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Icosahedron-Dodecahedron Duality Demonstration

Icosahedron-Dodecahedron Duality Demonstration
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2023-03-14 00:38
PolygonVectorSolid GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example illustrates the dual relationship between an icosahedron and a dodecahedron, where each vertex corresponds to a face and each face contains a vertex. Key mathematical concepts include polygons, solid geometry, geometric construction, and symmetry. The construction utilizes GeoGebra commands such as OrthogonalLine (orthogonal line), Circle (circle), Intersect (intersection), Point (point), Angle (angle), If (conditional), Ellipse (ellipse), punto3d (3D point), Baricentro (centroid), Vector (vector), Translate (translation), Line (line), Segment (segment), Polygon (polygon), Midpoint (midpoint), Baricentro2 (centroid 2), Rotate (rotation), CircleArc (arc); objects include O_0 (origin), Central (center), e (unit vector), d (distance), H (height), D1 (diameter), ang (angle), t (parameter), t5, t1, t2, t4, t3, t6, ver1, ver2, ver3, ver4, ver5, carasIco (icosahedron faces), verEjes (vertices), carasDod (dodecahedron faces), verticesDod (dodecahedron vertices), verticesIco (icosahedron vertices), aristasDod (dodecahedron edges), aristasIco (icosahedron edges), fi (golden ratio), L (segment), M (point), O_r (rotation center), mira2, B, mira1, A, U, I12, I10, I08, I04, D09. Suitable for high school students and math enthusiasts studying polyhedron properties and dual principles.

GeoGebra Commands & Parameters

  1. OrthogonalLine(Central, xAxis) → e
  2. Circle(Central, 10.5) → d
  3. Intersect(d, e, 2) → H
  4. Point(d) → D1
  5. Angle(D1, Central, H) → ang
  6. If(t < 1, (abs(sgn(t)) * (sgn(t) - 1) / -2), Min[t - 1, 1]) → t2
  7. If(t < 3, (abs(sgn(t)) * (sgn(t) - 1) / -2), Min[t - 3, 1]) → t4
  8. If(t < 2, 0, Min[t - 2, 1]) → t3
  9. If(t5 < 4, 0, t5 - 4) → t6
  10. If(t1 > 0 ∧ t2 ≟ 0, true, false) → ver1
  11. If(t > 1 ∧ t ≤ 2, true, false) → ver2
  12. If(t3 > 0 ∧ t3 ≤ 1 ∧ t4 ≟ 0, true, false) → ver3
  13. If(t4 > 0 ∧ t6 ≟ 0, true, false) → ver4
  14. If(t6 > 0, true, false) → ver5
  15. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  16. Point(mira2) → B
  17. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  18. Point(mira1) → A
  19. punto3d(A, B, M, O_0, O_r, (-fi), 0, -1, U, L) → I12
  20. punto3d(A, B, M, O_0, O_r, fi, 0, -1, U, L) → I10
  21. punto3d(A, B, M, O_0, O_r, -1, (-fi), 0, U, L) → I08
  22. punto3d(A, B, M, O_0, O_r, 0, -1, (-fi), U, L) → I04
  23. Baricentro(I12, I04, I08) → D09
  24. punto3d(A, B, M, O_0, O_r, 0, 1, (-fi), U, L) → I02
  25. Baricentro(I04, I02, I10) → D16
  26. Baricentro(I02, I12, I04) → D17
  27. punto3d(A, B, M, O_0, O_r, -1, fi, 0, U, L) → I07
  28. Baricentro(I02, I07, I12) → D18
  29. punto3d(A, B, M, O_0, O_r, 0, 1, fi, U, L) → I01
  30. punto3d(A, B, M, O_0, O_r, 1, fi, 0, U, L) → I05
  31. Baricentro(I01, I07, I05) → D13
  32. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  33. Angle(M, O_r, B) → θ
  34. Angle(U, O_r, A) → φ
  35. Vector(O_r, O_0) → u
  36. Translate(Xr, u) → X0
  37. Translate(Zr, u) → Z0
  38. Translate(Yr, u) → Y0
  39. Vector(O_0, Z0) → v
  40. Vector(O_0, Y0) → w
  41. Vector(O_0, X0) → z
  42. Circle(O_0, L) → c
  43. Line(O_0, Y0) → a
  44. Line(O_0, yAxis) → b
  45. Line(O_0, X0) → g
  46. Line(Z0, O_0) → j
  47. Translate(P1r, u) → P1
  48. Segment(Mira3, Mira2) → k
  49. Segment((-1, 0), U) → l
  50. Segment((0, -1), M) → m
  51. punto3d(A, B, M, O_0, O_r, 0, -1, fi, U, L) → I03
  52. punto3d(A, B, M, O_0, O_r, 1, (-fi), 0, U, L) → I06
  53. punto3d(A, B, M, O_0, O_r, fi, 0, 1, U, L) → I09
  54. punto3d(A, B, M, O_0, O_r, (-fi), 0, 1, U, L) → I11
  55. Baricentro(I07, I05, I02) → D19
  56. Baricentro(I10, I05, I02) → D20
  57. Baricentro(I01, I11, I07) → D12
  58. Baricentro(I07, I11, I12) → D11
  59. Baricentro(I03, I01, I11) → D04
  60. Baricentro(I11, I12, I08) → D10
  61. Baricentro(I03, I11, I08) → D03
  62. Baricentro(I06, I08, I04) → D08
  63. Baricentro(I06, I08, I03) → D02
  64. Baricentro(I04, I06, I10) → D07
  65. Angle(D19a, M02, D13a) → angM02
  66. Angle(D15b, M01, D14c) → angM01
  67. Angle(D08b, M07, D02b) → angM07
  68. Angle(D20a, M12, D15c) → angM12
  69. Angle(D10c, M06, D03b) → angM06
  70. Angle(D01a, M08, D05b) → angM08
  71. Angle(D16b, M11, D07a) → angM11
  72. Angle(D12b, M04, D04a) → angM04
  73. Angle(D11b, M03, D12a) → angM03
  74. Angle(D06a, M09, D01b) → angM09
  75. Angle(D17a, M10, D18a) → angM10
  76. Angle(D09a, M05, D10a) → angM05
  77. Angle(D14a, D14, D05a) → angD14
  78. Angle(D13c, D13, D12a) → angD13
  79. Angle(D05a, D05, D04a) → angD05
  80. Angle(D01c, D01, D05c) → angD01
  81. Polygon(D06b, D15b, D14c, D05c, D01c) → p01, p01h15, p01h14, p01h05, p01h01, p01h06
  82. Polygon(D14b, D13a, D19a, D20b, D15a) → p02, p02h14, p02h13, p02h19, p02h20, p02h15
  83. Polygon(D11b, D18b, D19c, D13c, D12a) → p03, p03h11, p03h18, p03h19, p03h13, p03h12
  84. Polygon(D04a, D05a, D14a, D13b, D12b) → p04, p04h05, p04h14, p04h13, p04h12, p04h04
  85. Polygon(D18c, D11a, D10a, D09a, D17b) → p05, p05h18, p05h11, p05h10, p05h09, p05h17
  86. Polygon(D04b, D03b, D10c, D11c, D12c) → p06, p06h04, p06h03, p06h10, p06h11, p06h12
  87. Polygon(D08b, D02b, D03c, D10b, D09c) → p07, p07h02, p07h03, p07h10, p07h09, p07h08
  88. Polygon(D02a, D03a, D04c, D05b, D01a) → p08, p08h02, p08h03, p08h04, p08h05, p08h01
  89. Polygon(D08c, D02c, D01b, D06a, D07b) → p09, p09h08, p09h02, p09h01, p09h06, p09h07
  90. Polygon(D17a, D16a, D20c, D19b, D18a) → p10, p10h17, p10h16, p10h20, p10h19, p10h18
  91. Polygon(D07a, D08a, D09b, D17c, D16b) → p11, p11h08, p11h09, p11h17, p11h16, p11h07
  92. Polygon(D07c, D16c, D20a, D15c, D06c) → p12, p12h16, p12h20, p12h15, p12h06, p12h07
  93. Polygon(D14c, D05a, D14a, D13a, D14b, D15b) → h14, h05h14, p04h14', h13h14, p02h14', h14h15, p01h14'
  94. Polygon(D01a, D05b, D04a, D05a, D14c, D05c) → h05, p08h05', h04h05, p04h05', h05h14', p01h05', h01h05
  95. Polygon(D01a, D02a, D01b, D06a, D01c, D05c) → h01, p08h01', h01h02, p09h01', h01h06, p01h01', h01h05'
  96. Polygon(D01c, D06a, D07b, D06c, D15c, D06b) → h06, h01h06', p09h06', h06h07, p12h06', h06h15, p01h06'
  97. Polygon(D06b, D15b, D14b, D15a, D20a, D15c) → h15, p01h15', h14h15', p02h15', h15h20, p12h15', h06h15'
  98. Polygon(D14a, D13b, D12a, D13c, D19a, D13a) → h13, p04h13', h12h13, p03h13', h13h19, p02h13', h13h14'
  99. Polygon(D20b, D19a, D13c, D19c, D18a, D19b) → h19, p02h19', h13h19', p03h19', h18h19, p10h19', h19h20
  100. Polygon(D20a, D16c, D20c, D19b, D20b, D15a) → h20, p12h20', h16h20, p10h20', h19h20', p02h20', h15h20'
  101. Angle(D06c, D06, D15c) → angD06
  102. Angle(D15a, D15, D14b) → angD15
  103. Angle(D20c, D20, D19b) → angD20
  104. Angle(D19c, D19, D13c) → angD19
  105. Angle(D10b, D10, D03c) → angD10
  106. Angle(D03c, D03, D02b) → angD03
  107. Angle(D09b, D09, D08a) → angD09
  108. Angle(D11a, D11, D10a) → angD11
  109. Angle(D12c, D12, D04b) → angD12
  110. Angle(D18c, D18, D11a) → angD18
  111. Angle(D04b, D04, D03b) → angD04
  112. Angle(D07b, D07, D08c) → angD07
  113. Angle(D08b, D08, D09c) → angD08
  114. Angle(D16b, D16, D17c) → angD16
  115. Angle(D17c, D17, D09b) → angD17
  116. Angle(D02c, D02, D01b) → angD02
  117. Polygon(D11c, D10c, D03c, D10b, D09a, D10a) → h10, p06h10', h03h10, p07h10', h09h10, p05h10', h10h11
  118. Polygon(D10c, D03b, D04c, D03a, D02b, D03c) → h03, p06h03', h03h04, p08h03', h02h03, p07h03', h03h10'
  119. Polygon(D17b, D09a, D10b, D09c, D08a, D09b) → h09, p05h09', h09h10', p07h09', h08h09, p11h09', h09h17
  120. Polygon(D12b, D13b, D12a, D11b, D12c, D04b) → h12, p04h12', h12h13', p03h12', h11h12, p06h12', h04h12
  121. Polygon(D11c, D12c, D11b, D18b, D11a, D10a) → h11, p06h11', h11h12', p03h11', h11h18, p05h11', h10h11'
  122. Polygon(D18b, D19c, D18a, D17a, D18c, D11a) → h18, p03h18', h18h19', p10h18', h17h18, p05h18', h11h18'
  123. Polygon(D05b, D04c, D03b, D04b, D12b, D04a) → h04, p08h04', h03h04', p06h04', h04h12', p04h04', h04h05'
  124. Polygon(D07c, D06c, D07b, D08c, D07a, D16b) → h07, p12h07', h06h07', p09h07', h07h08, p11h07', h07h16
  125. Polygon(D07a, D08a, D09c, D08b, D02c, D08c) → h08, p11h08', h08h09', p07h08', h02h08, p09h08', h07h08'
  126. Polygon(D16a, D20c, D16c, D07c, D16b, D17c) → h16, p10h16', h16h20', p12h16', h07h16', p11h16', h16h17
  127. Polygon(D16a, D17a, D18c, D17b, D09b, D17c) → h17, p10h17', h17h18', p05h17', h09h17', p11h17', h16h17'
  128. Polygon(D03a, D02a, D01b, D02c, D08b, D02b) → h02, p08h02', h01h02', p09h02', h02h08', p07h02', h02h03'
  129. Polygon(E05b, E05a, E05c) → t05, e05c, e05b, e05a
  130. Polygon(E03a, E03c, E03b) → t03, e03b, e03a, e03c
  131. Polygon(E13a, E13c, E13b) → t13, e13b, e13a, e13c
  132. Polygon(E19a, E19b, E19c) → t19, e19c, e19a, e19b
  133. Polygon(E01c, E01b, E01a) → t01, e01a, e01c, e01b
  134. Polygon(E15c, E15a, E15b) → t15, e15b, e15c, e15a
  135. Polygon(E07c, E07a, E07b) → t07, e07b, e07c, e07a
  136. Polygon(E14c, E14b, E14a) → t14, e14a, e14c, e14b
  137. Polygon(E02b, E02c, E02a) → t02, e02a, e02b, e02c
  138. Polygon(E11b, E11c, E11a) → t11, e11a, e11b, e11c
  139. Polygon(E10c, E10a, E10b) → t10, e10b, e10c, e10a
  140. Polygon(E12a, E12b, E12c) → t12, e12c, e12a, e12b
  141. Polygon(E16a, E16c, E16b) → t16, e16b, e16a, e16c
  142. Polygon(E20b, E20c, E20a) → t20, e20a, e20b, e20c
  143. Polygon(E06a, E06b, E06c) → t06, e06c, e06a, e06b
  144. Polygon(E18b, E18a, E18c) → t18, e18c, e18b, e18a
  145. Polygon(E08b, E08a, E08c) → t08, e08c, e08b, e08a
  146. Polygon(E17c, E17b, E17a) → t17, e17a, e17c, e17b
  147. Polygon(E09b, E09a, E09c) → t09, e09c, e09b, e09a
  148. Polygon(E04a, E04b, E04c) → t04, e04c, e04a, e04b
  149. Polygon(E15a, E14c, E14a, E13a, E13c, E19b, E19a, E20b, E20a, E15b) → d01, d0103, d01t15, d01t14, d0106, d01t13, d0102, d01t19, d0107, d01t20, d0111
  150. Polygon(E13c, E13b, E12a, E12c, E11c, E11b, E18c, E18b, E19c, E19b) → d02, d02t13, d0102', d0206, d02t12, d0209, d02t11, d0208, d02t18, d0207, d02t19
  151. Polygon(E14c, E15a, E15c, E06a, E06c, E01a, E01b, E05b, E05a, E14b) → d03, d0103', d03t14, d03t15, d0311, d03t06, d0304, d03t01, d0310, d03t05, d0306
  152. Polygon(E06c, E01a, E01c, E02c, E02a, E08c, E08b, E07c, E07a, E06b) → d04, d0304', d04t06, d04t01, d0410, d04t02, d0405, d04t08, d0412, d04t07, d0411
  153. Polygon(E08c, E02a, E02b, E03a, E03b, E10a, E10b, E09c, E09b, E08a) → d05, d0405', d05t08, d05t02, d0510, d05t03, d0509, d05t10, d0508, d05t09, d0512
  154. Polygon(E05c, E04b, E04c, E12b, E12a, E13b, E13a, E14a, E14b, E05a) → d06, d0610, d06t05, d06t04, d0609, d06t12, d0206', d06t13, d0106', d06t14, d0306'
  155. Polygon(E20b, E19a, E19c, E18b, E18a, E17b, E17a, E16c, E16a, E20c) → d07, d0107', d07t20, d07t19, d0207', d07t18, d0708, d07t17, d0712, d07t16, d0711
  156. Polygon(E11a, E10c, E10b, E09c, E09a, E17c, E17b, E18a, E18c, E11b) → d08, d0809, d08t11, d08t10, d0508', d08t09, d0812, d08t17, d0708', d08t18, d0208'
  157. Polygon(E03c, E04a, E04c, E12b, E12c, E11c, E11a, E10c, E10a, E03b) → d09, d0910, d09t03, d09t04, d0609', d09t12, d0209', d09t11, d0809', d09t10, d0509'
  158. Polygon(E01c, E01b, E05b, E05c, E04b, E04a, E03c, E03a, E02b, E02c) → d10, d10t01, d0410', d0310', d10t05, d0610', d10t04, d0910', d10t03, d0510', d10t02
  159. Polygon(E20a, E15b, E15c, E06a, E06b, E07a, E07b, E16b, E16a, E20c) → d11, d0111', d11t20, d11t15, d0311', d11t06, d0411', d11t07, d1112, d11t16, d0711'
  160. Polygon(E07c, E08b, E08a, E09b, E09a, E17c, E17a, E16c, E16b, E07b) → d12, d0412', d12t07, d12t08, d0512', d12t09, d0812', d12t17, d0712', d12t16, d1112'
  161. Angle(E19b, F01, E13c) → angF01
  162. Angle(E18b, F07, E19c) → angF07
  163. Angle(E18c, F02, E11b) → angF02
  164. Angle(E04c, F09, E12b) → angF09
  165. Angle(E12b, F06, E04c) → angF06
  166. Angle(E15a, F03, E14c) → angF03
  167. Angle(E20a, F11, E15b) → angF11
  168. Angle(E02c, F10, E01c) → angF10
  169. Angle(E08c, F05, E02a) → angF05
  170. Angle(E09c, F08, E10b) → angF08
  171. Angle(E07a, F04, E06b) → angF04
  172. Angle(E16b, F12, E07b) → angF12
  173. Angle(E14c, D14, E14a) → angT14
  174. Angle(E13c, D13, E13b) → angT13
  175. Angle(E15b, D15, E15a) → angT15
  176. Angle(E19c, D19, E19b) → angT19
  177. Angle(E20c, D20, E20b) → angT20
  178. Angle(E01c, D01, E01a) → angT01
  179. Angle(E02a, D02, E02c) → angT02
  180. Angle(E08b, D08, E08c) → angT08
  181. Angle(E07b, D07, E07a) → angT07
  182. Angle(E06b, D06, E06a) → angT06
  183. Angle(E04a, D04, E04b) → angT04
  184. Angle(E12c, D12, E12b) → angT12
  185. Angle(E16b, D16, E16c) → angT16
  186. Angle(E17a, D17, E17c) → angT17
  187. Angle(E09c, D09, E09a) → angT09
  188. Angle(E18a, D18, E18c) → angT18
  189. Angle(E11b, D11, E11a) → angT11
  190. Angle(E10c, D10, E10b) → angT10
  191. Angle(E03b, D03, E03a) → angT03
  192. Angle(E05c, D05, E05b) → angT05
  193. Baricentro(G16a, G16c, G16b) → TR16
  194. Angle(G16a, TR16, G16b) → angTR16
  195. Polygon(G16a, G16c, G16b) → tr16, g16b, g16a, g16c
  196. Polygon(G17b, G17c, G17a) → tr17, g17b, g17c, g17a
  197. Polygon(G09a, G09c, G09b) → tr09, g09b, g09a, g09c
  198. Baricentro(G09a, G09c, G09b) → TR09
  199. Angle(G09a, TR09, G09b) → angTR09
  200. Baricentro(G17a, G17b, G17c) → TR17
  201. Angle(G17a, TR17, G17c) → angTR17
  202. Baricentro(G15a, G15b, G15c) → TR15
  203. Angle(G15a, TR15, G15c) → angTR15
  204. Baricentro(G14c, G14a, G14b) → TR14
  205. Angle(G14c, TR14, G14b) → angTR14
  206. Baricentro(G13b, G13a, G13c) → C
  207. Baricentro(G13b, G13a, G13c) → D
  208. Angle(G13b, D, G13c) → angTR13
  209. Polygon(G15a, G15b, G15c) → tr15, g15c, g15a, g15b
  210. Polygon(G14c, G14a, G14b) → tr14, g14b, g14c, g14a
  211. Polygon(G13c, G13b, G13a) → tr13, g13a, g13c, g13b
  212. Polygon(G01a, G01b, G01c) → tr01, g01c, g01a, g01b
  213. Polygon(G02a, G02b, G02c) → tr02, g02c, g02a, g02b
  214. Polygon(G03a, G03b, G03c) → tr03, g03c, g03a, g03b
  215. Polygon(G04a, G04b, G04c) → tr04, g04c, g04a, g04b
  216. Polygon(G05a, G05b, G05c) → tr05, g05c, g05a, g05b
  217. Polygon(G06a, G06b, G06c) → tr06, g06c, g06a, g06b
  218. Polygon(G07a, G07b, G07c) → tr07, g07c, g07a, g07b
  219. Polygon(G08a, G08b, G08c) → tr08, g08c, g08a, g08b
  220. Polygon(G10a, G10b, G10c) → tr10, g10c, g10a, g10b
  221. Polygon(G11a, G11b, G11c) → tr11, g11c, g11a, g11b
  222. Polygon(G12a, G12b, G12c) → tr12, g12c, g12a, g12b
  223. Polygon(G18a, G18b, G18c) → tr18, g18c, g18a, g18b
  224. Polygon(G19a, G19b, G19c) → tr19, g19c, g19a, g19b
  225. Polygon(G20a, G20b, G20c) → tr20, g20c, g20a, g20b
  226. Baricentro(G01a, G01b, G01c) → TR01
  227. Baricentro(G02a, G02b, G02c) → TR02
  228. Baricentro(G03a, G03b, G03c) → TR03
  229. Baricentro(G04a, G04b, G04c) → TR04
  230. Baricentro(G05a, G05b, G05c) → TR05
  231. Baricentro(G06a, G06b, G06c) → TR06
  232. Baricentro(G07a, G07b, G07c) → TR07
  233. Baricentro(G08a, G08b, G08c) → TR08
  234. Baricentro(G10a, G10b, G10c) → TR10
  235. Baricentro(G11a, G11b, G11c) → TR11
  236. Baricentro(G12a, G12b, G12c) → TR12
  237. Baricentro(G13a, G13b, G13c) → TR13
  238. Baricentro(G18a, G18b, G18c) → TR18
  239. Baricentro(G19a, G19b, G19c) → TR19
  240. Baricentro(G20a, G20b, G20c) → TR20
  241. Angle(G06b, TR06, G06c) → angTR06
  242. Angle(G01a, TR01, G01c) → angTR01
  243. Angle(G05a, TR05, G05c) → angTR05
  244. Angle(G11a, TR11, G11b) → angTR11
  245. Angle(G18a, TR18, G18c) → angTR18
  246. Angle(G12a, TR12, G12c) → angTR12
  247. Angle(G07a, TR07, G07b) → angTR07
  248. Angle(G08b, TR08, G08a) → angTR08
  249. Angle(G02a, TR02, G02c) → angTR02
  250. Angle(G10a, TR10, G10c) → angTR10
  251. Angle(G03a, TR03, G03c) → angTR03
  252. Angle(G19b, TR19, G19a) → angTR19
  253. Angle(G20c, TR20, G20b) → angTR20
  254. Angle(G04c, TR04, G04a) → angTR04
  255. Angle(G19c, MD01, G19a) → angMD01
  256. Angle(G13b, MD06, G13a) → angMD06
  257. Angle(G11c, MD02, G11b) → angMD02
  258. Angle(G17b, MD07, G17c) → angMD07
  259. Angle(G07a, MD11, G07c) → angMD11
  260. Angle(G01b, MD03, G01c) → angMD03
  261. Angle(G09a, MD12, G09c) → angMD12
  262. Angle(G09c, MD08, G09b) → angMD08
  263. Angle(G12a, MD09, G12b) → angMD09
  264. Angle(G05a, MD10, G05b) → angMD10
  265. Angle(G06b, MD04, G06a) → angMD04
  266. Angle(G10a, MD05, G10b) → angMD05
  267. Polygon(G16b, G17b, G17a, G09c, G09a, G08c, G08b, G07b, G07c, G16a) → dec12, md0712, g16a_1, g17b_1, md0812, g09c_1, md0512, g08c_1, md0412, g07b_1, md1112
  268. Polygon(G17c, G18b, G18a, G11c, G11a, G10c, G10b, G09b, G09c, G17a) → dec08, md0708, g17a_2, g18b_1, md0208, g11c_1, md0809, g10c_1, md0508, g09b_1, g09c_2
  269. Polygon(G08c, G09a, G09b, G10b, G10a, G03b, G03a, G02c, G02b, G08a) → dec05, g08c_2, g08a_1, g09a_2, g09b_2, g10b_2, md0509, g03b_1, md0510, g02c_1, md0405
  270. Polygon(G06b, G07a, G07b, G08b, G08a, G02b, G02a, G01a, G01c, G06a) → dec04, md0411, g06a_1, g07a_1, g07b_2, g08b_2, g08a_2, g02b_2, md0410, g01a_1, md0304
  271. Polygon(G05a, G01b, G01a, G02a, G02c, G03a, G03c, G04a, G04b, G05b) → dec10, md0310, g05b_1, g01b_1, g01a_2, g02a_2, g02c_2, g03a_2, md0910, g04a_1, md0610
  272. Polygon(G12b, G11b, G11a, G10c, G10a, G03b, G03c, G04a, G04c, G12a) → dec09, md0209, g12a_1, g11b_1, g11a_2, g10c_2, g10a_2, g03b_2, g03c_2, g04a_2, md0609
  273. Polygon(G05b, G04b, G04c, G12a, G12c, G13a, G13b, G14b, G14a, G05c) → dec06, g05b_2, g05c_1, g04b_2, g04c_2, g12a_2, md0206, g13a_1, md0106, g14b_1, md0306
  274. Polygon(G19a, G13c, G13b, G14b, G14c, G15b, G15c, G20b, G20c, G19c) → dec01, md0102, g19c_1, g13c_1, g13b_2, g14b_2, md0103, g15b_2, md0111, g20b_1, md0107
  275. Polygon(G16a, G07c, G07a, G06b, G06c, G15a, G15c, G20b, G20a, G16c) → dec11, g16a_2, g16c_1, g07c_2, g07a_2, g06b_2, md0311, g15a_2, g15c_2, g20b_2, md0711
  276. Polygon(G05a, G05c, G14a, G14c, G15b, G15a, G06c, G06a, G01c, G01b) → dec03, g05a_2, g01b_2, g05c_2, g14a_2, g14c_1, g15b_1, g15a_1, g06c_1, g06a_2, g01c_2
  277. Polygon(G19b, G19a, G13c, G13a, G12c, G12b, G11b, G11c, G18a, G18c) → dec02, g19b_1, md0207, g19a_2, g13c_2, g13a_2, g12c_2, g12b_2, g11b_2, g11c_2, g18a_2
  278. Polygon(G17b, G16b, G16c, G20a, G20c, G19c, G19b, G18c, G18b, G17c) → dec07, g17b_2, g17c_2, g16b_2, g16c_2, g20a_2, g20c_2, g19c_2, g19b_2, g18c_2, g18b_2
  279. Midpoint(D14, D05) → H01
  280. Midpoint(D14, D13) → H02
  281. Midpoint(D14, D15) → H03
  282. Midpoint(D12, D13) → H04
  283. Midpoint(D19, D13) → H05
  284. Midpoint(D19, D18) → H06
  285. Midpoint(D19, D20) → H07
  286. Midpoint(D15, D20) → H08
  287. Midpoint(D16, D20) → H09
  288. Midpoint(D06, D15) → H10
  289. Midpoint(D11, D18) → H11
  290. Midpoint(D17, D18) → H12
  291. Midpoint(D11, D12) → H13
  292. Midpoint(D04, D12) → H14
  293. Midpoint(D06, D07) → H15
  294. Midpoint(D06, D01) → H16
  295. Midpoint(D05, D01) → H17
  296. Midpoint(D05, D04) → H18
  297. Midpoint(D03, D04) → H19
  298. Midpoint(D01, D02) → H20
  299. Midpoint(D07, D16) → H21
  300. Midpoint(D17, D16) → H22
  301. Midpoint(D17, D09) → H23
  302. Midpoint(D11, D10) → H24
  303. Midpoint(D07, D08) → H25
  304. Midpoint(D02, D03) → H26
  305. Midpoint(D02, D08) → H27
  306. Midpoint(D03, D10) → H28
  307. Midpoint(D09, D10) → H29
  308. Midpoint(D09, D08) → H30
  309. Angle(J12d, F12, J12e) → angP12
  310. Angle(J08b, F08, J08c) → angP08
  311. Angle(J05c, F05, J05d) → angP05
  312. Angle(J09a, F09, J09b) → angP09
  313. Angle(J10a, F10, J10b) → angP10
  314. Angle(J04c, F04, J04d) → angP04
  315. Angle(J07c, F07, J07d) → angP07
  316. Polygon(J12b, J12a, J12e, J12d, J12c) → pen12, j12b, j12a, j12e, j12d, j12c
  317. Polygon(J08b, J08a, J08e, J08d, J08c) → pen08, j08b, j08a, j08e, j08d, j08c
  318. Polygon(J05b, J05a, J05e, J05d, J05c) → pen05, j05b, j05a, j05e, j05d, j05c
  319. Polygon(J09b, J09a, J09e, J09d, J09c) → pen09, j09b, j09a, j09e, j09d, j09c
  320. Polygon(J10a, J10e, J10d, J10c, J10b) → pen10, j10a, j10e, j10d, j10c, j10b
  321. Polygon(J04b, J04a, J04e, J04d, J04c) → pen04, j04b, j04a, j04e, j04d, j04c
  322. Polygon(J12e, J08a, J08b, J05e, J05a, J12d) → poly1, j12e_1, j08a_1, j08b_1, j05e_1, j05a_1, j12d_1
  323. Polygon(J12c, J12d, J05a, J05b, J04e, J04a) → poly2, j12c_1, j12d_2, j05a_2, j05b_1, j04e_1, j04a_1
  324. Polygon(J04e, J05b, J05c, J10d, J10e, J04d) → poly3, j04e_2, j05b_2, j05c_1, j10d_1, j10e_1, j04d_1
  325. Polygon(J09a, J10c, J10d, J05c, J05d, J09e) → poly4, j09a_1, j10c_1, j10d_2, j05c_2, j05d_1, j09e_1
  326. Polygon(J09d, J08c, J08b, J05e, J05d, J09e) → poly5, j09d_1, j08c_1, j08b_2, j05e_2, j05d_2, j09e_2
  327. Polygon(J07c, J07b, J07a, J07e, J07d) → pen07, j07c, j07b, j07a, j07e, j07d
  328. Polygon(J12a, J12e, J08a, J08e, J07d, J07c) → poly6, j12a_1, j12e_2, j08a_2, j08e_1, j07d_1, j07c_1
  329. Polygon(J11b, J11a, J11e, J11d, J11c) → pen11, j11b, j11a, j11e, j11d, j11c
  330. Angle(J11c, F11, J11d) → angP11
  331. Angle(J01c, F01, J01d) → angP01
  332. Angle(J02e, F02, J02a) → angP02
  333. Angle(J03a, F03, J03b) → angP03
  334. Angle(J06b, F06, J06c) → angP06
  335. Polygon(J07b, J07c, J12a, J12b, J11d, J11e) → poly7, j07b_1, j07c_2, j12a_2, j12b_1, j11d_1, j11e_1
  336. Polygon(J01c, J01b, J01a, J01e, J01d) → pen01, j01c, j01b, j01a, j01e, j01d
  337. Polygon(J07a, J07b, J11e, J11a, J01c, J01d) → poly8, j07a_1, j07b_2, j11e_2, j11a_1, j01c_1, j01d_1
  338. Polygon(J02d, J02c, J02b, J02a, J02e) → pen02, j02d, j02c, j02b, j02a, j02e
  339. Polygon(J03b, J03c, J03d, J03e, J03a) → pen03, j03b, j03c, j03d, j03e, j03a
  340. Polygon(J06a, J06e, J06d, J06c, J06b) → pen06, j06a, j06e, j06d, j06c, j06b
  341. Polygon(J01a, J06b, J06a, J03c, J03b, J01b) → poly9, j01a_1, j06b_1, j06a_1, j03c_1, j03b_1, j01b_1
  342. Polygon(J09b, J06d, J06c, J02b, J02a, J09c) → poly10, j09b_1, j06d_1, j06c_1, j02b_1, j02a_1, j09c_1
  343. Polygon(J02e, J02a, J09c, J09d, J08c, J08d) → poly11, j02e_1, j02a_2, j09c_2, j09d_2, j08c_2, j08d_1
  344. Polygon(J11b, J11a, J01c, J01b, J03b, J03a) → poly12, j11b_1, j11a_2, j01c_2, j01b_2, j03b_2, j03a_1
  345. Polygon(J11c, J04b, J04c, J03e, J03a, J11b) → poly13, j11c_1, j04b_1, j04c_1, j03e_1, j03a_2, j11b_2
  346. Polygon(J02c, J02b, J06c, J06b, J01a, J01e) → poly14, j02c_1, j02b_2, j06c_2, j06b_2, j01a_2, j01e_1
  347. Polygon(J07e, J02d, J02c, J01e, J01d, J07a) → poly15, j07e_1, j02d_1, j02c_2, j01e_2, j01d_2, j07a_2
  348. Polygon(J10a, J10e, J04d, J04c, J03e, J03d) → poly16, j10a_1, j10e_2, j04d_2, j04c_2, j03e_2, j03d_1
  349. Polygon(J10b, J06e, J06a, J03c, J03d, J10a) → poly17, j10b_1, j06e_1, j06a_2, j03c_2, j03d_2, j10a_2
  350. Polygon(J04a, J04b, J11c, J11d, J12b, J12c) → poly18, j04a_2, j04b_2, j11c_2, j11d_2, j12b_2, j12c_2
  351. Polygon(J10c, J09a, J09b, J06d, J06e, J10b) → poly19, j10c_2, j09a_2, j09b_2, j06d_2, j06e_2, j10b_2
  352. Polygon(J02d, J07e, J07d, J08e, J08d, J02e) → poly20, j02d_2, j07e_2, j07d_2, j08e_2, j08d_2, j02e_2
  353. Baricentro2(J10b, J06e, J10a, J03d, J03c, J06a) → BA17
  354. Baricentro2(J10a, J03d, J03e, J04c, J04d, J10e) → BA16
  355. Angle(J03c, BA17, J06a) → angBA17
  356. Angle(J04c, BA16, J03e) → angBA16
  357. Baricentro2(J06a, J03c, J03b, J01b, J01a, J06b) → BA09
  358. Angle(J03b, BA09, J01b) → angBA09
  359. Baricentro2(J03b, J01b, J03a, J11b, J11a, J01c) → BA12
  360. Angle(J11a, BA12, J01c) → angBA12
  361. Baricentro2(J11b, J03a, J03e, J04c, J04b, J11c) → BA13
  362. Angle(J04b, BA13, J11c) → angBA13
  363. Baricentro2(J09d, J08c, J09c, J02a, J02e, J08d) → BA11
  364. Angle(J02e, BA11, J08d) → angBA11
  365. Baricentro2(J08d, J02e, J08e, J07d, J07e, J02d) → BA20
  366. Angle(J02d, BA20, J07e) → angBA20
  367. Baricentro2(J02a, J09c, J02b, J06c, J06d, J09b) → BA10
  368. Angle(J02a, BA10, J09c) → angBA10
  369. Baricentro2(J01a, J06b, J06c, J02b, J02c, J01e) → BA14
  370. Angle(J01e, BA14, J02c) → angBA14
  371. Baricentro2(J01e, J02c, J01d, J07a, J02d, J07e) → BA15
  372. Angle(J01d, BA15, J07a) → angBA15
  373. Baricentro2(J11a, J01c, J01d, J07a, J07b, J11e) → BA08
  374. Angle(J11e, BA08, J07b) → angBA08
  375. Baricentro2(J04b, J11c, J04a, J12c, J12b, J11d) → BA18
  376. Angle(J11c, BA18, J04b) → angBA18
  377. Baricentro2(J11e, J07b, J07c, J12a, J12b, J11d) → BA07
  378. Angle(J12a, BA07, J07c) → angBA07
  379. Baricentro2(J06e, J10b, J10c, J09a, J09b, J06d) → BA19
  380. Angle(J10b, BA19, J06e) → angBA19
  381. Baricentro2(J09a, J10c, J10d, J05c, J05d, J09e) → BA04
  382. Angle(J09e, BA04, J05d) → angBA04
  383. Baricentro2(J12a, J07c, J07d, J08e, J08a, J12e) → BA06
  384. Angle(J12e, BA06, J08a) → angBA06
  385. Baricentro2(J10e, J04d, J10d, J05c, J04e, J05b) → BA03
  386. Angle(J04d, BA03, J10e) → angBA03
  387. Baricentro2(J12e, J08a, J08b, J05e, J05a, J12d) → BA01
  388. Angle(J05e, BA01, J08b) → angBA01
  389. Baricentro2(J05a, J12d, J12c, J04a, J04e, J05b) → BA02
  390. Angle(J05a, BA02, J12d) → angBA02
  391. Baricentro2(J05e, J08b, J08c, J09d, J09e, J05d) → BA05
  392. Angle(J05d, BA05, J09e) → angBA05
  393. Polygon(K09, K03, K01) → poly21, k01, k09, k03
  394. Polygon(K09, K05, K01) → poly22, k01_1, k09_1, k05
  395. Polygon(K01, K07, K05) → poly23, k05_1, k01_2, k07
  396. Polygon(K09, K10, K05) → poly24, k05_2, k09_2, k10
  397. Polygon(K10, K02, K05) → poly25, k05_3, k10_1, k02
  398. Polygon(K02, K05, K07) → poly26, k07_1, k02_1, k05_4
  399. Polygon(K09, K06, K10) → poly27, k10_2, k09_3, k06
  400. Polygon(K06, K09, K03) → poly28, k03_1, k06_1, k09_4
  401. Polygon(K03, K01, K11) → poly29, k11, k03_2, k01_3
  402. Polygon(K01, K11, K07) → poly30, k07_2, k01_4, k11_1
  403. Polygon(K06, K08, K03) → poly31, k03_3, k06_2, k08
  404. Polygon(K03, K08, K11) → poly32, k11_2, k03_4, k08_1
  405. Polygon(K06, K08, K04) → poly33, k04, k06_3, k08_2
  406. Polygon(K06, K10, K04) → poly34, k04_1, k06_4, k10_3
  407. Polygon(K10, K04, K02) → poly35, k02_2, k10_4, k04_2
  408. Polygon(K02, K12, K04) → poly36, k04_3, k02_3, k12
  409. Polygon(K12, K07, K02) → poly37, k02_4, k12_1, k07_3
  410. Polygon(K11, K07, K12) → poly38, k12_2, k11_3, k07_4
  411. Polygon(K08, K12, K04) → poly39, k04_4, k08_3, k12_3
  412. Polygon(K08, K11, K12) → poly40, k12_4, k08_4, k11_4
  413. Baricentro(K01, K05, K07) → E
  414. Angle(K07, E, K05) → ang33
  415. Baricentro(K04, K08, K12) → F
  416. Angle(K12, F, K04) → ang39
  417. Baricentro(K06, K03, K08) → G
  418. Angle(K08, G, K06) → ang31
  419. Baricentro(K08, K03, K11) → H_1
  420. Angle(K11, H_1, K08) → ang32
  421. Baricentro(K12, K08, K11) → I
  422. Angle(K08, I, K11) → ang40
  423. Baricentro(K06, K09, K03) → J
  424. Angle(K06, J, K09) → ang28
  425. Baricentro(K06, K09, K10) → K
  426. Angle(K09, K, K06) → ang27
  427. Baricentro(K04, K02, K12) → N
  428. Angle(K02, N, K04) → ang36
  429. Baricentro(K10, K06, K04) → O
  430. Angle(K04, O, K10) → ang34
  431. Baricentro(K10, K02, K04) → P
  432. Angle(K04, P, K02) → ang35
  433. Baricentro(K03, K01, K11) → Q
  434. Baricentro(K01, K11, K07) → R
  435. Baricentro(K01, K09, K05) → S
  436. Baricentro(K09, K10, K05) → T
  437. Baricentro(K01, K03, K09) → V
  438. Baricentro(K01, K05, K07) → W
  439. Baricentro(K07, K12, K02) → Z
  440. Baricentro(K05, K07, K02) → A1
  441. Baricentro(K05, K02, K10) → B1
  442. Angle(K01, V, K03) → ang21
  443. Angle(K01, Q, K11) → ang29
  444. Angle(K11, R, K01) → ang30
  445. Baricentro(K11, K07, K12) → C1
  446. Angle(K07, C1, K12) → ang38
  447. Angle(K12, Z, K07) → ang37
  448. Angle(K05, W, K07) → ang23
  449. Angle(K09, S, K05) → ang22
  450. Angle(K10, T, K05) → ang24
  451. Angle(K10, B1, K02) → ang25
  452. Angle(K02, A1, K07) → ang26
  453. Rotate(H, 324°, O_0) → E1
  454. Rotate(H, 275.99999999999994°, O_0) → F1
  455. Rotate(H, 252°, O_0) → G1
  456. Rotate(H, 218.00000000000003°, O_0) → H1
  457. Rotate(H, 180°, O_0) → I1
  458. Rotate(H, 140°, O_0) → J1
  459. Rotate(H, 108°, O_0) → K1
  460. Rotate(H, 78°, O_0) → L1
  461. Rotate(H, 36°, O_0) → M1
  462. CircleArc(O_0, M1, E1) → f
  463. CircleArc(O_0, E1, M1) → h

AI Drawing Prompt Reference

Create a centered geometric demonstration of an icosahedron–dodecahedron duality in 3D. Start from a central origin with orthogonal reference axes, define a circle with a fixed radius around the center, and find the height intersection point with an orthogonal line; mark the associated angles. Construct two ellipses along perpendicular directions as guide tracks, take their intersection to obtain key base points, then use golden-ratio-based parameters to place several symmetric 3D points from these bases. Compute the centroids of selected triples to serve as dual face centers. Finally, draw a segment connecting two specific intersection points of the ellipse guides and include a circle and coordinate-direction lines through the origin to visualize the symmetry and vector-based placement.
The prompt above was compiled from the original drawing commands and is for reference only.