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Open Cube to Octahedron

Open Cube to Octahedron
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2020-03-14 17:46
Coordinate GeometrySolid GeometryTransformation
This example demonstrates the process of unfolding a cube into an octahedron through geometric transformations. Key math concepts include solid geometry, transformations, vectors, and coordinate geometry. The construction utilizes GeoGebra commands and objects such as Polygon, Point, Cube, Midpoint, Line, Rotate, Plane, Mirror, Intersect, and PointIn. Suitable for high school and above students studying spatial geometry and 3D shape construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4, xOyPlane) → poly1, f, g, h, i, C, D
  2. Point(Circle[B, Distance[A, B], Segment[A, B]]) → E
  3. Cube(A, B, E) → a, F, faceEFJI, faceGHIJ, edgeAF, edgeAB, edgeBE, edgeEF, edgeAG, edgeFJ, edgeGJ, edgeGH, G, edgeBH, edgeHI, edgeEI, edgeIJ, H, I, J, faceABEF, faceAFJG, faceABHG, faceBEIH
  4. Midpoint(edgeAB) → K
  5. Midpoint(edgeBE) → L
  6. Midpoint(edgeEF) → M
  7. Midpoint(edgeAF) → N
  8. Midpoint(edgeAG) → O
  9. Midpoint(edgeBH) → P
  10. Midpoint(edgeEI) → Q
  11. Midpoint(edgeFJ) → R
  12. Midpoint(edgeGH) → S
  13. Midpoint(edgeHI) → T
  14. Midpoint(edgeIJ) → U
  15. Midpoint(edgeGJ) → V
  16. Midpoint(S, U) → W
  17. Midpoint(K, M) → Z
  18. Midpoint(U, M) → A_1
  19. Midpoint(P, Q) → B_1
  20. Midpoint(O, P) → C_1
  21. Midpoint(O, R) → D_1
  22. Midpoint(W, U) → E_1
  23. Midpoint(W, T) → F_1
  24. Midpoint(U, A_1) → G_1
  25. Midpoint(M, A_1) → H_1
  26. Midpoint(B_1, Q) → I_1
  27. Midpoint(T, B_1) → J_1
  28. Polygon(F_1, E_1, U, I, T) → poly2, f_1, e_1, u, i_1, t
  29. Midpoint(Q, A_1) → K_1
  30. Polygon(U, G_1, K_1, Q, I) → poly3, u_1, g_1, k_1, q, i_2
  31. Polygon(T, J_1, I_1, Q, I) → poly4, t_1, j_1, j, q_1, i_3
  32. Polygon(E_1, G_1, U) → t1, u_2, k, l
  33. Polygon(F_1, J_1, T) → t2, t_2, m, n
  34. Polygon(K_1, I_1, Q) → t3, q_2, p, r
  35. Polygon(E_1, G_1, K_1, I_1, J_1, F_1) → poly5, s, c, d, e, h_1, l_1
  36. Line(Q, P) → m_1
  37. Rotate(poly4, (-α), m_1) → poly4'
  38. Rotate(poly2, (-α), m_1) → poly2'
  39. Rotate(poly3, (-α), m_1) → poly3'
  40. Rotate(t2, (-α), m_1) → t2'
  41. Rotate(poly5, (-α), m_1) → poly5'
  42. Rotate(t1, (-α), m_1) → t1'
  43. Plane(V, D_1, W) → p_1
  44. Plane(R, D_1, A_1) → r_1
  45. Mirror(t2, r_1) → t2'_1
  46. Mirror(t1, r_1) → t1'_1
  47. Mirror(poly2, r_1) → poly2'_1
  48. Mirror(poly3, r_1) → poly3'_1
  49. Mirror(poly4, r_1) → poly4'_1
  50. Mirror(t1', p_1) → t1''
  51. Mirror(t2', p_1) → t2''
  52. Mirror(poly2', p_1) → poly2''
  53. Mirror(poly3', p_1) → poly3''
  54. Mirror(poly4', p_1) → poly4''
  55. Mirror(poly5', p_1) → poly5''
  56. Plane(N, D_1, Z) → p_2
  57. Plane(U, S, K) → r_2
  58. Mirror(poly3'', r_2) → poly3'''
  59. Mirror(poly4'', r_2) → poly4'''
  60. Mirror(poly2'', r_2) → poly2'''
  61. Mirror(t1'', r_2) → t1'''
  62. Rotate(t3, (-α), m_1) → t3'
  63. Mirror(t3', p_1) → t3''
  64. Mirror(poly5'', r_2) → poly5'''
  65. Mirror(t1', r_2) → t1''_1
  66. Mirror(poly3', r_2) → poly3''_1
  67. Mirror(L, r_2) → L'
  68. Mirror(poly5', r_2) → poly5''_1
  69. Mirror(poly2', r_2) → poly2''_1
  70. Mirror(poly4', r_2) → poly4''_1
  71. Midpoint(A_1, R) → L_1
  72. Mirror(t3, r_2) → t3'_1
  73. Mirror(poly3'_1, r_2) → poly6
  74. Midpoint(B_1, L) → M_1
  75. Midpoint(L', Z) → N_1
  76. Midpoint(Z, L) → O_1
  77. Midpoint(M, Z) → P_1
  78. Polygon(K_1, H_1, P_1, O_1, M_1, I_1) → poly7, n_1, s_1, a_1, o_1, b_1, c_1
  79. Mirror(t2'_1, r_2) → t4
  80. Mirror(poly7, r_2) → poly7'
  81. Mirror(poly2'_1, r_2) → poly8
  82. Mirror(t3'_1, p_2) → t5
  83. Mirror(poly7', p_2) → poly7''
  84. Mirror(poly8, p_2) → poly8'
  85. Mirror(poly6, p_2) → poly6'
  86. Mirror(poly4'_1, p_2) → poly9
  87. Mirror(poly3'_1, p_2) → poly10
  88. Mirror(poly2'_1, p_2) → poly11
  89. Mirror(poly7, p_1) → poly7'_1
  90. Line(Z, L') → d_1
  91. Midpoint(K, Z) → Q_1
  92. Polygon(K, Q_1, O_1, L, B) → poly12, k_2, f_2, g_2, l_2, b_2
  93. Polygon(L, O_1, P_1, M, E) → poly13, l_3, h_2, j_2, m_2, e_2
  94. Polygon(D, M, P_1, N_1, L') → poly14, d_2, m_3, n_2, s_2, l'
  95. Polygon(L', N_1, Q_1, K, A) → poly15, a_2, c_2, f_3, k_3, a_3
  96. Mirror(poly4'_1, r_2) → poly16
  97. Mirror(poly9, r_2) → poly9'
  98. Mirror(t3, p_1) → t3'_2
  99. Mirror(p_1, p_2) → p_3
  100. Mirror(t3'', r_2) → t3'''
  101. Mirror(t3', r_2) → t3''_1
  102. Rotate(poly5', β, d_1) → poly5''_2
  103. Rotate(t1', β, d_1) → t1''_2
  104. Rotate(poly3', β, d_1) → poly3''_2
  105. Rotate(t3', β, d_1) → t3''_2
  106. Rotate(poly4', β, d_1) → poly4''_2
  107. Rotate(poly2', β, d_1) → poly2''_2
  108. Rotate(t3, β, d_1) → t3'_3
  109. Rotate(poly3'_1, β, d_1) → poly17
  110. Rotate(poly7, β, d_1) → poly7'_2
  111. Rotate(poly13, β, d_1) → poly13'
  112. Mirror(poly5''_2, p_1) → poly18
  113. Mirror(t1''_2, p_1) → t6
  114. Mirror(poly3''_2, p_1) → poly19
  115. Mirror(poly17, p_1) → poly17'
  116. Mirror(t3''_2, p_1) → t7
  117. Mirror(t3'_3, p_1) → t8
  118. Mirror(poly7'_2, p_1) → poly20
  119. Mirror(poly18, r_2) → poly18'
  120. Mirror(t6, r_2) → t6'
  121. Mirror(t7, r_2) → t7'
  122. Mirror(t8, r_2) → t8'
  123. Mirror(poly19, r_2) → poly19'
  124. Mirror(poly17', r_2) → poly17''
  125. Mirror(poly20, r_2) → poly20'
  126. Mirror(poly3''_2, r_2) → poly21
  127. Mirror(poly17, r_2) → poly17'_1
  128. Mirror(t3''_2, r_2) → t9
  129. Mirror(t3'_3, r_2) → t10
  130. Mirror(poly7'_2, r_2) → poly22
  131. Mirror(poly5''_2, r_2) → poly23
  132. Mirror(t1''_2, r_2) → t11
  133. Mirror(t1'', r_2) → t1'''_1
  134. Mirror(t1', r_2) → t1''_3
  135. Rotate(poly2''', (-β), d_1) → poly24
  136. Rotate(poly15, (-β), d_1) → poly15'
  137. Rotate(poly12, (-β), d_1) → poly12'
  138. Rotate(poly2'', (-β), d_1) → poly2'''_1
  139. Rotate(poly2''_1, β, d_1) → poly25
  140. Rotate(poly14, β, d_1) → poly14'
  141. Rotate(poly4'_1, β, d_1) → poly26
  142. Rotate(poly9, (-β), d_1) → poly9'_1
  143. Rotate(poly4'', (-β), d_1) → poly4'''_1
  144. Rotate(poly16, β, d_1) → poly16'
  145. Rotate(poly4''_1, β, d_1) → poly27
  146. Rotate(poly9', (-β), d_1) → poly9''
  147. Rotate(poly4''', (-β), d_1) → poly28
  148. Point(zAxis) → R_1
  149. Line(R_1, m_1) → g_3
  150. Rotate(poly20, (-γ), g_3) → poly20'_1
  151. Rotate(t8, (-γ), g_3) → t8'_1
  152. Rotate(t7, (-γ), g_3) → t7'_1
  153. Rotate(poly18, (-γ), g_3) → poly18'_1
  154. Rotate(poly17', (-γ), g_3) → poly17''_1
  155. Rotate(poly19, (-γ), g_3) → poly19'_1
  156. Rotate(poly17, (-γ), g_3) → poly17'_2
  157. Rotate(poly3''_2, (-γ), g_3) → poly29
  158. Rotate(t6, (-γ), g_3) → t6'_1
  159. Rotate(t1''_2, (-γ), g_3) → t12
  160. Rotate(t8', γ, g_3) → t8''
  161. Rotate(poly20', γ, g_3) → poly20''
  162. Rotate(t7', γ, g_3) → t7''
  163. Rotate(poly18', γ, g_3) → poly18''
  164. Rotate(poly17'', γ, g_3) → poly17'''
  165. Rotate(poly19', γ, g_3) → poly19''
  166. Rotate(poly17'_1, γ, g_3) → poly30
  167. Rotate(poly21, γ, g_3) → poly21'
  168. Rotate(t6', γ, g_3) → t6''
  169. Rotate(t11, γ, g_3) → t11'
  170. Rotate(poly5''_2, (-γ), g_3) → poly31
  171. Rotate(poly7'_2, (-γ), g_3) → poly32
  172. Rotate(t3''_2, (-γ), g_3) → t13
  173. Rotate(t3'_3, (-γ), g_3) → t14
  174. Rotate(poly22, γ, g_3) → poly22'
  175. Rotate(t10, γ, g_3) → t10'
  176. Rotate(t9, γ, g_3) → t9'
  177. Rotate(poly23, γ, g_3) → poly23'
  178. Midpoint(E, L) → S_1
  179. Midpoint(L', A) → T_1
  180. Midpoint(L, B) → U_1
  181. Midpoint(L', S) → V_1
  182. Midpoint(D, L') → W_1
  183. Mirror(N_1, L') → Z_1
  184. Mirror(O_1, L) → A_2
  185. Polygon(Z_1, T_1, N_1, W_1) → q1, z_1, h_3, j_3, w_1
  186. Polygon(O_1, U_1, A_2, S_1) → q2, n_3, q_3, r_3, s_3
  187. Rotate(q2, (-γ), g_3) → q2'
  188. Rotate(q2, (-γ), g_3) → q2'_1
  189. Rotate(q1, γ, g_3) → q1'_1
  190. Rotate(q2, (-γ), g_3) → q2'_2
  191. Intersect(d_1, poly32) → D_2
  192. Intersect(d_1, poly4'''_1) → E_2
  193. Mirror(D_2, E_2) → F_2
  194. Mirror(N_1, L') → B_2
  195. PointIn(poly31) → C_2
  196. PointIn(poly20'_1) → G_2
  197. Polygon(F_2, D_2, G_2) → t15, t_3, b_3, c_3
  198. Polygon(F_2, C_2, D_2) → t16, d_3, e_3, f_4
  199. Mirror(t15, r_2) → t15'
  200. Mirror(t16, r_2) → t16'
  201. Mirror(q2'_2, r_2) → q3
  202. Intersect(g_3, poly22') → H_2
  203. PointIn(poly32) → I_2
  204. PointIn(poly22') → J_2
  205. PointIn(poly20'') → K_2
  206. Polygon(I_2, K_2, H_2) → t17, g_4, h_4, i_4
  207. Polygon(K_2, J_2, H_2) → t18, j_4, k_4, l_4
  208. Mirror(q2'_2, r_2) → q4
  209. Mirror(q2'_2, r_2) → q5