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Christmas Tree Construction

Christmas Tree Construction
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2020-12-19 16:24
PolygonSequenceSolid GeometryTransformationGeometric Construction
This example dynamically demonstrates the 3D geometric construction of a Christmas tree using GeoGebra. It involves core mathematical concepts such as solid geometry, polygon construction, transformations, and sequences. Key GeoGebra commands and objects used in the construction include Line, Ray, Mirror, Polygon, Sequence, Circle, Cylinder, CurveCartesian, Element, Point, Prism, Cube, and PolyLine. It is suitable for high school geometry instruction and learning geometric constructions.

GeoGebra Commands & Parameters

  1. Line(K_2, yAxis) → i_2
  2. Ray(O, B) → a
  3. Mirror(M, a) → M'
  4. Polygon(B, A, K, L, M, D, C, E, F, G, H, I, J) → poly1, b, j, k_{3}, l, m, a_2, d, c_1, e, f, g_1, h, i
  5. Mirror(B, a) → B'
  6. Mirror(M, a) → M'_1
  7. Polygon(B, N, A_1, C_1, M, Q, P, R, S, T, U, V, W) → poly2, b_2, w_1, a_3, c_2, m_1, n, q, p, r, s, t, u_1, v_1
  8. Line(O, zAxis) → f_1
  9. Sequence(Rotate[poly1, i, f_1], i, 0, (2 * π), (2 * π / o)) → l1
  10. Circle(O, 0.05, f_1) → c
  11. Cylinder(c, 0.25) → a_1, e_2, b_1
  12. CurveCartesian(x(O) + ((g_2 * ℯ^((k * i))) * cos(i)), y(O) + ((g_2 * ℯ^((k * i))) * sin(i)), z(O) - ((g_2 * ℯ^((k * i))) * cot(α)) + 2, i, 0, (5.75 * π)) → f_2
  13. Sequence((x(O) + ((g_2 * ℯ^((k * i))) * cos(i)), y(O) + ((g_2 * ℯ^((k * i))) * sin(i)), z(O) - ((g_2 * ℯ^((k * i))) * cot(α)) + 2), i, 0, (5.75 * π), π / 5) → l2
  14. Element(l2, A1) → B1
  15. Element(l2, A2) → B2
  16. Element(l2, A3) → B3
  17. Element(l2, A4) → B4
  18. Element(l2, A5) → B5
  19. Element(l2, A6) → B6
  20. Element(l2, A7) → B7
  21. Element(l2, A8) → B8
  22. Element(l2, A9) → B9
  23. Element(l2, A10) → B10
  24. Element(l2, A11) → B11
  25. Element(l2, A12) → B12
  26. Element(l2, A13) → B13
  27. Element(l2, A14) → B14
  28. Element(l2, A15) → B15
  29. Element(l2, A16) → B16
  30. Element(l2, A17) → B17
  31. Element(l2, A18) → B18
  32. Element(l2, A19) → B19
  33. Element(l2, A20) → B20
  34. Element(l2, A21) → B21
  35. Element(l2, A22) → B22
  36. Element(l2, A23) → B23
  37. Element(l2, A24) → B24
  38. Element(l2, A25) → B25
  39. Element(l2, A26) → B26
  40. Element(l2, A27) → B27
  41. Element(l2, A28) → B28
  42. Element(l2, A29) → B29
  43. Element(l2, A30) → B30
  44. Element(l2, A31) → B31
  45. Element(l2, A32) → B32
  46. Element(l2, A33) → B33
  47. Element(l2, A34) → B34
  48. Element(l2, A35) → B35
  49. Element(l2, A36) → B36
  50. Element(l2, A37) → B37
  51. Element(l2, A38) → B38
  52. Element(l2, A39) → B39
  53. Element(l2, A40) → B40
  54. Element(l2, A41) → B41
  55. Element(l2, A42) → B42
  56. Element(l2, A43) → B43
  57. Element(l2, A44) → B44
  58. Element(l2, A45) → B45
  59. Element(l2, A46) → B46
  60. Element(l2, A47) → B47
  61. Element(l2, A48) → B48
  62. Element(l2, A49) → B49
  63. Element(l2, A50) → B50
  64. Element(l2, A51) → B51
  65. Element(l2, A52) → B52
  66. Element(l2, A53) → B53
  67. Element(l2, A54) → B54
  68. Element(l2, A55) → B55
  69. Element(l2, A56) → B56
  70. Element(l2, A57) → B57
  71. Element(l2, A58) → B58
  72. Element(l2, A59) → B59
  73. Element(l2, A60) → B60
  74. Element(l2, A61) → B61
  75. Element(l2, A62) → B62
  76. Element(l2, A63) → B63
  77. Element(l2, A64) → B64
  78. Element(l2, A65) → B65
  79. Element(l2, A66) → B66
  80. Element(l2, A67) → B67
  81. Element(l2, A68) → B68
  82. Element(l2, A69) → B69
  83. Element(l2, A70) → B70
  84. Element(l2, A71) → B71
  85. Element(l2, A72) → B72
  86. Element(l2, A73) → B73
  87. Element(l2, A74) → B74
  88. Element(l2, A75) → B75
  89. Element(l2, A76) → B76
  90. Element(l2, A77) → B77
  91. Element(l2, A78) → B78
  92. Element(l2, A79) → B79
  93. Element(l2, A80) → B80
  94. Element(l2, A81) → B81
  95. Element(l2, A82) → B82
  96. Element(l2, A83) → B83
  97. Element(l2, A84) → B84
  98. Element(l2, A85) → B85
  99. CurveCartesian(x(O) + ((g_2 * ℯ^((k * i))) * cos(i)), y(O) - ((g_2 * ℯ^((k * i))) * sin(i)), z(O) - ((g_2 * ℯ^((k * i))) * cot(α)) + 2, i, 0, (5.75 * π)) → h_2
  100. Point(l1) → Z
  101. Point(l1) → B_1
  102. Point(l1) → E_2
  103. Point(l1) → F_2
  104. Point(l1) → G_2
  105. Point(l1) → H_2
  106. Point(l1) → I_2
  107. Circle(J_2 + (0, 0, 0.1), 0.1, yAxis) → p_2
  108. Sequence(Rotate[K_2 + (0.1, 0, 0), i, i_2], i, 0, (2 * π), (2 * π / 6)) → l3
  109. Polygon(Element[l3, 1], Element[l3, 3], Element[l3, 5]) → t1, j_2, l_2, n_2
  110. Polygon(Element[l3, 2], Element[l3, 4], Element[l3, 6]) → t2, h_1, i_1, j_1
  111. Polygon(D_1, D_1 + (0.5, 0, 0), D_1 + (0.5, 0.25, 0), D_1 + (0, 0.25, 0)) → q1, d_1, l_1, n_1, p_1
  112. Prism(q1, 0.25) → u, E_1, arête1, arête2, arête3, arête4, arête5, arête7, arête8, arête6, F_1, G_1, H_1, face1, face3, face4, face2, face5
  113. Polygon(I_1, J_1, 4, xOyPlane) → poly3, q_1, r_1, s_1, t_1, K_1, L_1
  114. Point(Circle[N_1, Distance[M_1, N_1], Segment[M_1, N_1]]) → O_1
  115. Cube(M_1, N_1, O_1) → e_1, P_1, face10, face11, arête10, arête9, arête12, arête14, arête11, arête16, arête18, arête17, Q_1, arête13, arête19, arête15, arête20, R_1, S_1, T_1, face6, face8, face7, face9
  116. Polygon(U_1, U_1 + (0.5, 0, 0), U_1 + (0.5, 0, 0.25), U_1 + (0, 0, 0.25)) → q2, k_2, m_2, q_2, r_2
  117. Prism(q2, -0.05) → o_1, V_1, arête21, arête22, arête23, arête24, arête25, arête27, arête28, arête26, W_1, Z_1, A_2, face12, face14, face15, face13, face16
  118. PolyLine(B27, G_2, Z, H_2, I_2) → s_2
  119. Point(l1) → B_2
  120. Point(l1) → C_2
  121. Point(l1) → D_2
  122. Point(l1) → L_2
  123. Point(l1) → M_2
  124. Point(l1) → N_2
  125. Point(l1) → O_2
  126. Point(l1) → P_2
  127. Point(l1) → Q_2
  128. Point(l1) → R_2
  129. Point(l1) → S_2
  130. Point(l1) → T_2
  131. Point(l1) → U_2
  132. Point(l1) → V_2
  133. Point(l1) → W_2
  134. Point(l1) → Z_2
  135. Point(l1) → A_3
  136. Point(l1) → B_3
  137. Point(l1) → C_3
  138. Point(l1) → D_3

AI Drawing Prompt Reference

Create a 3D “Christmas tree construction” style geometric model centered around the origin. Define a vertical axis and a horizontal reference line, then use a mirror operation about a reference plane to build a symmetric side shape. First, form an initial multi-vertex polygon representing one or more base layers of the tree. Use a rotation about the vertical axis to generate repeated segments of this structure around the axis, producing stacked ring-like bands. At the origin, draw a small circle that serves as the base of a cylinder, and extrude it to create the main solid. Add a parameterized 3D curve (a spiral/rotational-trajectory type) rising along the height direction, sample it at multiple parameter values to obtain key layered points that guide how the curve relates to the solid.
The prompt above was compiled from the original drawing commands and is for reference only.