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Overlapping Quarters of a Cube Demo

Overlapping Quarters of a Cube Demo
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2019-01-22 20:37
VectorCoordinate GeometrySolid GeometryOptimizationGeometric Construction
This example demonstrates the overlapping quarters of a cube, illustrating geometric relationships and spatial reasoning. Key math concepts include solid geometry, linear programming, polygon construction, vector operations, and quadratic equations. It is constructed using commands like Intersect, Point, Polygon, Midpoint, Segment, Rotate, Circle, Vector, Translate, and Prism. Suitable for high school students and math enthusiasts exploring spatial geometry and optimization problems.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  4. Midpoint(i) → E
  5. Midpoint(h) → F
  6. Midpoint(f) → H
  7. Midpoint(F, H) → I
  8. Midpoint(E, I) → L
  9. Midpoint(F, I) → J
  10. Midpoint(D, I) → B_1
  11. Midpoint(B_1, L) → N_1
  12. Midpoint(N_1, B_1) → A_1
  13. Midpoint(L, N_1) → T
  14. Polygon(T, A_1, 4) → poly5, p_1, q_1, r_1, s_1, C_1, F_1
  15. Midpoint(I, C) → D_1
  16. Midpoint(J, D_1) → P_1
  17. Midpoint(J, P_1) → M
  18. Midpoint(P_1, D_1) → O
  19. Polygon(M, O, 4) → poly4, m, n, p, q, R, S
  20. Segment(F, H) → j
  21. Midpoint(F, J) → N
  22. Rotate(E, (-(90°)), I) → E'
  23. Rotate(B_1, (-(90°)), I) → Z_1
  24. Rotate(L, (-(90°)), I) → L'
  25. Rotate(D, (-(90°)), I) → D'
  26. Rotate(D', (-(90°)), I) → D''
  27. Rotate(E', (-(180°)), I) → E''_1
  28. Rotate(D', (-(180°)), I) → D''_1
  29. Polygon(E, L, R, S, M, J, F, D, T, F_1, C_1, A_1, B_1, I, D_1, O) → poly2, l, l_1, m_1, n_1, t_1, a_1, b_1, d_1, s, t, e, f_1, h_1, i_1, j_1, k_1
  30. Circle(I, r) → c_1
  31. Segment(I, Z_1) → g_1
  32. Intersect(c_1, i_1, 1) → G
  33. Intersect(c_1, j_1, 1) → K
  34. Vector(I, G) → u
  35. Vector(I, K) → v
  36. Rotate(E, 90°, I) → E'_1
  37. Rotate(L, 90°, I) → L'_1
  38. Rotate(T, 90°, I) → T'
  39. Rotate(F_1, 90°, I) → P
  40. Rotate(C_1, 90°, I) → Q
  41. Rotate(A_1, 90°, I) → U
  42. Rotate(B_1, 90°, I) → V
  43. Rotate(I, 90°, I) → I'
  44. Rotate(D_1, 90°, I) → W
  45. Rotate(O, 90°, I) → O'
  46. Rotate(R, 90°, I) → R'
  47. Rotate(S, 90°, I) → S'
  48. Rotate(M, 90°, I) → M'
  49. Rotate(J, 90°, I) → J'
  50. Rotate(F, 90°, I) → F'
  51. Rotate(D, 90°, I) → D'_1
  52. Polygon(E'_1, L'_1, R', S', M', J', F', D'_1, T', P, Q, U, V, I', W, O') → poly2', e'_1, o', r', s', m', j', f', d'_1, l'_1, t', p_2, q_2, u_1, v_1, i', w
  53. Rotate(E, 180°, I) → E'_2
  54. Rotate(L, 180°, I) → L'_2
  55. Rotate(T, 180°, I) → T'_1
  56. Rotate(F_1, 180°, I) → Z
  57. Rotate(C_1, 180°, I) → E_1
  58. Rotate(A_1, 180°, I) → G_1
  59. Rotate(B_1, 180°, I) → H_1
  60. Rotate(I, 180°, I) → I'_1
  61. Rotate(D_1, 180°, I) → I_1
  62. Rotate(O, 180°, I) → O'_1
  63. Rotate(R, 180°, I) → R'_1
  64. Rotate(S, 180°, I) → S'_1
  65. Rotate(M, 180°, I) → M'_1
  66. Rotate(J, 180°, I) → J'_1
  67. Rotate(F, 180°, I) → F'_1
  68. Rotate(D, 180°, I) → D'_2
  69. Polygon(E'_2, L'_2, R'_1, S'_1, M'_1, J'_1, F'_1, D'_2, T'_1, Z, E_1, G_1, H_1, I'_1, I_1, O'_1) → poly2'_1, e'_2, o'_1, r'_1, s'_1, m'_1, j'_1, f'_1, d'_2, l'_2, t'_1, z_1, e_1, g_2, h_2, i'_1, i_2
  70. Rotate(E'_1, 180°, I) → J_1
  71. Rotate(L'_1, 180°, I) → K_1
  72. Rotate(T', 180°, I) → T''
  73. Rotate(P, 180°, I) → P'
  74. Rotate(Q, 180°, I) → Q'
  75. Rotate(U, 180°, I) → U'
  76. Rotate(V, 180°, I) → V'
  77. Rotate(I', 180°, I) → I''
  78. Rotate(W, 180°, I) → W'
  79. Rotate(O', 180°, I) → O''
  80. Rotate(R', 180°, I) → R''
  81. Rotate(S', 180°, I) → S''
  82. Rotate(M', 180°, I) → M''
  83. Rotate(J', 180°, I) → J''
  84. Rotate(F', 180°, I) → F''
  85. Rotate(D'_1, 180°, I) → L_1
  86. Polygon(J_1, K_1, R'', S'', M'', J'', F'', L_1, T'', P', Q', U', V', I'', W', O'') → poly2'', j_2, o'', r'', s'', m'', j'', f'', l_2, k_2, t'', p', q', u', v', i'', w'
  87. Translate(J_1, v) → M_1
  88. Translate(K_1, v) → O_1
  89. Translate(T'', v) → T'''
  90. Translate(P', v) → P''
  91. Translate(Q', v) → Q''
  92. Translate(U', v) → U''
  93. Translate(V', v) → V''
  94. Translate(I'', v) → I'''
  95. Translate(W', v) → W''
  96. Translate(O'', v) → O'''
  97. Translate(R'', v) → R'''
  98. Translate(S'', v) → S'''
  99. Translate(M'', v) → M'''
  100. Translate(J'', v) → J'''
  101. Translate(F'', v) → F'''
  102. Translate(L_1, v) → Q_1
  103. Polygon(M_1, O_1, R''', S''', M''', J''', F''', Q_1, T''', P'', Q'', U'', V'', I''', W'', O''') → poly2''', m_2, o''', r''', s''', m''', j''', f''', q_3, o_1, t''', p'', q'', u'', v'', i''', w''
  104. Translate(E'_2, u) → R_1
  105. Translate(L'_2, u) → S_1
  106. Translate(T'_1, u) → T_1
  107. Translate(Z, u) → Z'
  108. Translate(E_1, u) → U_1
  109. Translate(G_1, u) → V_1
  110. Translate(H_1, u) → W_1
  111. Translate(I'_1, u) → A_2
  112. Translate(I_1, u) → B_2
  113. Translate(O'_1, u) → C_2
  114. Translate(R'_1, u) → D_2
  115. Translate(S'_1, u) → E_2
  116. Translate(M'_1, u) → F_2
  117. Translate(J'_1, u) → G_2
  118. Translate(F'_1, u) → H_2
  119. Translate(D'_2, u) → I_2
  120. Translate(E'_2, Vector[(-u)]) → J_2
  121. Translate(L'_2, Vector[(-u)]) → K_2
  122. Translate(T'_1, Vector[(-u)]) → L_2
  123. Translate(Z, Vector[(-u)]) → Z'_1
  124. Translate(E_1, Vector[(-u)]) → M_2
  125. Translate(G_1, Vector[(-u)]) → N_2
  126. Translate(H_1, Vector[(-u)]) → O_2
  127. Translate(I'_1, Vector[(-u)]) → P_2
  128. Translate(I_1, Vector[(-u)]) → Q_2
  129. Translate(O'_1, Vector[(-u)]) → R_2
  130. Translate(R'_1, Vector[(-u)]) → S_2
  131. Translate(S'_1, Vector[(-u)]) → T_2
  132. Translate(M'_1, Vector[(-u)]) → U_2
  133. Translate(J'_1, Vector[(-u)]) → V_2
  134. Translate(F'_1, Vector[(-u)]) → W_2
  135. Translate(D'_2, Vector[(-u)]) → Z_2
  136. Polygon(J_2, K_2, S_2, T_2, U_2, V_2, W_2, Z_2, L_2, Z'_1, M_2, N_2, O_2, P_2, Q_2, R_2) → poly3, j_3, r_2, s_2, t_2, u_2, v_2, w_2, z_2, k_3, l_3, z'_1, m_3, n_2, o_2, p_3, q_4
  137. Translate(E'_1, Vector[(-v)]) → A_3
  138. Translate(L'_1, Vector[(-v)]) → B_3
  139. Translate(T', Vector[(-v)]) → T''_1
  140. Translate(P, Vector[(-v)]) → P'_1
  141. Translate(Q, Vector[(-v)]) → Q'_1
  142. Translate(U, Vector[(-v)]) → U'_1
  143. Translate(V, Vector[(-v)]) → V'_1
  144. Translate(I', Vector[(-v)]) → I''_1
  145. Translate(W, Vector[(-v)]) → W'_1
  146. Translate(O', Vector[(-v)]) → O''_1
  147. Translate(R', Vector[(-v)]) → R''_1
  148. Translate(S', Vector[(-v)]) → S''_1
  149. Translate(M', Vector[(-v)]) → M''_1
  150. Translate(J', Vector[(-v)]) → J''_1
  151. Translate(F', Vector[(-v)]) → F''_1
  152. Translate(D'_1, Vector[(-v)]) → C_3
  153. Polygon(A_3, B_3, R''_1, S''_1, M''_1, J''_1, F''_1, C_3, T''_1, P'_1, Q'_1, U'_1, V'_1, I''_1, W'_1, O''_1) → poly2''_1, a_3, o''_1, r''_1, s''_1, m''_1, j''_1, f''_1, c_3, b_3, t''_1, p'_1, q'_1, u'_1, v'_1, i''_1, w'_1
  154. Translate(E, u) → E'_3
  155. Translate(L, u) → L'_3
  156. Translate(T, u) → T'_2
  157. Translate(F_1, u) → D_3
  158. Translate(C_1, u) → E_3
  159. Translate(A_1, u) → F_3
  160. Translate(B_1, u) → G_3
  161. Translate(I, u) → I'_2
  162. Translate(D_1, u) → H_3
  163. Translate(O, u) → O'_2
  164. Translate(R, u) → R'_2
  165. Translate(S, u) → S'_2
  166. Translate(M, u) → M'_2
  167. Translate(J, u) → J'_2
  168. Translate(F, u) → F'_2
  169. Translate(D, u) → D'_3
  170. Polygon(E'_3, L'_3, R'_2, S'_2, M'_2, J'_2, F'_2, D'_3, T'_2, D_3, E_3, F_3, G_3, I'_2, H_3, O'_2) → poly2'_2, e'_3, o'_2, r'_2, s'_2, m'_2, j'_2, f'_2, d'_3, l'_3, t'_2, d_3, e_3, f_3, g_3, i'_2, h_3
  171. Prism(poly2''', 8) → a, I_3, R_3, S_3, T_3, U_3, V_3, W_3, Z_3, face1, face3, face4, J_3, face5, face6, face7, face8, face9, face10, face11, face12, face13, face14, K_3, face15, face16, face2, face17, edge1, edge2, edge3, edge4, edge5, edge6, L_3, edge7, edge8, edge9, edge10, edge11, edge12, edge13, edge14, edge15, edge16, M_3, edge17, edge19, edge20, edge21, edge22, edge23, edge24, edge25, edge26, edge27, N_3, edge28, edge29, edge30, edge31, edge32, edge18, O_3, P_3, Q_3
  172. Prism(poly3, 8) → b, A_4, J_4, K_4, L_4, M_4, N_4, O_4, P_4, face18, face20, face21, B_4, face22, face23, face24, face25, face26, face27, face28, face29, face30, face31, C_4, face32, face33, face19, face34, edge33, edge34, edge35, edge36, edge37, edge38, D_4, edge39, edge40, edge41, edge42, edge43, edge44, edge45, edge46, edge47, edge48, E_4, edge49, edge51, edge52, edge53, edge54, edge55, edge56, edge57, edge58, edge59, F_4, edge60, edge61, edge62, edge63, edge64, edge50, G_4, H_4, I_4
  173. Prism(poly2''_1, 8) → c, Q_4, B_5, C_5, D_5, E_5, F_5, G_5, H_5, face35, face37, face38, R_4, face39, face40, face41, face42, face43, face44, face45, face46, face47, face48, S_4, face49, face50, face36, face51, edge65, edge66, edge67, edge68, edge69, edge70, T_4, edge71, edge72, edge73, edge74, edge75, edge76, edge77, edge78, edge79, edge80, U_4, edge81, edge83, edge84, edge85, edge86, edge87, edge88, edge89, edge90, edge91, V_4, edge92, edge93, edge94, edge95, edge96, edge82, W_4, Z_4, A_5
  174. Prism(poly2'_2, 8) → d, I_5, R_5, S_5, T_5, U_5, V_5, W_5, Z_5, face52, face54, face55, J_5, face56, face57, face58, face59, face60, face61, face62, face63, face64, face65, K_5, face66, face67, face53, face68, edge97, edge98, edge99, edge100, edge101, edge102, L_5, edge103, edge104, edge105, edge106, edge107, edge108, edge109, edge110, edge111, edge112, M_5, edge113, edge115, edge116, edge117, edge118, edge119, edge120, edge121, edge122, edge123, N_5, edge124, edge125, edge126, edge127, edge128, edge114, O_5, P_5, Q_5

AI Drawing Prompt Reference

Draw a demonstration of overlapping quarters of a cube. Start by creating a point A at the intersection of the x-axis and y-axis, then create a point B on the x-axis. Construct a quadrilateral with A and B as vertices, and find the midpoints E, F, H, and I of the sides. Continue constructing other key points and quadrilaterals through midpoint and rotation operations, ultimately forming a complex overlapping shape with multiple quadrilaterals and a circle.
The prompt above was compiled from the original drawing commands and is for reference only.