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Creating Nets of the Cube

Creating Nets of the Cube
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2016-12-24 13:09
PolygonSolid GeometryGeometric Construction
This example demonstrates how to create a net of a cube using GeoGebra. It involves core mathematical concepts such as solid geometry, polygon construction, and geometric transformations. Key construction steps include using commands like Polygon, Rotate, Vector, Midpoint, OrthogonalLine, Plane, Mirror, and Segment. Suitable for middle school students and above learning solid geometry and geometric construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Rotate(Polygon[B, E, F, C], (-α), b) → poly2
  3. Rotate(C, (-α), b) → C'
  4. Rotate(F, (-α), b) → F'
  5. Rotate(E, (-α), b) → E'
  6. Rotate(F, (-α), b) → F'_1
  7. Rotate(B, (-α), b) → B'
  8. Rotate(E, (-α), b) → E'_1
  9. Vector(C, Rotate[F, 90°, C]) → u
  10. Rotate(Polygon[A, B, B - u, A - u], (-α), a) → poly5
  11. Midpoint(A, C') → G
  12. OrthogonalLine(G, poly1) → l
  13. Rotate(poly5, 180°, l) → poly5'
  14. Rotate(poly2, 180°, l) → poly2'
  15. Midpoint(B', F'_1) → H
  16. OrthogonalLine(H, poly2) → m
  17. Rotate(B - u, (-α), a) → I
  18. Midpoint(A, I) → J
  19. OrthogonalLine(J, poly5) → n
  20. Rotate(poly1, (-(180°)), n) → poly1'
  21. Rotate(poly2, (-(180°)), n) → poly2'_1
  22. Rotate(poly5', (-(180°)), n) → poly5''
  23. Vector(B, Rotate[A, 90°, B]) → v
  24. Rotate(Rotate[Polygon[B, E, E + v, B + v], (-α), Segment[B, E]], (-α), b) → poly6
  25. Rotate(poly6, 180°, m) → poly6'
  26. Rotate(poly1, 180°, m) → poly1'_1
  27. Rotate(A, 180°, m) → A'
  28. Midpoint(E'_1, A') → K
  29. OrthogonalLine(K, poly1'_1) → i
  30. Rotate(poly2, 180°, i) → poly2'_2
  31. Vector(A, Rotate[D, -90°, A]) → w
  32. Rotate(Rotate[Rotate[Polygon[E, E + w, E + w + v, E + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly3
  33. Rotate(Rotate[Rotate[Rotate[Polygon[E + v, E + v + w, E + v + w - u, E + v - u], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly4
  34. Plane(l, m) → e
  35. Mirror(poly3, e) → poly3'
  36. Mirror(poly4, e) → poly4'
  37. Rotate(poly6', 180°, l) → poly6''
  38. Rotate(poly6, 180°, l) → poly6'_1
  39. Rotate(poly3, 180°, l) → poly3'_2
  40. Rotate(poly1', 180°, l) → poly1''
  41. Rotate(poly5'', 180°, l) → poly5'''
  42. Rotate(Rotate[Rotate[E + w + v, α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → L
  43. Midpoint(E'_1, L) → M
  44. OrthogonalLine(M, poly3) → f
  45. Rotate(poly6, (-(180°)), f) → poly6'_2
  46. Rotate(poly6'_2, 180°, l) → poly7
  47. Mirror(poly6'_2, e) → poly8
  48. Rotate(poly8, 180°, l) → poly8'
  49. Rotate(poly3', 180°, l) → poly3''
  50. Mirror(poly2'_1, e) → poly9
  51. Rotate(poly1'_1, 180°, l) → poly10
  52. Segment(C', F'_1) → g
  53. Segment(F'_1, E'_1) → h
  54. Segment(B', E'_1) → j
  55. Rotate(poly6, 180°, i) → poly6'_3
  56. Rotate(poly6', 180°, i) → poly6''_1
  57. Rotate(poly5, 180°, m) → poly5'_1
  58. Rotate(poly5', 180°, m) → poly5''_1
  59. Rotate(poly5'_1, 180°, l) → poly11
  60. Rotate(poly5''_1, 180°, l) → poly12
  61. Rotate(Rotate[Rotate[Rotate[Rotate[Polygon[O, N, N + B - A, O + B - A], α, Segment[O, N]], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly13
  62. Mirror(poly13, e) → poly13'
  63. Rotate(poly6', (-(180°)), n) → poly6''_2
  64. Mirror(poly6''_2, e) → poly14
  65. Rotate(F'_1, 180°, i) → P
  66. Rotate(E'_1, 180°, i) → Q
  67. Segment(P, Q) → h'
  68. Rotate(A, 180°, m) → A'_1
  69. Rotate(B, 180°, m) → B'_1
  70. Segment(A'_1, B'_1) → a'
  71. Rotate(C, 180°, m) → C'_1
  72. Rotate(D, 180°, m) → D'
  73. Segment(C'_1, D') → c'
  74. Rotate(B', 180°, i) → B''
  75. Rotate(E'_1, 180°, i) → R
  76. Segment(B'', R) → j'
  77. Rotate(C', 180°, i) → C''
  78. Rotate(F'_1, 180°, i) → S
  79. Segment(C'', S) → g'
  80. Rotate(C', 180°, l) → C''_1
  81. Rotate(F'_1, 180°, l) → T
  82. Segment(C''_1, T) → g'_1
  83. Rotate(B', 180°, l) → B''_1
  84. Rotate(E'_1, 180°, l) → U
  85. Segment(B''_1, U) → j'_1
  86. Rotate(F'_1, 180°, l) → V
  87. Rotate(E'_1, 180°, l) → W
  88. Segment(V, W) → h'_1
  89. Rotate(C'_1, 180°, l) → Z
  90. Rotate(D', 180°, l) → D''
  91. Segment(Z, D'') → c''
  92. Rotate(A'_1, 180°, l) → A_1
  93. Rotate(B'_1, 180°, l) → B_1
  94. Segment(A_1, B_1) → a''
  95. Rotate(Rotate[Segment[E, E + v], (-α), Segment[B, E]], (-α), b) → k
  96. Mirror(k, e) → k'
  97. Rotate(k, 180°, l) → k'_1
  98. Rotate(k', 180°, l) → k''
  99. Rotate(Rotate[E + v, (-α), Segment[B, E]], (-α), b) → C_1
  100. Midpoint(B, C_1) → D_1
  101. OrthogonalLine(D_1, poly6) → p
  102. Rotate(poly2, (-(180°)), p) → poly2'_3
  103. Rotate(poly1, (-(180°)), p) → poly1'_2
  104. Mirror(poly2'_3, e) → poly15
  105. Mirror(poly1'_2, e) → poly16
  106. Rotate(Rotate[Segment[E + v, B + v], (-α), Segment[B, E]], (-α), b) → q
  107. Mirror(q, e) → q'
  108. Rotate(B, (-(180°)), p) → B'_2
  109. Rotate(C, (-(180°)), p) → C'_2
  110. Segment(B'_2, C'_2) → b'
  111. Mirror(B'_2, e) → E_1
  112. Mirror(C'_2, e) → F_1
  113. Segment(E_1, F_1) → b''
  114. Rotate(Rotate[Rotate[Segment[E + w + v, E + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → r
  115. Mirror(B'_2, e) → G_1
  116. Mirror(O, e) → O'
  117. Rotate(Rotate[Rotate[Rotate[Segment[E + v + w, E + v + w - u], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → s
  118. Rotate(Rotate[Rotate[Segment[E + w, E + w + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → t
  119. Mirror(t, e) → t'
  120. Mirror(r, e) → r'
  121. Mirror(s, e) → s'
  122. Rotate(poly5', (-(180°)), p) → poly5''_2
  123. Mirror(poly5''_2, e) → poly17
  124. Rotate(C, (-(180°)), p) → C'_3
  125. Rotate(D, (-(180°)), p) → D'_1
  126. Segment(C'_3, D'_1) → c'_1
  127. Mirror(C'_3, e) → H_1
  128. Mirror(D'_1, e) → I_1
  129. Segment(H_1, I_1) → f_1
  130. Rotate(Segment[B - u, A - u], (-α), a) → g_1
  131. Rotate(t, 180°, l) → t'_1
  132. Rotate(t', 180°, l) → t''
  133. Rotate(Segment[A, A - u], (-α), a) → poly18
  134. Rotate(Rotate[Polygon[A, A - u, A - u - w, A - w], (-α), Segment[A, A - u]], (-α), a) → poly19
  135. Rotate(Rotate[Rotate[Polygon[A - u, A - u + v, A - u + v - w, A - u - w], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → poly20
  136. Rotate(Rotate[Segment[A - u - w, A - w], (-α), Segment[A, A - u]], (-α), a) → h_1
  137. Rotate(Rotate[Segment[A - u, A - u - w], (-α), Segment[A, A - u]], (-α), a) → i_1
  138. Rotate(Rotate[Rotate[Segment[A - u + v - w, A - u - w], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → j_1
  139. Rotate(Rotate[Rotate[Polygon[A - w, A - w - u, (2 * A) - w - u - B, (2 * A) - w - B], (-α), Segment[A - w, A - w - u]], (-α), Segment[A, A - u]], (-α), a) → poly21
  140. Rotate(Rotate[Rotate[Rotate[Polygon[A - u + v - w, A - u - w, A - u - w + B - E, A - u + v - w + B - E], α, Segment[A - u + v - w, A - u - w]], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → poly22
  141. Mirror(poly19, e) → poly19'
  142. Mirror(h_1, e) → k_1
  143. Mirror(poly21, e) → poly21'
  144. Mirror(poly22, e) → poly22'
  145. Mirror(j_1, e) → l_1
  146. Mirror(poly20, e) → poly20'
  147. Mirror(i_1, e) → m_1
  148. Rotate(C, (-(180°)), n) → C'_4
  149. Rotate(D, (-(180°)), n) → D'_2
  150. Segment(C'_4, D'_2) → c'_2
  151. Rotate(B, (-(180°)), n) → B'_3
  152. Rotate(C, (-(180°)), n) → C'_5
  153. Segment(B'_3, C'_5) → b'_1
  154. Rotate(C', (-(180°)), n) → C''_2
  155. Rotate(F'_1, (-(180°)), n) → J_1
  156. Segment(C''_2, J_1) → g'_2
  157. Mirror(poly18, e) → poly18'
  158. Mirror(g_1, e) → n_1
  159. Mirror(B'_3, e) → K_1
  160. Mirror(C'_5, e) → L_1
  161. Segment(K_1, L_1) → p_1
  162. Mirror(C'_4, e) → M_1
  163. Mirror(D'_2, e) → N_1
  164. Segment(M_1, N_1) → q_1
  165. Mirror(C''_2, e) → O_1
  166. Mirror(J_1, e) → P_1
  167. Segment(O_1, P_1) → r_1

AI Drawing Prompt Reference

Draw a net of a cube consisting of six connected rectangles, with three rectangles aligned in a straight line and the other three rectangles adjacent to these three, forming a continuous planar shape. Start by drawing three rectangles, then rotate one rectangle and connect the adjacent rectangles, finally adjust the positions to make all rectangles connect into a complete net.
The prompt above was compiled from the original drawing commands and is for reference only.