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Cube Area via Geometric Translation

Cube Area via Geometric Translation
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2023-10-01 20:56
VectorSolid GeometryTransformationGeometric Construction
This example demonstrates a spatial geometry setup where a cube is constructed and then translated to reveal relationships connected to cube-related areas (e.g., face/section area in a 3D configuration). The key mathematical ideas include solid geometry (faces of a cube and their areas), vector-based translation, and constructing consistent geometry by computing intersections. Construction approach in GeoGebra: Intersect is used to locate critical intersection points; Vector and Point represent directions and vertices; Cube creates the 3D cube; Translate performs the geometric transformation (shifting the cube/figures); and Textfield provides labels or parameter input for explanation. The scene is built from multiple named points and vectors (e.g., A, B, u, C, v, w, …) that support the area visualization. Suitable for middle school through early university learners for lessons on 3D geometry, exploratory modeling, and diagram-driven understanding of area correspondences in space.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xOyPlane) → A
  2. Vector(A, B) → u
  3. Vector(A, C) → v
  4. Vector(A, D) → w
  5. Vector(A, E) → a
  6. Vector(A, S) → c
  7. Vector(A, O) → d
  8. Vector(A, K) → e
  9. Vector(A, G) → f
  10. Vector(A, T) → g
  11. Vector(A, P) → h
  12. Vector(A, L) → i
  13. Vector(A, H) → j
  14. Vector(A, R) → k
  15. Vector(A, N) → l
  16. Vector(A, J) → m
  17. Vector(A, F) → n
  18. Vector(A, U) → p
  19. Vector(A, Q) → q
  20. Vector(A, M) → r
  21. Vector(A, I) → s
  22. Point(xAxis) → V
  23. Point(xAxis) → W
  24. Point(Circle[W, Distance[V, W], Segment[V, W]]) → Z
  25. Cube(V, W, Z) → o, A_1, face5, face6, edge1, edgeVW_{1}, edgeWZ, edge4, edge2, edge6, edge8, edge7, B_1, edge3, edge9, edge5, edge10, C_1, D_1, E_1, face1, face3, face2, face4
  26. Intersect(zAxis, xAxis) → F_1
  27. Point(Circle[F_1, Distance[W, F_1], Segment[W, F_1]]) → G_1
  28. Cube(W, F_1, G_1) → t, H_1, face11, face12, edge12, edge11, edge14, edge16, edge13, edge18, edge20, edge19, I_1, edge15, edge21, edge17, edge22, J_1, K_1, L_1, face7, face9, face8, face10
  29. Point(xAxis) → M_1
  30. Point(Circle[M_1, Distance[F_1, M_1], Segment[F_1, M_1]]) → N_1
  31. Cube(F_1, M_1, N_1) → a_1, O_1, face17, face18, edge24, edge23, edge26, edge28, edge25, edge30, edge32, edge31, P_1, edge27, edge33, edge29, edge34, Q_1, R_1, S_1, face13, face15, face14, face16
  32. Point(xAxis) → T_1
  33. Point(Circle[T_1, Distance[M_1, T_1], Segment[M_1, T_1]]) → U_1
  34. Cube(M_1, T_1, U_1) → b_1, V_1, face23, face24, edge36, edge35, edge38, edge40, edge37, edge42, edge44, edge43, W_1, edge39, edge45, edge41, edge46, Z_1, A_2, B_2, face19, face21, face20, face22
  35. Point(Circle[V, Distance[W, V], Segment[W, V]]) → C_2
  36. Cube(W, V, C_2) → c_1, D_2, face29, face30, edge49, edgeVW, edge47, edge51, edge50, edge53, edge55, edge54, E_2, edge48, edge56, edge52, edge57, F_2, G_2, H_2, face25, face28, face26, face27
  37. Point(Circle[W, Distance[F_1, W], Segment[F_1, W]]) → I_2
  38. Cube(F_1, W, I_2) → d_1, J_2, face35, face36, edge61, edge58, edge59, edge63, edge62, edge65, edge67, edge66, K_2, edge60, edge68, edge64, edge69, L_2, M_2, N_2, face31, face34, face32, face33
  39. Point(Circle[F_1, Distance[M_1, F_1], Segment[M_1, F_1]]) → O_2
  40. Cube(M_1, F_1, O_2) → e_1, P_2, face41, face42, edge73, edge70, edge71, edge75, edge74, edge77, edge79, edge78, Q_2, edge72, edge80, edge76, edge81, R_2, S_2, T_2, face37, face40, face38, face39
  41. Point(Circle[M_1, Distance[T_1, M_1], Segment[T_1, M_1]]) → U_2
  42. Cube(T_1, M_1, U_2) → f_1, V_2, face47, face48, edge85, edge82, edge83, edge87, edge86, edge89, edge91, edge90, W_2, edge84, edge92, edge88, edge93, Z_2, A_3, B_3, face43, face46, face44, face45
  43. Point(Circle[U_2, Distance[V_2, U_2], Segment[V_2, U_2]]) → C_3
  44. Cube(V_2, U_2, C_3) → g_1, D_3, face53, face54, edge97, edge94, edge95, edge99, edge98, edge101, edge103, edge102, E_3, edge96, edge104, edge100, edge105, F_3, G_3, H_3, face49, face52, face50, face51
  45. Point(Circle[O_2, Distance[U_2, O_2], Segment[U_2, O_2]]) → I_3
  46. Cube(U_2, O_2, I_3) → h_1, J_3, face59, face60, edge109, edge106, edge107, edge111, edge110, edge113, edge115, edge114, K_3, edge108, edge116, edge112, edge117, L_3, M_3, N_3, face55, face58, face56, face57
  47. Point(Circle[I_2, Distance[O_2, I_2], Segment[O_2, I_2]]) → O_3
  48. Cube(O_2, I_2, O_3) → i_1, P_3, face65, face66, edge121, edge118, edge119, edge123, edge122, edge125, edge127, edge126, Q_3, edge120, edge128, edge124, edge129, R_3, S_3, T_3, face61, face64, face62, face63
  49. Point(Circle[C_2, Distance[I_2, C_2], Segment[I_2, C_2]]) → U_3
  50. Cube(I_2, C_2, U_3) → j_1, V_3, face71, face72, edge133, edge130, edge131, edge135, edge134, edge137, edge139, edge138, W_3, edge132, edge140, edge136, edge141, Z_3, A_4, B_4, face67, face70, face68, face69
  51. Point(Circle[U_1, Distance[N_1, U_1], Segment[N_1, U_1]]) → C_4
  52. Cube(N_1, U_1, C_4) → k_1, D_4, face77, face78, edge143, edge142, edge145, edge147, edge144, edge149, edge151, edge150, E_4, edge146, edge152, edge148, edge153, F_4, G_4, H_4, face73, face75, face74, face76
  53. Point(Circle[N_1, Distance[G_1, N_1], Segment[G_1, N_1]]) → I_4
  54. Cube(G_1, N_1, I_4) → l_1, J_4, face83, face84, edge155, edge154, edge157, edge159, edge156, edge161, edge163, edge162, K_4, edge158, edge164, edge160, edge165, L_4, M_4, N_4, face79, face81, face80, face82
  55. Point(Circle[G_1, Distance[Z, G_1], Segment[Z, G_1]]) → O_4
  56. Cube(Z, G_1, O_4) → m_1, P_4, face89, face90, edge167, edge166, edge169, edge171, edge168, edge173, edge175, edge174, Q_4, edge170, edge176, edge172, edge177, R_4, S_4, T_4, face85, face87, face86, face88
  57. Point(Circle[Z, Distance[A_1, Z], Segment[A_1, Z]]) → U_4
  58. Cube(A_1, Z, U_4) → n_1, V_4, face95, face96, edge181, edge178, edge179, edge183, edge182, edge185, edge187, edge186, W_4, edge180, edge188, edge184, edge189, Z_4, A_5, B_5, face91, face94, face92, face93
  59. Translate(face18, g) → face18'
  60. Translate(face16, g) → face16'
  61. Translate(face14, g) → face14'
  62. Translate(face15, g) → face15'
  63. Translate(face17, g) → face17'
  64. Translate(face13, g) → face13'
  65. Translate(face36, k) → face36'
  66. Translate(face35, k) → face35'
  67. Translate(face34, k) → face34'
  68. Translate(face33, k) → face33'
  69. Translate(face32, k) → face32'
  70. Translate(face31, k) → face31'
  71. Translate(face42, p) → face42'
  72. Translate(face41, p) → face41'
  73. Translate(face40, p) → face40'
  74. Translate(face37, p) → face37'
  75. Translate(face39, p) → face39'
  76. Translate(face41, p) → face41'_{1}
  77. Translate(face12, c) → face12'
  78. Translate(face10, c) → face10'
  79. Translate(face8, c) → face8'
  80. Translate(face11, c) → face11'
  81. Translate(face9, c) → face9'
  82. Translate(face7, c) → face7'
  83. Translate(face24, h) → face24'
  84. Translate(face22, h) → face22'
  85. Translate(face20, h) → face20'
  86. Translate(face23, h) → face23'
  87. Translate(face21, h) → face21'
  88. Translate(face19, h) → face19'
  89. Translate(face30, l) → face30'
  90. Translate(face29, l) → face29'
  91. Translate(face28, l) → face28'
  92. Translate(face27, l) → face27'
  93. Translate(face26, l) → face26'
  94. Translate(face25, l) → face25'
  95. Translate(face60, q) → face60'
  96. Translate(face59, q) → face59'
  97. Translate(face58, q) → face58'
  98. Translate(face56, q) → face56'
  99. Translate(face57, q) → face57'
  100. Translate(face55, q) → face55'
  101. Translate(face90, d) → face90'
  102. Translate(face86, d) → face86'
  103. Translate(face88, d) → face88'
  104. Translate(face89, d) → face89'
  105. Translate(face87, d) → face87'
  106. Translate(face85, d) → face85'
  107. Translate(face47, q) → face47'
  108. Translate(face46, q) → face46'
  109. Translate(face48, q) → face48'
  110. Translate(face44, q) → face44'
  111. Translate(face45, q) → face45'
  112. Translate(face43, q) → face43'
  113. Translate(face65, l) → face65'
  114. Translate(face66, l) → face66'
  115. Translate(face64, l) → face64'
  116. Translate(face63, l) → face63'
  117. Translate(face62, l) → face62'
  118. Translate(face61, l) → face61'
  119. Translate(face84, h) → face84'
  120. Translate(face82, h) → face82'
  121. Translate(face80, h) → face80'
  122. Translate(face83, h) → face83'
  123. Translate(face81, h) → face81'
  124. Translate(face79, h) → face79'
  125. Translate(face6, d) → face6'
  126. Translate(face2, d) → face2'
  127. Translate(face4, d) → face4'
  128. Translate(face3, d) → face3'
  129. Translate(face5, d) → face5'
  130. Translate(face1, d) → face1'
  131. Translate(face78, i) → face78'
  132. Translate(face76, i) → face76'
  133. Translate(face74, i) → face74'
  134. Translate(face54, r) → face54'
  135. Translate(face53, r) → face53'
  136. Translate(face52, r) → face52'
  137. Translate(face70, m) → face70'
  138. Translate(face71, m) → face71'
  139. Translate(face72, m) → face72'
  140. Translate(face92, e) → face92'
  141. Translate(face93, e) → face93'
  142. Translate(face96, e) → face96'
  143. Translate(face51, r) → face51'
  144. Translate(face50, r) → face50'
  145. Translate(face75, i) → face75'
  146. Translate(face77, i) → face77'
  147. Translate(face94, e) → face94'
  148. Translate(face95, e) → face95'
  149. Translate(face69, m) → face69'
  150. Translate(face68, m) → face68'
  151. Translate(face67, m) → face67'
  152. Translate(face91, e) → face91'
  153. Translate(face73, i) → face73'
  154. Translate(face49, r) → face49'
  155. Translate(face38, p) → face38'
  156. Textfield(side) → InputBox1
  157. Textfield(side_{1}) → InputBox2
  158. Textfield(side_{2}) → InputBox1_{2}
  159. Textfield(side_{2}) → InputBox1_{3}
  160. Textfield(side_{2}) → InputBox1_{1}

AI Drawing Prompt Reference

Draw a cube and then translate it using spatial geometric transformations to reveal relationships connected to cube-related areas. Construct a cube first, then use vectors to determine directions and find key positions by calculating intersections. Translate the cube along the vectors to form a new cube. The final image should include the original and translated cubes, along with their intersections and faces, to illustrate the relationship between cube surface areas under spatial transformations.
The prompt above was compiled from the original drawing commands and is for reference only.