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Regular Polyhedra – Two Projections: Monge and Axonometric

Regular Polyhedra – Two Projections: Monge and Axonometric
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2012-09-14 00:29
PolygonCoordinate GeometrySolid GeometryTransformationGeometric Construction
① This example shows the cube and its dual regular polyhedra rendered in both Monge (orthographic) and axonometric (oblique) projections. ② Core concepts include solid geometry of regular polyhedra, dual polyhedra, and projection transformations (Monge and axonometric). ③ Construction highlights: the model is built with GeoGebra commands Translate, Mirror, Midpoint, Polygon, Intersect, and Segment, using base objects i, j, T, e1‑e4, h, OO and their transformed copies OO', OO'', etc. ④ Suitable for high‑school solid geometry lessons, university geometry courses, and anyone interested in geometric modeling.

GeoGebra Commands & Parameters

  1. Translate(OO, e1) → OO'_1
  2. Translate(OO, e2) → OO'_2
  3. Translate(OO, e3) → OO'_3
  4. Translate(OO, e4) → OO'_4
  5. Translate(OO, h) → OO'_5
  6. Mirror(OO'_5, OO) → OO''_1
  7. Midpoint(OO'_2, OO'_1) → A_3
  8. Mirror(OO, A_3) → OO'_6
  9. Mirror(OO'_6, OO'_2) → OO''_2
  10. Mirror(OO'_6, OO'_1) → OO''_3
  11. Mirror(OO''_3, OO'_4) → OO'''_1
  12. Midpoint(OO'_5, OO'_2) → B_2
  13. Mirror(OO, B_2) → OO'_7
  14. Mirror(OO'_7, OO'_5) → OO''_4
  15. Mirror(OO'_7, OO) → OO''_5
  16. Mirror(OO''_4, OO) → C_2
  17. Midpoint(OO'_5, OO'_1) → D_3
  18. Mirror(OO, D_3) → OO'
  19. Mirror(OO', OO'_5) → OO''_6
  20. Mirror(OO', OO) → OO''
  21. Mirror(OO'', OO''_1) → OO'''
  22. Midpoint(C_2, OO'') → E_1
  23. Mirror(OO''_1, E_1) → B
  24. Mirror(B, OO'') → C
  25. Mirror(C, OO''_5) → D
  26. Mirror(D, OO''') → A
  27. Mirror(A, OO'_6) → F
  28. Mirror(F, OO'_7) → G
  29. Mirror(G, OO''_6) → H
  30. Mirror(H, OO''_4) → E
  31. Polygon(E, H, G, F) → многоугольник1, h''', h'', h', h_1
  32. Polygon(B, C, D, A) → многоугольник2, f, f', f'', f'''
  33. Polygon(G, H, C, B) → многоугольник3, h'_1, h''_1, f'_1, f_1
  34. Polygon(F, G, B, A) → многоугольник4, h_2, h'_2, f_2, f'''_1
  35. Polygon(E, F, A, D) → многоугольник5, h'''_1, h_3, f'''_2, f''_1
  36. Polygon(C, H, E, D) → многоугольник6, f'_2, h''_2, h'''_2, f''_2
  37. Midpoint(G, C) → F_1
  38. Polygon(C, H, E, D, A, F, G, B) → многоугольник7, c_1, c1, d1, d_1, a_1, a1, b1, b_1
  39. Polygon(D, A, F, E) → многоугольник8, d, a_2, a1_1, d1_1
  40. Polygon(E, F, A, D) → многоугольник9, d1_2, a1_2, a_3, d_2
  41. Polygon(F, G, B, A) → многоугольник10, a1_3, b1_1, b_2, a_4
  42. Polygon(F, G, B, A) → многоугольник11, a1_4, b1_2, b_3, a_5
  43. Polygon(B, A, D, C) → многоугольник12, b_4, a_6, d_3, c_2
  44. Polygon(B, C, D, A) → многоугольник13, b_5, c_3, d_4, a_7
  45. Intersect(yAxis, xAxis) → D_2
  46. Polygon(D_2, A_2, 4) → mnohoúhelník1, d_5, a_8, g, k, F_2, E_2
  47. Polygon(B_1, C_1, 4) → mnohoúhelník2, a_9, d_6, l, m, D_1, A_1
  48. Midpoint(E_2, A_2) → I
  49. Midpoint(C_1, A_1) → J
  50. Midpoint(k) → K
  51. Midpoint(a_8) → L
  52. Midpoint(g) → M
  53. Midpoint(d_5) → N
  54. Midpoint(a_9) → O
  55. Midpoint(d_6) → P
  56. Midpoint(m) → Q
  57. Midpoint(l) → R
  58. Polygon(OO'_5, OO'_4, OO'_1) → mnohoúhelník3, oo'_1, oo'_5, oo'_4
  59. Intersect(oo'_4, oo'_1) → S
  60. Intersect(oo'_4, oo'_5) → U
  61. Polygon(S, U, OO'_2) → mnohoúhelník4, oo'_2, s, u
  62. Intersect(oo'_5, oo'_1) → V
  63. Polygon(V, OO''_1, U) → mnohoúhelník5, u_1, v, oo''_1
  64. Intersect(u, s) → W
  65. Intersect(v, u_1) → Z
  66. Polygon(U, W, Z) → mnohoúhelník6, z, u_2, w
  67. Polygon(S, OO'_3, V) → mnohoúhelník7, v_1, s_1, oo'_3
  68. Intersect(s_1, v_1) → G_1
  69. Polygon(G_1, Z, W) → mnohoúhelník8, w_1, g_1, z_1
  70. Polygon(V, G_1, Z) → mnohoúhelník9, z_2, v_2, g_2
  71. Polygon(S, G, W) → mnohoúhelník18, w_2, s_2, g_3
  72. Polygon(S, E, V) → mnohoúhelník19, v_3, s_3, e_1
  73. Intersect(s_3, v_3) → H_1
  74. Polygon(H_1, V, U) → mnohoúhelník20, u_3, h_4, v_4
  75. Polygon(U, S, H_1) → mnohoúhelník21, h_5, u_4, s_4
  76. Polygon(U, Z, A) → mnohoúhelník22, a_{10}, u_5, z_3
  77. Intersect(z_3, u_5) → I_1
  78. Polygon(Z, W, I_1) → mnohoúhelník23, i_5, z_4, w_3
  79. Polygon(U, W, I_1) → mnohoúhelník24, i_6, u_6, w_4
  80. Polygon(C, V, Z) → mnohoúhelník25, z_5, c_4, v_5
  81. Intersect(v_5, z_5) → J_1
  82. Polygon(V, G_1, J_1) → mnohoúhelník26, j_5, v_6, g_4
  83. Polygon(G_1, Z, J_1) → mnohoúhelník27, j_6, g_5, z_6
  84. Intersect(s_2, w_2) → K_1
  85. Polygon(S, K_1, G_1) → mnohoúhelník28, g_6, s_5, k_3
  86. Polygon(K_1, G_1, W) → mnohoúhelník29, w_5, k_4, g_7
  87. Polygon(W, G_1, B) → mnohoúhelník30, b_6, w_6, g_8
  88. Intersect(g_8, w_6) → L_1
  89. Polygon(Z, G_1, L_1) → mnohoúhelník31, l_3, z_7, g_9
  90. Polygon(W, Z, L_1) → mnohoúhelník32, l_4, w_7, z_8
  91. Polygon(D, V, Z) → mnohoúhelník33, z_9, d_7, v_7
  92. Intersect(v_7, z_9) → M_1
  93. Polygon(U, V, M_1) → mnohoúhelník34, m_3, u_7, v_8
  94. Polygon(U, Z, M_1) → mnohoúhelník35, m_4, u_8, z_{10}
  95. Polygon(H, V, S) → mnohoúhelník39, s_8, h_6, v_9
  96. Intersect(v_9, s_8) → O_1
  97. Polygon(O_1, S, G_1) → mnohoúhelník40, g_{10}, o_2, s_9
  98. Polygon(O_1, V, G_1) → mnohoúhelník41, g_{11}, o_3, v_{10}
  99. Polygon(F, U, S) → mnohoúhelník42, s_{10}, f_3, u_{11}
  100. Intersect(u_{11}, s_{10}) → P_1
  101. Polygon(P_1, S, W) → mnohoúhelník44, w_{11}, p_3, s_{11}
  102. Segment(E_2, A_2) → t_1
  103. Segment(F_2, D_2) → t_2
  104. Segment(C_1, A_1) → t_3
  105. Segment(D_1, B_1) → t_4
  106. Segment(M, I) → n
  107. Segment(I, N) → p
  108. Segment(K, I) → q
  109. Segment(I, L) → r
  110. Segment(M, L) → i_1
  111. Segment(L, N) → j_1
  112. Segment(N, K) → k_1
  113. Segment(K, M) → l_1
  114. Segment(O, P) → m_1
  115. Segment(P, R) → n_1
  116. Segment(R, Q) → p_1
  117. Segment(Q, O) → q_1
  118. Segment(O, J) → r_1
  119. Segment(J, P) → i_2
  120. Segment(J, R) → j_2
  121. Segment(J, Q) → k_2
  122. Polygon(S, W, G_1) → mnohoúhelník10, g_{12}, s_6, w_8

AI Drawing Prompt Reference

Draw a cube and its dual polyhedron with two projections: orthographic and oblique. Start by drawing the six faces of the cube, then use mirroring and translation to create the dual polyhedron. Next, draw the orthographic and oblique projections separately. Finally, connect key points to form polygons, showcasing the structure of the cube and its dual.
The prompt above was compiled from the original drawing commands and is for reference only.