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The Five Regular Divisions of the Sphere

The Five Regular Divisions of the Sphere
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2016-07-30 03:57
PolygonVectorSolid GeometryTransformationGeometric Construction
In collections Polyeder, whose coordinates are golden numbers
① This example demonstrates how to construct the five regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) as regular divisions on a sphere, showing their symmetric projections and geometric relationships on the spherical surface. ② Core mathematical concepts include regular polyhedra (Platonic solids), spherical geometry, symmetry, circumscribed and inscribed spheres, and vector/polygon constructions in 3D space. ③ Construction highlights: from the summary, it uses GeoGebra commands such as Line, Sphere, Intersect, Cube, Tetrahedron, Midpoint, Polygon, Segment, Circle, CircleArc, and creates multiple cubes (Cubo1 to Cubo8), icosahedra (Ico01 to Ico12), and face objects (e.g., caraABCO), with SelectedIndex and Element for dynamic selection, plus sliders (fi, Escala) to adjust scale and rotation. ④ Suitable for high school to university-level solid geometry teaching, and for mathematical outreach to illustrate the intrinsic connection between regular polyhedra and the sphere, enhancing understanding of 3D symmetry and spatial construction.

GeoGebra Commands & Parameters

  1. Line((0, 0, 0), Ico07_1) → j_4
  2. Line((0, 0, 0), Ico05_1) → m_4
  3. Line((0, 0, 0), Ico03_1) → n_3
  4. Sphere((0, 0, 0), Cubo1) → t
  5. Intersect(j_4, t) → Ico06, Ico07
  6. Intersect(m_4, t) → Ico08, Ico05
  7. Intersect(n_3, t) → Ico02, Ico03
  8. Line((0, 0, 0), Ico09_1) → n_4
  9. Intersect(n_4, t) → Ico12, Ico09
  10. Line((0, 0, 0), Ico01_1) → i_4
  11. Intersect(i_4, t) → Ico04, Ico01
  12. Line((0, 0, 0), Ico10_1) → l_4
  13. Intersect(l_4, t) → Ico11, Ico10
  14. Cube(Cubo1, Cubo2, Cubo3) → a, Cubo4, caraBCGF, caraDEFG, aristaCO, aristaAO, aristaAB, aristaBC, aristaDO, aristaCG, aristaDG, aristaDE, Cubo5, aristaAE, aristaEF, aristaBF, aristaFG, Cubo6, Cubo7, Cubo8, caraABCO, caraCGDO, caraAEDO, caraABFE
  15. Tetrahedron(Cubo5, Cubo7, Cubo2) → b, H, aristaAH, aristaFH, caraADF, caraADH, caraDFH, caraAFH, aristaAD, aristaDF, aristaAF, aristaDH
  16. Midpoint(aristaDF) → I
  17. Midpoint(aristaAF) → Octa1_1
  18. Line((0, 0, 0), Octa1_1) → a_2
  19. Intersect(a_2, t) → Octa2, Octa1
  20. Midpoint(aristaAD) → Octa4_1
  21. Line((0, 0, 0), Octa4_1) → b_3
  22. Intersect(b_3, t) → Octa5, Octa4
  23. Polygon(Octa4_1, Octa1_1, I) → polígono1, i, m, j
  24. Intersect(aristaDF, m) → Octa6_1
  25. Line((0, 0, 0), Octa6_1) → a_3
  26. Intersect(a_3, t) → Octa3, Octa6
  27. Midpoint(aristaDH) → Octa2_1
  28. Midpoint(aristaAH) → Octa3_1
  29. Midpoint(aristaFH) → Octa5_1
  30. Polygon(Octa4, Octa2, Octa6) → polígono2, k, i_1, m_1
  31. Polygon(Octa2, Octa3, Octa4) → polígono3, l, k_1, m_2
  32. Polygon(Octa3, Octa1, Octa4) → polígono4, j_1, l_1, m_3
  33. Polygon(Octa1, Octa5, Octa3) → polígono5, n, j_2, l_2
  34. Polygon(Octa5, Octa1, Octa6) → polígono6, n_1, i_2, j_3
  35. Polygon(Octa2, Octa5, Octa6) → polígono8, c, i_3, k_3
  36. Polygon(Octa3, Octa2, Octa5) → polígono7, k_2, n_2, l_3
  37. Segment(Ico01_1, Ico05_1) → d
  38. Segment(Ico02_1, Ico04_1) → h
  39. Segment(Ico02_1, Ico05_1) → p
  40. Segment(Ico02_1, Ico07_1) → q
  41. Segment(Ico02_1, Ico10_1) → r
  42. Segment(Ico02_1, Ico12_1) → s
  43. Segment(Ico04_1, Ico06_1) → d_1
  44. Segment(Ico04_1, Ico08_1) → e_1
  45. Segment(Ico04_1, Ico10_1) → f_1
  46. Segment(Ico04_1, Ico12_1) → g_1
  47. Segment(Ico05_1, Ico07_1) → h_1
  48. Segment(Ico05_1, Ico09_1) → p_1
  49. Segment(Ico05_1, Ico10_1) → q_1
  50. Segment(Ico06_1, Ico08_1) → r_1
  51. Segment(Ico06_1, Ico09_1) → s_1
  52. Segment(Ico06_1, Ico10_1) → t_1
  53. Segment(Ico07_1, Ico12_1) → b_2
  54. Segment(Ico08_1, Ico12_1) → c_2
  55. Segment(Ico11_1, Ico12_1) → e_2
  56. Polygon(Ico01, Ico03, Ico11) → polígono10, a_1, a_4, b_4
  57. Polygon(Ico03, Ico09, Ico01) → polígono11, i_5, j_5, l_5
  58. Polygon(Ico11, Ico01, Ico07) → polígono9, j_{10}, k_{10}, l_{10}
  59. Polygon(Ico08, Ico06, Ico03) → polígono12, m_5, n_5, a_5
  60. Polygon(Ico08, Ico03, Ico11) → polígono13, b_5, i_6, j_6
  61. Polygon(Ico03, Ico06, Ico09) → polígono14, l_6, m_6, n_6
  62. Polygon(Ico10, Ico09, Ico06) → polígono15, a_6, b_6, i_7
  63. Polygon(Ico06, Ico04, Ico10) → polígono16, j_7, l_7, m_7
  64. Polygon(Ico08, Ico06, Ico04) → polígono17, n_7, a_7, b_7
  65. Polygon(Ico11, Ico08, Ico12) → polígono18, i_8, j_8, l_8
  66. Polygon(Ico12, Ico11, Ico07) → polígono19, m_8, n_8, r_8
  67. Polygon(Ico12, Ico08, Ico04) → polígono20, s_8, t_8, a_8
  68. Polygon(Ico12, Ico02, Ico04) → polígono21, b_8, f_9, g_9
  69. Polygon(Ico02, Ico12, Ico07) → polígono22, h_9, i_9, j_9
  70. Polygon(Ico10, Ico05, Ico09) → polígono23, k_9, l_9, m_9
  71. Polygon(Ico05, Ico02, Ico10) → polígono24, n_9, p_9, q_9
  72. Polygon(Ico04, Ico10, Ico02) → polígono25, r_9, s_9, t_9
  73. Polygon(Ico05, Ico01, Ico09) → polígono26, a_9, b_9, c_9
  74. Polygon(Ico07, Ico02, Ico05) → polígono27, d_9, e_9, f_{10}
  75. Polygon(Ico05, Ico07, Ico01) → polígono28, g_{10}, h_{10}, i_{10}
  76. Polygon(Dod09, Dod11, Cubo8, Dod01, Cubo7) → polígono29, dod09, dod11, cubo8, dod01, cubo7
  77. Polygon(Dod09, Cubo6, Dod02, Cubo5, Dod11) → polígono30, dod09_1, cubo6, dod02, cubo5, dod11_1
  78. Polygon(Dod10, Dod12, Cubo1, Dod04, Cubo2) → polígono31, dod10, dod12, cubo1, dod04, cubo2
  79. Polygon(Dod12, Cubo4, Dod03, Cubo3, Dod10) → polígono32, dod12_1, cubo4, dod03, cubo3, dod10_1
  80. Polygon(Dod02, Cubo6, Dod07, Cubo2, Dod04) → polígono33, dod02_1, cubo6_1, dod07, cubo2_1, dod04_1
  81. Polygon(Dod03, Dod01, Cubo7, Dod05, Cubo3) → polígono34, dod03_1, dod01_1, cubo7_1, dod05, cubo3_1
  82. Polygon(Dod07, Dod05, Cubo7, Dod09, Cubo6) → polígono35, dod07_1, dod05_1, cubo7_2, dod09_2, cubo6_2
  83. Polygon(Dod07, Dod05, Cubo3, Dod10, Cubo2) → polígono36, dod07_2, dod05_2, cubo3_2, dod10_2, cubo2_2
  84. Polygon(Dod08, Dod06, Cubo8, Dod11, Cubo5) → polígono37, dod08, dod06, cubo8_1, dod11_2, cubo5_1
  85. Polygon(Dod08, Cubo1, Dod04, Dod02, Cubo5) → polígono38, dod08_1, cubo1_1, dod04_2, dod02_2, cubo5_2
  86. Polygon(Dod06, Dod08, Cubo1, Dod12, Cubo4) → polígono39, dod06_1, dod08_2, cubo1_2, dod12_2, cubo4_1
  87. Polygon(Dod06, Cubo4, Dod03, Dod01, Cubo8) → polígono40, dod06_2, cubo4_2, dod03_2, dod01_2, cubo8_2
  88. Circle(Octa1, Octa2, Octa6) → g
  89. CircleArc((0, 0, 0), Cubo7, Cubo2) → c_1
  90. CircleArc((0, 0, 0), Cubo7, Cubo4) → d_2
  91. CircleArc((0, 0, 0), Cubo5, Cubo7) → f_2
  92. CircleArc((0, 0, 0), Cubo5, Cubo2) → g_2
  93. CircleArc((0, 0, 0), Cubo5, Cubo4) → h_2
  94. CircleArc((0, 0, 0), Cubo2, Cubo4) → p_2
  95. CircleArc((0, 0, 0), Cubo6, Cubo7) → q_2
  96. CircleArc((0, 0, 0), Cubo8, Cubo7) → r_2
  97. CircleArc((0, 0, 0), Cubo8, Cubo5) → s_2
  98. CircleArc((0, 0, 0), Cubo6, Cubo5) → t_2
  99. CircleArc((0, 0, 0), Cubo7, Cubo3) → c_3
  100. CircleArc((0, 0, 0), Cubo4, Cubo3) → d_3
  101. CircleArc((0, 0, 0), Cubo4, Cubo8) → e_3
  102. CircleArc((0, 0, 0), Cubo4, Cubo1) → f_3
  103. CircleArc((0, 0, 0), Cubo5, Cubo1) → g_3
  104. CircleArc((0, 0, 0), Cubo2, Cubo1) → h_3
  105. CircleArc((0, 0, 0), Cubo2, Cubo6) → p_3
  106. CircleArc((0, 0, 0), Cubo2, Cubo3) → q_3
  107. CircleArc((0, 0, 0), Dod09, Dod11) → r_3
  108. CircleArc((0, 0, 0), Cubo8, Dod11) → s_3
  109. CircleArc((0, 0, 0), Cubo8, Dod01) → t_3
  110. CircleArc((0, 0, 0), Cubo7, Dod01) → c_4
  111. CircleArc((0, 0, 0), Cubo7, Dod09) → d_4
  112. CircleArc((0, 0, 0), Cubo6, Dod09) → e_4
  113. CircleArc((0, 0, 0), Cubo6, Dod02) → f_4
  114. CircleArc((0, 0, 0), Cubo5, Dod02) → g_4
  115. CircleArc((0, 0, 0), Cubo5, Dod11) → h_4
  116. CircleArc((0, 0, 0), Cubo5, Dod08) → k_4
  117. CircleArc((0, 0, 0), Dod06, Dod08) → p_4
  118. CircleArc((0, 0, 0), Dod06, Cubo8) → q_4
  119. CircleArc((0, 0, 0), Dod06, Cubo4) → r_4
  120. CircleArc((0, 0, 0), Dod03, Cubo4) → s_4
  121. CircleArc((0, 0, 0), Dod03, Dod01) → t_4
  122. CircleArc((0, 0, 0), Dod03, Cubo3) → c_5
  123. CircleArc((0, 0, 0), Dod05, Cubo3) → d_5
  124. CircleArc((0, 0, 0), Dod05, Cubo7) → e_5
  125. CircleArc((0, 0, 0), Dod05, Dod07) → f_5
  126. CircleArc((0, 0, 0), Cubo6, Dod07) → g_5
  127. CircleArc((0, 0, 0), Cubo2, Dod07) → h_5
  128. CircleArc((0, 0, 0), Cubo2, Dod04) → k_5
  129. CircleArc((0, 0, 0), Dod02, Dod04) → p_5
  130. CircleArc((0, 0, 0), Cubo1, Dod04) → q_5
  131. CircleArc((0, 0, 0), Cubo1, Dod08) → r_5
  132. CircleArc((0, 0, 0), Cubo1, Dod12) → s_5
  133. CircleArc((0, 0, 0), Cubo4, Dod12) → t_5
  134. CircleArc((0, 0, 0), Dod10, Dod12) → c_6
  135. CircleArc((0, 0, 0), Dod10, Cubo3) → d_6
  136. CircleArc((0, 0, 0), Dod10, Cubo2) → e_6
  137. Circle(Octa1, Octa2, Octa5) → f
  138. Circle(Octa3, Octa4, Octa6) → e
  139. Polygon(Octa4, Octa1, Octa6) → polígono1', octa6, octa4, octa1
  140. CircleArc((0, 0, 0), Ico08, Ico12) → c_7
  141. CircleArc((0, 0, 0), Ico11, Ico01) → c_8
  142. CircleArc((0, 0, 0), Ico04, Ico12) → d_7
  143. CircleArc((0, 0, 0), Ico09, Ico01) → d_8
  144. CircleArc((0, 0, 0), Ico04, Ico02) → e_7
  145. CircleArc((0, 0, 0), Ico03, Ico01) → e_8
  146. CircleArc((0, 0, 0), Ico03, Ico09) → f_6
  147. CircleArc((0, 0, 0), Ico12, Ico02) → f_7
  148. CircleArc((0, 0, 0), Ico05, Ico09) → f_8
  149. CircleArc((0, 0, 0), Ico06, Ico09) → g_6
  150. CircleArc((0, 0, 0), Ico12, Ico11) → g_7
  151. CircleArc((0, 0, 0), Ico10, Ico09) → g_8
  152. CircleArc((0, 0, 0), Ico06, Ico03) → h_6
  153. CircleArc((0, 0, 0), Ico07, Ico11) → h_7
  154. CircleArc((0, 0, 0), Ico10, Ico05) → h_8
  155. CircleArc((0, 0, 0), Ico11, Ico03) → k_6
  156. CircleArc((0, 0, 0), Ico07, Ico12) → k_7
  157. CircleArc((0, 0, 0), Ico10, Ico02) → k_8
  158. CircleArc((0, 0, 0), Ico11, Ico08) → p_6
  159. CircleArc((0, 0, 0), Ico07, Ico02) → p_7
  160. CircleArc((0, 0, 0), Ico10, Ico06) → p_8
  161. CircleArc((0, 0, 0), Ico03, Ico08) → q_6
  162. CircleArc((0, 0, 0), Ico05, Ico02) → q_7
  163. CircleArc((0, 0, 0), Ico10, Ico04) → q_8
  164. CircleArc((0, 0, 0), Ico06, Ico08) → r_6
  165. CircleArc((0, 0, 0), Ico05, Ico07) → r_7
  166. CircleArc((0, 0, 0), Ico06, Ico04) → s_6
  167. CircleArc((0, 0, 0), Ico01, Ico07) → s_7
  168. CircleArc((0, 0, 0), Ico08, Ico04) → t_6
  169. CircleArc((0, 0, 0), Ico01, Ico05) → t_7
  170. Sphere((0, 0, 0), Cubo1) → t'
  171. SelectedIndex(lista1) → op
  172. Element(ángulos, op) → ang

AI Drawing Prompt Reference

Draw a sphere centered at the origin, and show the five regular polyhedra as symmetric projections on its surface. First, from the center, cast multiple rays to intersect the sphere, producing vertices that form an icosahedron. Then, construct a cube inside the sphere)Skip, and generate several rotated copies of this cube to form a family of cubes. Next, use midpoints of certain cube edges to define vertices of an octahedron, and connect these vertices to form its faces. Finally, draw segments between icosahedron vertices to reveal its edges. Use sliders to adjust scale and rotation for dynamic viewing of the polyhedra relative to the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.