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One Cube to Four Cubes: Geometric Transformation

One Cube to Four Cubes: Geometric Transformation
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2023-05-02 05:04
Solid GeometryTransformationGeometric Construction
This example demonstrates the transformation of a cube into four smaller cubes through rotation. Key mathematical concepts include geometric transformations, solid geometry, and spatial visualization. Construction involves GeoGebra commands such as Point, Cube, Rotate, Midpoint, and Polygon to create and manipulate the cube and its rotated expansion. Suitable for middle or high school mathematics education to understand the relationship between three-dimensional geometry and two-dimensional unfolding.

GeoGebra Commands & Parameters

  1. Point(xAxis) → D
  2. Point(xAxis) → C
  3. Point(Circle[I, Distance[B, I], Segment[B, I]]) → O
  4. Cube(B, I, O) → a, P, faceOPTS, faceQRST, edgeBP, edgeBI, edgeIO, edgeOP, edgeBQ, edgePT, edgeQT, edgeQR, Q, edgeIR, edgeRS, edgeOS, edgeST, R, S, T, faceBIOP, faceBPTQ, faceBIRQ, faceIOSR
  5. Midpoint(Q, I) → U
  6. Midpoint(R, I) → V
  7. Midpoint(S, Q) → W
  8. Midpoint(Q, R) → Z
  9. Midpoint(S, R) → A_1
  10. Midpoint(Q, B) → B_1
  11. Midpoint(R, O) → C_1
  12. Polygon(B_1, B, A, U) → q1, b_1, b, a_1, u
  13. Polygon(U, V, I, A) → q2, u_1, v, i, a_2
  14. Polygon(Z, U, V, R) → q3, z_1, u_2, v_1, r
  15. Point(edgeRS) → D_1
  16. Polygon(W, Z, R, D_1) → q4, w, z_2, r_1, d_1
  17. Polygon(R, V, C_1, D_1) → q5, r_2, v_2, c_1, f
  18. Line(B, I) → g
  19. Rotate(q1, α, g) → q1'
  20. Rotate(q2, α, g) → q2'
  21. Rotate(q3, α, g) → q3'
  22. Rotate(q5, α, g) → q5'
  23. Polygon(F_1, G, H, E_1) → q6, f_1, g_1, h, e_1
  24. Line(G, L) → j
  25. Polygon(E_1, H, J, G_1) → q7, k, h_1, j_1, l
  26. Line(O, I) → m
  27. Rotate(q7, (-(α - 90°)), m) → q7'
  28. Polygon(A, B, F, E) → q8, a_3, b_2, f_2, e
  29. Polygon(I, A, E, J) → q9, i_1, a_4, e_2, j_2
  30. Polygon(E, G, H, J) → q10, e_3, g_2, h_2, j_3
  31. Rotate(q4, α, g) → q4'
  32. Line(I, J) → n
  33. Polygon(F_1, G, H, E_1) → q11, p, g_3, h_3, q
  34. Rotate(q11, (-(α - 90°)), n) → q11'
  35. Polygon(J, H, L, M) → q12, j_4, h_4, l_1, m_1
  36. Polygon(E_1, H, L, H_1) → q13, s, h_5, l_2, t
  37. Line(G, H) → c
  38. Rotate(q13, α - 180°, c) → q13'
  39. Polygon(C, B, A, D) → q14, c_2, b_3, a_5, d
  40. Rotate(C, 90°, zAxis) → C'
  41. Rotate(B, 90°, zAxis) → B'
  42. Rotate(A, 90°, zAxis) → A'
  43. Rotate(D, 90°, zAxis) → D'
  44. Rotate(A, 90°, zAxis) → A'_{1}
  45. Rotate(B, 90°, zAxis) → B'_{1}
  46. Rotate(F, 90°, zAxis) → F'
  47. Rotate(E, 90°, zAxis) → E'
  48. Rotate(A, 180°, zAxis) → A'_{2}
  49. Rotate(B, 180°, zAxis) → B'_{2}
  50. Rotate(F, 180°, zAxis) → F'_{1}
  51. Rotate(E, 180°, zAxis) → E'_{1}
  52. Rotate(A, 270°, zAxis) → A'_{3}
  53. Rotate(B, 270°, zAxis) → B'_{3}
  54. Rotate(F, 270°, zAxis) → F'_{2}
  55. Rotate(E, 270°, zAxis) → E'_{2}
  56. Rotate(C, 90°, zAxis) → C'_{1}
  57. Rotate(B, 90°, zAxis) → B'_{4}
  58. Rotate(A, 90°, zAxis) → A'_{4}
  59. Rotate(D, 90°, zAxis) → D'_{1}
  60. Polygon(C'_{1}, B'_{4}, A'_{4}, D'_{1}) → q14', c'_{1}, b'_{4}, a'_{4}, d'_{1}
  61. Rotate(C, 180°, zAxis) → C'_{2}
  62. Rotate(B, 180°, zAxis) → B'_{5}
  63. Rotate(A, 180°, zAxis) → A'_{5}
  64. Rotate(D, 180°, zAxis) → D'_{2}
  65. Polygon(C'_{2}, B'_{5}, A'_{5}, D'_{2}) → q14'_{1}, c'_{2}, b'_{5}, a'_{5}, d'_{2}
  66. Rotate(C, 270°, zAxis) → C'_{3}
  67. Rotate(B, 270°, zAxis) → B'_{6}
  68. Rotate(A, 270°, zAxis) → A'_{6}
  69. Rotate(D, 270°, zAxis) → D'_{3}
  70. Polygon(C'_{3}, B'_{6}, A'_{6}, D'_{3}) → q14'_{2}, c'_{3}, b'_{6}, a'_{6}, d'_{3}
  71. Rotate(A, 90°, zAxis) → A'_{7}
  72. Rotate(B, 90°, zAxis) → B'_{7}
  73. Rotate(F, 90°, zAxis) → F'_{3}
  74. Rotate(E, 90°, zAxis) → E'_{3}
  75. Polygon(A'_{7}, B'_{7}, F'_{3}, E'_{3}) → q8', a'_{7}, b'_{7}, f'_{3}, e'_{3}
  76. Rotate(A, 180°, zAxis) → A'_{8}
  77. Rotate(B, 180°, zAxis) → B'_{8}
  78. Rotate(F, 180°, zAxis) → F'_{4}
  79. Rotate(E, 180°, zAxis) → E'_{4}
  80. Polygon(A'_{8}, B'_{8}, F'_{4}, E'_{4}) → q8'_{1}, a'_{8}, b'_{8}, f'_{4}, e'_{4}
  81. Rotate(A, 270°, zAxis) → A'_{9}
  82. Rotate(B, 270°, zAxis) → B'_{9}
  83. Rotate(F, 270°, zAxis) → F'_{5}
  84. Rotate(E, 270°, zAxis) → E'_{5}
  85. Polygon(A'_{9}, B'_{9}, F'_{5}, E'_{5}) → q8'_{2}, a'_{9}, b'_{9}, f'_{5}, e'_{5}
  86. Rotate(q1', 90°, zAxis) → q1''
  87. Rotate(q1', 180°, zAxis) → q1''_{1}
  88. Rotate(q1', 270°, zAxis) → q1''_{2}
  89. Rotate(q9, 90°, zAxis) → q9'
  90. Rotate(q9, 180°, zAxis) → q9'_{1}
  91. Rotate(q9, 270°, zAxis) → q9'_{2}
  92. Rotate(q2', 90°, zAxis) → q2''
  93. Rotate(q2', 180°, zAxis) → q2''_{1}
  94. Rotate(q2', 270°, zAxis) → q2''_{2}
  95. Rotate(q10, 90°, zAxis) → q10'
  96. Rotate(q10, 180°, zAxis) → q10'_{1}
  97. Rotate(q10, 270°, zAxis) → q10'_{2}
  98. Rotate(q3', 90°, zAxis) → q3''
  99. Rotate(q3', 180°, zAxis) → q3''_{1}
  100. Rotate(q3', 270°, zAxis) → q3''_{2}
  101. Rotate(q12, 90°, zAxis) → q12'
  102. Rotate(q12, 180°, zAxis) → q12'_{1}
  103. Rotate(q12, 270°, zAxis) → q12'_{2}
  104. Rotate(q7', 90°, zAxis) → q7''
  105. Rotate(q7', 180°, zAxis) → q7''_{1}
  106. Rotate(q7', 270°, zAxis) → q7''_{2}
  107. Rotate(q5', 90°, zAxis) → q5''
  108. Rotate(q5', 180°, zAxis) → q5''_{1}
  109. Rotate(q5', 270°, zAxis) → q5''_{2}
  110. Rotate(q13', 90°, zAxis) → q13''
  111. Rotate(q13', 180°, zAxis) → q13''_{1}
  112. Rotate(q13', 270°, zAxis) → q13''_{2}
  113. Rotate(q11', 90°, zAxis) → q11''
  114. Rotate(q11', 180°, zAxis) → q11''_{1}
  115. Rotate(q11', 270°, zAxis) → q11''_{2}
  116. Rotate(q4', 90°, zAxis) → q4''
  117. Rotate(q4', 180°, zAxis) → q4''_{1}
  118. Rotate(q4', 270°, zAxis) → q4''_{2}
  119. Polygon(L, H, K, N) → q15, l_3, h_6, k_1, n_1
  120. Rotate(q15, 90°, zAxis) → q15'
  121. Rotate(q15, 180°, zAxis) → q15'_{1}
  122. Rotate(q15, 270°, zAxis) → q15'_{2}
  123. Line(F'_{3}, I) → p_1
  124. Rotate(C, (-(α - 270°)), p_1) → C'_{4}
  125. Rotate(B, (-(α - 270°)), p_1) → B'_{10}
  126. Rotate(A, (-(α - 270°)), p_1) → A'_{10}
  127. Rotate(D, (-(α - 270°)), p_1) → D'_{4}
  128. Polygon(C'_{4}, B'_{10}, A'_{10}, D'_{4}) → q14'_{3}, c'_{4}, b'_{10}, a'_{10}, d'_{4}
  129. Rotate(A, (-(α - 360°)), yAxis) → A'_{11}
  130. Rotate(B, (-(α - 360°)), yAxis) → B'_{11}
  131. Rotate(F, (-(α - 360°)), yAxis) → F'_{6}
  132. Rotate(E, (-(α - 360°)), yAxis) → E'_{6}
  133. Polygon(A'_{11}, B'_{11}, F'_{6}, E'_{6}) → q8'_{3}, a'_{11}, b'_{11}, f'_{6}, e'_{6}
  134. Polygon(B_1, B, A, U) → q16, q_1, b_4, a_6, u_3
  135. Rotate(q16, (-(α - 360°)), yAxis) → q16'
  136. Polygon(U, A, E, I_1) → q17, u_4, a_7, e_4, s_1
  137. Line(F, M) → t_1
  138. Rotate(q17, α - 450°, t_1) → q17'
  139. Rotate(q9, α - 450°, t_1) → q9'_{3}
  140. Polygon(U, A, I, V) → q18, u_5, a_8, i_2, v_3
  141. Rotate(q18, α - 450°, t_1) → q18'
  142. Polygon(I_1, E, G, F_1) → q19, k_2, e_5, g_4, m_2
  143. Line(F'_{4}, B'_{8}) → n_2
  144. Polygon(G_1, I_1, F_1, E_1) → q20, p_2, q_2, s_2, t_2
  145. Rotate(q20, (-(α - 540°)), n_2) → q20'
  146. Rotate(q19, (-(α - 540°)), n_2) → q19'
  147. Polygon(I_1, E, J, G_1) → q21, d_2, e_6, j_5, f_3
  148. Rotate(q21, (-(α - 540°)), n_2) → q21'
  149. Rotate(q10, (-(α - 540°)), n_2) → q10'_{3}
  150. Polygon(H_1, L, M, J_1) → q22, i_3, l_4, m_3, k_3
  151. Line(G, H) → n_3
  152. Rotate(q22, α - 630°, n_3) → q22'
  153. Rotate(q12, α - 630°, n_3) → q12'_{3}
  154. Polygon(G_1, J, H, E_1) → q23, p_3, j_6, h_7, q_3
  155. Rotate(q23, α - 630°, n_3) → q23'
  156. Polygon(G_1, E_1, H_1, J_1) → q24, r_3, s_3, t_3, c_3
  157. Rotate(q24, α - 630°, n_3) → q24'
  158. Polygon(G_1, J, M, J_1) → q25, d_3, j_7, m_4, f_4
  159. Rotate(q25, α - 630°, n_3) → q25'
  160. Rotate(q14'_{3}, 90°, zAxis) → q26
  161. Rotate(q14'_{3}, 180°, zAxis) → q27
  162. Rotate(q14'_{3}, 270°, zAxis) → q28
  163. Rotate(q16', 90°, zAxis) → q16''
  164. Rotate(q16', 180°, zAxis) → q16''_{1}
  165. Rotate(q16', 270°, zAxis) → q16''_{2}
  166. Rotate(q8'_{3}, 90°, zAxis) → q29
  167. Rotate(q8'_{3}, 180°, zAxis) → q30
  168. Rotate(q8'_{3}, 270°, zAxis) → q31
  169. Rotate(q9'_{3}, 90°, zAxis) → q32
  170. Rotate(q9'_{3}, 180°, zAxis) → q33
  171. Rotate(q9'_{3}, 270°, zAxis) → q34
  172. Rotate(q17', 90°, zAxis) → q17''
  173. Rotate(q17', 180°, zAxis) → q17''_{1}
  174. Rotate(q17', 270°, zAxis) → q17''_{2}
  175. Rotate(q10'_{3}, 90°, zAxis) → q35
  176. Rotate(q10'_{3}, 180°, zAxis) → q36
  177. Rotate(q10'_{3}, 270°, zAxis) → q37
  178. Rotate(q18', 90°, zAxis) → q18''
  179. Rotate(q18', 180°, zAxis) → q18''_{1}
  180. Rotate(q18', 270°, zAxis) → q18''_{2}
  181. Rotate(q21', 90°, zAxis) → q21''
  182. Rotate(q21', 180°, zAxis) → q21''_{1}
  183. Rotate(q21', 270°, zAxis) → q21''_{2}
  184. Rotate(q20', 90°, zAxis) → q20''
  185. Rotate(q20', 180°, zAxis) → q20''_{1}
  186. Rotate(q20', 270°, zAxis) → q20''_{2}
  187. Rotate(q19', 90°, zAxis) → q19''
  188. Rotate(q19', 180°, zAxis) → q19''_{1}
  189. Rotate(q19', 270°, zAxis) → q19''_{2}
  190. Rotate(q12'_{3}, 90°, zAxis) → q38
  191. Rotate(q12'_{3}, 180°, zAxis) → q39
  192. Rotate(q12'_{3}, 270°, zAxis) → q40
  193. Rotate(q23', 90°, zAxis) → q23''
  194. Rotate(q23', 180°, zAxis) → q23''_{1}
  195. Rotate(q23', 270°, zAxis) → q23''_{2}
  196. Rotate(q22', 90°, zAxis) → q22''
  197. Rotate(q22', 180°, zAxis) → q22''_{1}
  198. Rotate(q22', 270°, zAxis) → q22''_{2}
  199. Rotate(q25', -90°, zAxis) → q25''
  200. Rotate(q25', -180°, zAxis) → q25''_{1}
  201. Rotate(q25', -270°, zAxis) → q25''_{2}
  202. Rotate(q24', 90°, zAxis) → q24''
  203. Rotate(q24', 180°, zAxis) → q24''_{1}
  204. Rotate(q24', 270°, zAxis) → q24''_{2}

AI Drawing Prompt Reference

Draw a cube and then transform it into four smaller cubes through rotation to demonstrate geometric transformation. Start by creating a cube, find midpoints of key points, connect these to form new polygons, and finally rotate parts of the shape around the center of the cube to create four smaller cubes surrounding the original.
The prompt above was compiled from the original drawing commands and is for reference only.