MathPlotAI Logo MathPlotAI

Octahedron with Cutting Planes

Octahedron with Cutting Planes
Loading the math board and drawing, please wait…
2015-12-21 07:25
PolygonSolid GeometryGeometric Construction
This example demonstrates the intersection of an octahedron with multiple cutting planes, showcasing the resulting geometric shapes and lines. Key math concepts include planes, logic, quadrilaterals, variance, isosceles triangles, and geometric constructions. The construction utilizes GeoGebra commands such as Intersect, Point, Mirror, Circle, AngularBisector, Octahedron, Line, Segment, Polygon, Plane, IntersectPath, Sphere, PointIn, Midpoint, Ray, Rotate, and Star. It is suitable for high school and above, focusing on solid geometry, geometric constructions, and spatial reasoning.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → O
  2. Point(xAxis) → A
  3. Mirror(A, O) → B
  4. Mirror(C, O) → D
  5. Mirror(E, O) → F
  6. Circle(O, x(A), zAxis) → c
  7. Circle(O, x(A), yAxis) → d
  8. Circle(O, x(A), xAxis) → e
  9. AngularBisector(A, O, C) → a
  10. AngularBisector(C, O, E) → b
  11. AngularBisector(E, O, A) → f
  12. Octahedron(A, C, F) → g, G, face7, face8, edge1, edge2, edge3, edge4, edge5, edge6, edge7, edge8, H, edge9, edge10, edge11, edge12, I, face1, face2, face3, face4, face5, face6
  13. Mirror(αAngle, O) → αAngle'
  14. Mirror(βAngle, O) → βAngle'
  15. Mirror(γAngle, O) → γAngle'
  16. Line(γAngle, yAxis) → h
  17. Line(γAngle', yAxis) → i
  18. Segment(γAngle', γAngle) → j
  19. Circle(γAngle, x(A), j) → k
  20. Intersect(h, k) → J, K
  21. Segment(K, J) → l
  22. Circle(γAngle', x(A), j) → p
  23. Intersect(i, p) → L, M
  24. Segment(J, L) → m
  25. Segment(L, M) → n
  26. Segment(M, K) → q
  27. Polygon(M, L, J, K) → poly1, m_1, l_1, j_1, k_1
  28. Segment(βAngle, βAngle') → r
  29. Line(βAngle', xAxis) → s
  30. Line(βAngle, xAxis) → t
  31. Circle(βAngle', x(A), r) → c_1
  32. Circle(βAngle, x(A), r) → d_1
  33. Intersect(s, c_1) → N, P
  34. Intersect(t, d_1) → Q, R
  35. Segment(N, P) → a_1
  36. Segment(P, R) → b_1
  37. Segment(R, Q) → e_1
  38. Segment(Q, N) → f_1
  39. Polygon(Q, R, P, N) → poly2, q_1, r_1, p_1, n_1
  40. Segment(αAngle, αAngle') → g_1
  41. Line(αAngle', zAxis) → h_1
  42. Line(αAngle, zAxis) → i_1
  43. Circle(αAngle', x(A), g_1) → s_1
  44. Circle(αAngle, x(A), g_1) → t_1
  45. Intersect(i_1, t_1) → S, T
  46. Intersect(h_1, s_1) → U, V
  47. Segment(U, S) → a_2
  48. Segment(S, T) → b_2
  49. Segment(T, V) → c_2
  50. Segment(V, U) → d_2
  51. Polygon(S, T, V, U) → poly3, s_2, t_2, v, u_1
  52. Plane(T, V, U) → o_1
  53. IntersectPath(o_1, face1) → f_2
  54. IntersectPath(o_1, face3) → g_2
  55. Intersect(o_1, face3) → W, Z
  56. IntersectPath(o_1, face2) → h_2
  57. Intersect(o_1, face2) → A_1, B_1
  58. IntersectPath(o_1, poly3) → i_2
  59. Intersect(o_1, poly3) → C_1
  60. IntersectPath(o_1, face8) → j_2
  61. Intersect(o_1, face8) → D_1, E_1
  62. IntersectPath(o_1, face7) → k_2
  63. Intersect(o_1, face7) → F_1, G_1
  64. IntersectPath(o_1, face4) → l_2
  65. Intersect(o_1, face4) → H_1, I_1
  66. IntersectPath(o_1, face5) → m_2
  67. Intersect(o_1, face5) → J_1, K_1
  68. IntersectPath(o_1, face6) → n_2
  69. Intersect(o_1, face6) → L_1, M_1
  70. Plane(P, N, Q) → v_1
  71. IntersectPath(v_1, face5) → p_2
  72. Intersect(v_1, face5) → N_1, O_1
  73. IntersectPath(v_1, face4) → q_2
  74. Intersect(v_1, face4) → P_1, Q_1
  75. IntersectPath(v_1, face7) → r_2
  76. Intersect(v_1, face7) → R_1, S_1
  77. IntersectPath(v_1, face2) → a_3
  78. Intersect(v_1, face2) → T_1, U_1
  79. IntersectPath(v_1, face8) → b_3
  80. Intersect(v_1, face8) → V_1, W_1
  81. IntersectPath(v_1, face3) → c_3
  82. Intersect(v_1, face3) → Z_1, A_2
  83. IntersectPath(v_1, face1) → d_3
  84. Intersect(v_1, face1) → B_2, C_2
  85. IntersectPath(v_1, face6) → e_3
  86. Intersect(v_1, face6) → D_2, E_2
  87. Plane(M, L, J) → w_1
  88. IntersectPath(w_1, face4) → f_3
  89. Intersect(w_1, face4) → F_2, G_2
  90. IntersectPath(w_1, face3) → g_3
  91. Intersect(w_1, face3) → H_2, I_2
  92. IntersectPath(w_1, face6) → h_3
  93. Intersect(w_1, face6) → J_2, K_2
  94. IntersectPath(w_1, face7) → i_3
  95. Intersect(w_1, face7) → L_2, M_2
  96. IntersectPath(w_1, face8) → j_3
  97. Intersect(w_1, face8) → N_2, O_2
  98. IntersectPath(w_1, face5) → k_3
  99. Intersect(w_1, face5) → P_2, Q_2
  100. IntersectPath(w_1, face2) → l_3
  101. Intersect(w_1, face2) → R_2, S_2
  102. IntersectPath(w_1, face1) → m_3
  103. Intersect(w_1, face1) → T_2, U_2
  104. Sphere(O, (1.5 * x(A))) → c_1
  105. PointIn(c_1) → V_2
  106. PointIn(c_1) → W_2
  107. PointIn(c_1) → Z_2
  108. Plane(V_2, W_2, Z_2) → e_2
  109. IntersectPath(e_2, c_1) → p_3
  110. IntersectPath(e_2, face8) → n_3
  111. Intersect(e_2, face8) → A_3, B_3
  112. IntersectPath(e_2, g) → poly4, C_3, t_3, a_4, b_4, D_3, E_3, F_3, G_3, H_3, q_3, r_3, s_3
  113. IntersectPath(e_2, face4) → c_4
  114. Intersect(e_2, face4) → I_3, J_3
  115. IntersectPath(e_2, face3) → d_4
  116. Intersect(e_2, face3) → K_3, L_3
  117. IntersectPath(e_2, face5) → e_4
  118. IntersectPath(e_2, face7) → f_4
  119. Intersect(e_2, face7) → M_3, N_3
  120. IntersectPath(e_2, face6) → g_4
  121. Intersect(e_2, face6) → O_3, P_3
  122. IntersectPath(e_2, face1) → h_4
  123. Intersect(e_2, face1) → Q_3, R_3
  124. IntersectPath(e_2, face2) → i_4
  125. AngularBisector(C, A, G) → c_1
  126. AngularBisector(A, G, C) → j_4
  127. Intersect(j_4, c_1) → S_3
  128. Midpoint(G, C) → T_3
  129. Midpoint(S_3, T_3) → U_3
  130. Midpoint(U_3, T_3) → V_3
  131. Ray(O, S_3) → k_4
  132. Segment(S_3, V_3) → radStar
  133. Circle(S_3, radStar, k_4) → p_4
  134. Rotate(V_3, 60°, k_4) → W_3
  135. Rotate(W_3, 60°, k_4) → Z_3
  136. Rotate(Z_3, 60°, k_4) → A_4
  137. Rotate(A_4, 60°, k_4) → B_4
  138. Rotate(B_4, 60°, k_4) → C_4
  139. Segment(A_4, V_3) → Star12
  140. Segment(B_4, W_3) → Star13
  141. Segment(Z_3, C_4) → Star11
  142. Star(A, G, D, O) → Star41, Star42, Star43
  143. Midpoint(B, D) → D_4
  144. Plane(F, D_4, G) → o_2
  145. Mirror(Star42, o_2) → Star23
  146. Mirror(Star43, o_2) → Star22
  147. Mirror(Star41, o_2) → Star21
  148. Midpoint(D, A) → E_4
  149. Plane(F, E_4, G) → u_2
  150. Mirror(A_4, u_2) → F_4
  151. Mirror(V_3, u_2) → G_4
  152. Segment(F_4, G_4) → Star32
  153. Mirror(Z_3, u_2) → H_4
  154. Mirror(C_4, u_2) → I_4
  155. Segment(H_4, I_4) → Star33
  156. Mirror(B_4, u_2) → J_4
  157. Mirror(W_3, u_2) → K_4
  158. Segment(J_4, K_4) → Star31
  159. Midpoint(F, D) → L_4
  160. Midpoint(C, G) → M_4
  161. Plane(L_4, D_4, M_4) → v_2
  162. Mirror(Star41, v_2) → Star61
  163. Mirror(Star43, v_2) → Star62
  164. Mirror(Star42, v_2) → Star63
  165. Midpoint(F, A) → N_4
  166. Midpoint(B, G) → O_4
  167. Plane(N_4, E_4, O_4) → w_2
  168. Mirror(A_4, w_2) → P_4
  169. Mirror(V_3, w_2) → Q_4
  170. Segment(P_4, Q_4) → Star72
  171. Mirror(B_4, w_2) → R_4
  172. Mirror(W_3, w_2) → S_4
  173. Segment(R_4, S_4) → Star71
  174. Mirror(Z_3, w_2) → T_4
  175. Mirror(C_4, w_2) → U_4
  176. Segment(T_4, U_4) → Star73
  177. Plane(F, E_4, G) → o_3
  178. Mirror(R_4, o_3) → V_4
  179. Mirror(S_4, o_3) → W_4
  180. Segment(V_4, W_4) → Star51
  181. Mirror(T_4, o_3) → Z_4
  182. Mirror(U_4, o_3) → A_5
  183. Segment(Z_4, A_5) → Star52
  184. Mirror(P_4, o_3) → B_5
  185. Mirror(Q_4, o_3) → C_5
  186. Segment(B_5, C_5) → Star53
  187. Plane(F, D_4, G) → u_3
  188. Mirror(Star61, u_3) → Star81
  189. Mirror(Star62, u_3) → Star83
  190. Mirror(Star63, u_3) → Star82