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Old Lantern: A 3D Geometric Construction and Visualization

Old Lantern: A 3D Geometric Construction and Visualization
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2021-01-25 01:28
CirclePolygonSolid GeometryTransformationGeometric Construction
① This GeoGebra example constructs a three-dimensional model named 'Old Lantern', visually demonstrating a lantern-like shape composed of polygons, arcs, circles, and cylinders, integrating both planar and solid geometry interactions. ② Core mathematical concepts include solid geometry (cube, cylinder), polygon and arc construction, midpoint and centroid usage, rotation and transformation (Rotate), along with related topics from the original tags such as probability, quadratic equations, real numbers, calculus, and inequalities, though the primary visual focus is the geometric composition of the 3D object. ③ Construction highlights: Based on the construction summary, the example employs commands such as Polygon, Midpoint, Segment, Intersect, Centroid, CircumcircleArc, Circle, CircleArc, Line, Rotate, Join, Sequence, Cube, Ray, Point, Angle, and Cylinder, with numerous named objects (e.g., m1, A, B, poly1, f, g, h, i, C, D, E, etc.) to assemble the intricate lantern structure. ④ Target audience and use: Suitable for high school and early university mathematics education, serving as a visual aid for solid geometry, geometric construction, and transformations, and useful for exploring 3D modeling and mathematical visualization, or as a creative example combining art and mathematics.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Midpoint(C, A) → E
  3. Midpoint(f) → F
  4. Midpoint(A, F) → G
  5. Midpoint(F, B) → H
  6. Midpoint(h) → I
  7. Midpoint(I, C) → J
  8. Midpoint(I, D) → K
  9. Midpoint(g) → L
  10. Midpoint(i) → M
  11. Midpoint(C, L) → N
  12. Midpoint(L, B) → O
  13. Midpoint(A, M) → P
  14. Midpoint(M, D) → Q
  15. Segment(K, G) → j
  16. Segment(J, H) → k
  17. Segment(Q, N) → l
  18. Segment(P, O) → m
  19. Intersect(l, j) → R
  20. Intersect(m, j) → S
  21. Intersect(j, l) → T
  22. Intersect(j, poly1) → U, V
  23. Midpoint(k) → W
  24. Midpoint(l) → Z
  25. Centroid(poly1) → A_1
  26. Centroid(poly1) → B_1
  27. Midpoint(m) → C_1
  28. CircumcircleArc(G, C_1, H) → c
  29. Midpoint(j) → D_1
  30. CircumcircleArc(Q, D_1, P) → d
  31. CircumcircleArc(N, W, O) → e
  32. CircumcircleArc(K, Z, J) → p
  33. Circle(E, W) → q
  34. CircleArc(A, G, P) → r
  35. Intersect(m, poly1) → E_1, F_1
  36. Intersect(m, k) → G_1
  37. Intersect(l, k) → H_1
  38. Segment(S, H_1) → s
  39. Intersect(q, s) → K_1, L_1
  40. Midpoint(E, W) → M_1
  41. CircumcircleArc(H_1, M_1, G_1) → t
  42. Midpoint(D_1, E) → I_1
  43. Midpoint(Z, E) → J_1
  44. Midpoint(E, C_1) → N_1
  45. CircumcircleArc(R, J_1, H_1) → c_1
  46. CircumcircleArc(R, I_1, S) → d_1
  47. CircumcircleArc(S, N_1, G_1) → e_1
  48. Midpoint(W, L) → O_1
  49. CircumcircleArc(H_1, O_1, G_1) → f_1
  50. Midpoint(M, D_1) → P_1
  51. CircumcircleArc(R, P_1, S) → g_1
  52. CircleArc(B, O, H) → h_1
  53. CircleArc(C, J, N) → k_1
  54. CircleArc(D, Q, K) → p_1
  55. Midpoint(C_1, F) → Q_1
  56. Midpoint(Z, I) → R_1
  57. CircumcircleArc(R, R_1, H_1) → q_1
  58. CircumcircleArc(S, Q_1, G_1) → r_1
  59. Intersect(p, q_1) → S_1, T_1, ,
  60. Intersect(d, g_1) → U_1, V_1, ,
  61. Intersect(c, r_1) → W_1, Z_1, ,
  62. Intersect(e, f_1) → A_2, B_2, ,
  63. CircumcircleArc(B_2, L, A_2) → s_1
  64. CircumcircleArc(Z_1, F, W_1) → t_1
  65. CircumcircleArc(U_1, M, V_1) → c_2
  66. CircumcircleArc(T_1, I, S_1) → d_2
  67. Segment(A, C) → n
  68. Segment(D, B) → a
  69. Line(A, B) → b
  70. Rotate(l1, 90°, b) → l1'
  71. Line(E, zAxis) → i_1
  72. Rotate(l1', 90°, i_1) → l1''
  73. Rotate(l1', (2 * 90°), i_1) → l1''_{1}
  74. Rotate(l1', (3 * 90°), i_1) → l1''_{2}
  75. Midpoint(D_2, A) → H_2
  76. Midpoint(E_2, B) → I_2
  77. Polygon(A, B, 4) → poly2, j_1, l_1, m_1, n_1, J_2, K_2
  78. Join(Sequence[Sequence[(x(C_2) + ((ray * cos(i + π / 4)) * cos(j)), y(C_2) + ((ray * sin(i + π / 4)) * cos(j)), z(C_2) + (ray * sin(j))), i, 0, 6.28319, (2 * π / n1)], j, 0, 1.5708, π / m1]) → l2
  79. Sequence(Polygon[Element[l2, i], Element[l2, i + 1], Element[l2, i + n1 + 2], Element[l2, i + n1 + 1]], i, 1, Length[l2]) → l3
  80. Polygon(D_2, E_2, 4, xOyPlane) → poly3, a_1, b_1, f_2, g_2, L_2, M_2
  81. Line(D_2, A) → h_2
  82. Cube(poly1, true) → o, N_2, arête1, arête4, arête6, arête5, arête2, arête7, arête3, arête8, O_2, P_2, Q_2, face2, face1, face3, face4, face5
  83. Circle(H_2, Distance[A, B], i_1) → e_2
  84. Ray(R_2, H_2) → i_2
  85. Intersect(i_2, e_2) → S_2,
  86. CircleArc(H_2, I_2, S_2) → k_2
  87. Point(k_2) → T_2
  88. Angle(I_2, H_2, T_2) → α
  89. Rotate(l1', α, h_2) → l1''_{3}
  90. Polygon(A, B, O_2, N_2) → q1, j_2, l_2, m_2, n_2
  91. Rotate(q1, α, h_2) → q1'
  92. Midpoint(arête7) → U_2
  93. Midpoint(arête6) → V_2
  94. Circle(E, 0.65, i_1) → p_2
  95. Cylinder(p_2, 0.65) → u, q_2, v
  96. Circle(E, 0.6, i_1) → r_2
  97. Cylinder(r_2, 2) → w, w_{2}, w_{3}
  98. Segment(E + (0, 0, 2), E + (0, 0, 2.25)) → s_2
  99. CircumcircleArc(U_2, W_2, V_2) → t_2

AI Drawing Prompt Reference

Create a 3D geometric “Old Lantern” figure. Start by forming a polygonal main body in space, then construct several midpoints and connecting segments from its edges and vertices. Use intersections of those segments and the polygon boundary to obtain key points that act as anchors for curved features. Draw circles and circumcircles arcs using appropriate anchor points, and compute centroids of the polygon to guide additional arc construction. Apply rotations to transform selected parts so the arcs and polygonal faces wrap into a lantern-like silhouette. Integrate the curved framework with cylinder/solid elements so the final object is assembled from polygons, circular arcs, circles, and cylindrical solids, with layered interactions between planar parts and the 3D volume.
The prompt above was compiled from the original drawing commands and is for reference only.