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Geometric Construction for a Calendar Puzzle

Geometric Construction for a Calendar Puzzle
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2021-02-18 10:33
CirclePolygonSequenceTransformationGeometric Construction
In collections GeoGebra and Laser Cutting
This example demonstrates a “calendar puzzle” using a planar geometric layout, showing how intersection and symmetry constraints can encode number/date rules. The core mathematics combines logical reasoning with relationships among natural numbers and rational numbers (e.g., ratio-like constraints), realized through classical Euclidean constructions. The build is driven by GeoGebra commands and constructed objects: Intersect to obtain key intersection points, Point to define construction points, Segment and Polygon to form the assembled polygonal regions (poly1, poly2), CircleArc to create curved boundary arcs, OrthogonalLine to enforce perpendicular constraints, and Mirror to impose reflection symmetry. Sequence is used to generate structured sets of points/segments following a rule (e.g., l1–l6 and related marked elements), while intermediate objects such as q1, a–d, E–N and others represent auxiliary quantities that connect the logical conditions to the geometry. Suitable for secondary school geometry, inquiry-based learning, and early contest training, especially for students learning to translate a puzzle’s logic into geometric constraints in GeoGebra.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Point(yAxis) → D
  4. Polygon(A, B, C, D) → q1, a, b, c, d
  5. Sequence(Point[a, i / 8], i, 1, 7) → l1
  6. Sequence(Point[c, i / 8], i, 1, 7) → l2
  7. Sequence(Point[b, i / 7], i, 1, 6) → l3
  8. Sequence(Point[d, i / 7], i, 1, 6) → l4
  9. Sequence(Segment[l1(i), l2(8 - i)], i, 1, 7) → l5
  10. Sequence(Segment[l3(i), l4(7 - i)], i, 1, 7) → l6
  11. Point(l6) → E
  12. Point(c) → F
  13. Point(l6) → G
  14. Point(d) → H
  15. Point(d) → I
  16. Point(l6) → J
  17. Point(c) → K
  18. Polygon(F, G, H, I, J, K) → poly1, f, g, h, i, j, k
  19. Intersect(b, c) → L
  20. Point(b) → M
  21. Point(l6) → N
  22. Point(l6) → O
  23. Polygon(K, L, M, N, O, J) → poly2, k_1, l, m, n, o, j_1
  24. Point(l5) → P
  25. Point(l6) → Q
  26. Point(l6) → R
  27. Point(l6) → S
  28. Point(b) → T
  29. Polygon(P, Q, R, S, T, M, N, O) → poly3, p, q, r, s, t, m_1, n_1, o_1
  30. Point(l6) → U
  31. Point(b) → V
  32. Polygon(Q, U, V, T, S, R) → poly4, q_1, u, v, t_1, s_1, r_1
  33. Point(u) → W
  34. Point(a) → Z
  35. Point(a) → A_1
  36. Polygon(J, P, U, W, Z, A_1) → poly5, j_2, p_1, u_1, w, z_1, a_1
  37. Point(d) → B_1
  38. Point(l6) → C_1
  39. Point(l6) → D_1
  40. Point(l6) → E_1
  41. Point(l6) → F_1
  42. Point(a_1) → G_1
  43. Polygon(I, B_1, C_1, D_1, E_1, F_1, G_1, J) → poly6, i_1, b_1, c_1, d_1, e_1, f_1, g_1, j_3
  44. Point(l6) → H_1
  45. Point(d) → I_1
  46. Intersect(a, d) → J_1
  47. Polygon(F_1, H_1, I_1, J_1, A_1, G_1) → poly7, e, h_1, l_1, f_2, g_2, h_2
  48. Point(d) → K_1
  49. Polygon(K_1, E_1, H_1, I_1) → q2, i_2, k_2, l_2, m_2
  50. Segment(L_1, M_1) → n_2
  51. Segment(N_1, O_1) → p_2
  52. Segment(P_1, Q_1) → q_2
  53. Segment(R_1, S_1) → r_2
  54. Intersect(c, d) → T_1
  55. CircleArc(T_1, L_1, R_1) → c_2
  56. Point(d) → U_1
  57. CircleArc(U_1, S_1, P_1) → d_2
  58. Intersect(a, b) → V_1
  59. CircleArc(V_1, Q_1, O_1) → e_2
  60. Intersect(b, c) → W_1
  61. CircleArc(W_1, N_1, M_1) → s_2
  62. Point(yAxis) → Z_1
  63. OrthogonalLine(Z_1, yAxis) → t_2
  64. Mirror(C, t_2) → C'
  65. Mirror(D, t_2) → D'
  66. Mirror(D, t_2) → D'_{1}
  67. Mirror(A, t_2) → A'
  68. Mirror(B, b) → B'
  69. Mirror(A, t_2) → A'_{2}
  70. Mirror(C, t_2) → C'_{1}
  71. Mirror(B, t_2) → B'_{1}
  72. Mirror(C, t_2) → C'_{2}
  73. Mirror(A, t_2) → A'_{1}
  74. Mirror(B, t_2) → B'_{2}
  75. Mirror(L_1, t_2) → A_2
  76. Mirror(M_1, t_2) → B_2
  77. Mirror(P_1, t_2) → C_2
  78. Mirror(Q_1, t_2) → D_2
  79. Mirror(R_1, t_2) → E_2
  80. Mirror(S_1, t_2) → F_2
  81. Mirror(N_1, t_2) → G_2
  82. Mirror(O_1, t_2) → H_2
  83. Mirror(l6, t_2) → l6'

AI Drawing Prompt Reference

Draw a geometric layout composed of multiple polygons and circular arcs to demonstrate the encoding of date rules. Start by constructing a rectangle and generate several sets of points inside it following specific rules, connecting these points to form line segments. Use the intersections of these segments with the rectangle's boundaries to create two main polygonal regions, adding circular arcs as curved boundaries. Enforce symmetry using perpendicular lines and reflection operations. Finally, connect certain points and segments according to specific rules to form additional polygonal regions, making the layout more intricate and logical.
The prompt above was compiled from the original drawing commands and is for reference only.