly9, A0809, A0911, A0709, A0914, A0209, A0910, A0109, A0913'
  280. Polygon(T02c, T02b, T03b, T03c, T06c, T06b, T01a, T01c) → poly10, A0210, A1014, A0310, A1012, A0610, A1013', A0110, A0910'
  281. Polygon(T08b, T08c, T07c, T07b, T04b, T04c, T05c, T05b) → poly11, A0811, A0911', A0711, A1114, A0411, A1112, A0511, A1113'
  282. Polygon(T05a, T06a, T06c, T03c, T03a, T04a, T04c, T05c) → poly12, A1213', A0612, A1012', A0312, A1214, A0412, A1112', A0512
  283. Polygon(T08a, T01b, T01a, T06b, T06a, T05a, T05b, T08b) → poly13, A0913, A0113, A1013, A0613, A1213, A0513, A1113, A0813
  284. Polygon(Q01a, Q04a, Q03a, Q02a) → poly15, A1522, A1521, A1523, A1526
  285. Polygon(Q01b, Q06a, Q09a, Q05a) → poly16, A1626, A1627, A1628, A1622
  286. Polygon(Q08b, Q04b, Q05b, Q12a) → poly17, A1721, A1722, A1728, A1725
  287. Polygon(Q06b, Q02b, Q07a, Q10a) → poly18, A1826, A1823, A1824, A1827
  288. Polygon(Q07b, Q03b, Q08a, Q11a) → poly19, A1923, A1921, A1925, A1924
  289. Polygon(Q10b, Q11b, Q12b, Q09b) → poly20, A2024, A2025, A2028, A2027
  290. Polygon(Q04a, Q03a, Q03b, Q08a, Q08b, Q04b) → poly21, A1521', A2123, A1921', A2125, A1721', A2122
  291. Polygon(Q01b, Q01a, Q04a, Q04b, Q05b, Q05a) → poly22, A2226, A1522', A2122', A1722', A2228, A1622'
  292. Polygon(Q02b, Q07a, Q07b, Q03b, Q03a, Q02a) → poly23, A1823', A2324, A1923', A2123', A1523', A2326
  293. Polygon(Q07b, Q07a, Q10a, Q10b, Q11b, Q11a) → poly24, A2324', A1824', A2427, A2024', A2425, A1924'
  294. Polygon(Q08b, Q08a, Q11a, Q11b, Q12b, Q12a) → poly25, A2125', A1925', A2425', A2025', A2528, A1725'
  295. Polygon(Q01a, Q02a, Q02b, Q06b, Q06a, Q01b) → poly26, A1526', A2326', A1826', A2627, A1626', A2226'
  296. Polygon(Q06a, Q06b, Q10a, Q10b, Q09b, Q09a) → poly27, A2627', A1827', A2427', A2027', A2728, A1627'
  297. Polygon(Q05a, Q09a, Q09b, Q12b, Q12a, Q05b) → poly28, A1628', A2728', A2028', A2528', A1728', A2228'
  298. Angle(Q03a, M15, Q02a) → angM15
  299. Angle(Q06a, M16, Q09a) → angM16
  300. Angle(Q12a, M17, Q08b) → angM17
  301. Angle(Q02b, M18, Q07a) → angM18
  302. Angle(Q11a, M19, Q07b) → angM19
  303. Angle(Q10b, M20, Q11b) → angM20
  304. Angle(Q08a, M21, Q03b) → angM21
  305. Angle(Q04b, M22, Q04a) → angM22
  306. Angle(Q07a, M23, Q02b) → angM23
  307. Angle(Q07b, M24, Q11a) → angM24
  308. Angle(Q11a, M25, Q08a) → angM25
  309. Angle(Q02b, M26, Q06b) → angM26
  310. Angle(Q06b, M27, Q10a) → angM27
  311. Angle(Q09a, M28, Q09b) → angM28
  312. Segment(V1', V2') → cuboA1
  313. Segment(V2', V3') → cuboA2
  314. Segment(V3', V4') → cuboA3
  315. Segment(V4', V1') → cuboA4
  316. Segment(V1', V6') → cuboA5
  317. Segment(V6', V8') → cuboA6
  318. Segment(V8', V4') → cuboA7
  319. Segment(V2', V5') → cuboA8
  320. Segment(V5', V7') → cuboA9
  321. Segment(V7', V3') → cuboA10
  322. Segment(V6', V5') → cuboA11
  323. Segment(V7', V8') → cuboA12
  324. Intersect(d, e, 2) → E
  325. Intersect(d, e, 1) → F
  326. Rotate(H, 324°, C) → I
  327. Angle(I, C, H) → α
  328. Rotate(H, 252°, C) → J
  329. Angle(J, C, H) → β
  330. Rotate(H, 36°, C) → N
  331. Rotate(H, 108°, C) → K
  332. Angle(H, C, K) → γ
  333. Angle(H, C, N) → δ
  334. Rotate(H, 216°, C) → O
  335. Angle(O, C, H) → ε
  336. Rotate(H, 132°, C) → P
  337. Angle(P, C, H) → ζ
  338. Rotate(H, 77.99999999999999°, C) → Q
  339. Angle(Q, C, H) → η
  340. Rotate(H, 282°, C) → R
  341. Angle(R, C, H) → ι
  342. OrthogonalLine(C, yAxis) → h
  343. CircleArc(C, N, I) → f
  344. CircleArc(C, I, N) → p
  345. Midpoint(V2', V1') → Ma1
  346. Midpoint(V3', V7') → Ma10
  347. Midpoint(V6', V5') → Ma11
  348. Midpoint(V7', V8') → Ma12
  349. Midpoint(V3', V4') → Ma3
  350. Angle(V5', CuboM1, V7') → angCuboM1
  351. Angle(V1', CuboM2, V2') → angCuboM2
  352. Angle(V6', CuboM3, V5') → angCuboM3
  353. Angle(V7', CuboM4, V8') → angCuboM4
  354. Angle(V5', CuboM5, V6') → angCuboM5
  355. Angle(V6', CuboM6, V1') → angCuboM6
  356. Angle(octV1', octV4', octV5') → angOcta1
  357. Angle(octV2', octV1', octV5') → angOcta2
  358. Angle(octV3', octV2', octV5') → angOcta3
  359. Angle(octV4', octV3', octV5') → angOcta4
  360. Angle(octV1', octV6', octV4') → angOcta5
  361. Angle(octV2', octV6', octV1') → angOcta6
  362. Angle(octV3', octV6', octV2') → angOcta7
  363. Angle(octV4', octV6', octV3') → angOcta8
  364. Segment(T01a, T06b) → A1013b
  365. Segment(T08a, T01b) → A0913b
  366. Segment(T08c, T07c) → A0911b
  367. Segment(T02c, T01c) → A0910b
  368. Segment(T07a, T02a) → A0914b
  369. Segment(T03c, T06c) → A1012b
  370. Segment(T02b, T03b) → A1014b
  371. Segment(T04c, T05c) → A1112b
  372. Segment(T05b, T08b) → A1113b
  373. Segment(T07b, T04b) → A1114b
  374. Segment(T06a, T05a) → A1213b
  375. Segment(T03a, T04a) → A1214b

AI Drawing Prompt Reference

Draw a diagram illustrating the duality between a cube and an octahedron. Start by creating two perpendicular circles, find points on each circle, then use these points to construct the vertices of the cube and connect them to form the cube's edges. Next, determine the midpoints of the cube's edges and vertices, and use these midpoints to construct the vertices of the octahedron, connecting them to form the octahedron's edges, showcasing their dual relationship.
The prompt above was compiled from the original drawing commands and is for reference only.