MathPlotAI Logo MathPlotAI

Kürschák Hinged Puzzle Proof: Dodecagon and Square

Kürschák Hinged Puzzle Proof: Dodecagon and Square
Loading the math board and drawing, please wait…
2021-01-30 20:05
PolygonAngleTrigonometryTransformationGeometric Construction
This example visualizes a Kürschák-style hinged-puzzle argument showing how a regular dodecagon’s decomposition and rotation relationships can be matched to the geometric structure of a square. It emphasizes invariant geometry: by rotating and reflecting hinged parts, lengths and angles are preserved, turning a seemingly complex assembly into a verifiable planar proof. Key concepts include polygon geometry, construction of equilateral and isosceles triangle configurations, and proof by geometric transformations (symmetry and congruence). Construction uses GeoGebra commands such as Point to define the skeleton of vertices, Rotate for repositioning hinged pieces, Polygon to build the dodecagon/square outlines (e.g., poly1 and poly2), Flatten to unfold/lay parts into one plane, OrthogonalLine for perpendicular constructions, Intersect to compute key junction points, and Mirror plus LineBisector to enforce symmetry and equal-distance conditions. Suitable for high-school geometry, interactive proof demonstrations, and transformation-based geometry practice.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Rotate(A, (360° * 1 / 12)) → A_1
  3. Polygon(A, A_1, 12) → poly1, f, p_{2}, q, r, B, C, D, E, F, G, H, g, I, J, K, h, i, j, k, l, m, n
  4. Polygon(A, A_1, 3) → poly2, s, t, a, L
  5. Polygon(O, A, L) → t1, l_1, o, a_1
  6. Polygon(O, A_1, L) → t2, l_2, o_1, b
  7. Polygon(A_1, B, 3) → poly3, c, d, e, M
  8. Polygon(B, C, 3) → poly4, f_1, g_1, h_1, N
  9. Polygon(O, A_1, M) → t3, m_1, o_2, i_1
  10. Polygon(O, M, B) → t4, b_1, o_3, m_2
  11. Polygon(O, N, B) → t5, b_2, o_4, n_1
  12. Polygon(O, N, C) → t6, c_1, o_5, n_2
  13. Flatten(Sequence[Rotate[l1, (k * 90°)], k, 1, 3]) → l2
  14. OrthogonalLine(A, xAxis) → j_1
  15. OrthogonalLine(C, yAxis) → k_1
  16. Intersect(j_1, k_1) → P
  17. Mirror(P, xAxis) → P'
  18. Polygon(P', P, 4) → poly5, q_1, r_1, s_1, t_1, Q, R
  19. Rotate(O, ((-p(u)) * 270°), A) → O'
  20. Rotate(A, ((-p(u)) * 270°), A) → A'
  21. Rotate(L, ((-p(u)) * 270°), A) → L'
  22. Polygon(O', A', L') → t1', l', o', a'
  23. Polygon(L', A, 3) → poly6, p_1, d_1, e_1, S
  24. Mirror(A, a') → A'_{1}
  25. Rotate(S, (p(u) * 60°), O') → S'
  26. Rotate(L', (p(u) * 60°), O') → L''
  27. Polygon(O', L'', S') → t7, s', f_2, l''
  28. Mirror(L'', l'') → L'''
  29. Rotate(L''', (p(u) * (-360 + 180)°), S') → T
  30. Rotate(O', (p(u) * (-(180°))), S') → O''
  31. Polygon(O'', S', T) → t8, t_2, o'', g_2
  32. Polygon(T, S', 3) → poly7, h_2, i_2, j_2, U
  33. Rotate(U, (p(u) * 150°), T) → U'
  34. Rotate(S', (p(u) * 150°), T) → S''
  35. Polygon(U', T, S'') → t9, s'', u', t_3
  36. Rotate(U, (p(u) * 60°), O'') → SS
  37. Rotate(T, (p(u) * 60°), O'') → T'
  38. Polygon(O'', T', SS) → t10, u'_{1}, k_2, t'
  39. Rotate(O'', (p(u) * (-(180°))), SS) → O'''
  40. Rotate(poly6, ((p(u))^(2) * 150°), L') → poly6'
  41. LineBisector(O', O'') → q_2
  42. LineBisector(O'', O''') → r_2
  43. Mirror(poly6', q_2) → poly6''
  44. Mirror(O'', r_2) → O'''_{1}
  45. Mirror(S', r_2) → S''_{1}
  46. Mirror(T, r_2) → T'_{1}
  47. Polygon(O'''_{1}, S''_{1}, T'_{1}) → t8', t'_{1}, o'''_{1}, s''_{1}
  48. Mirror(O'', r_2) → O'''_{2}
  49. Mirror(T', r_2) → T''
  50. Mirror(SS, r_2) → SS'
  51. Polygon(O'''_{2}, T'', SS') → t10', ss', o'''_{2}, t''
  52. Mirror(poly6'', r_2) → poly6'''
  53. Rotate(l1, 0°) → l1'
  54. Flatten(Sequence[Rotate[{poly2, poly3, poly4}, (k * 90°)], k, 1, 3]) → lt
  55. Mirror(poly3, g) → poly3'
  56. Flatten(Sequence[Rotate[{poly3'}, (k * 90°)], k, 1, 3]) → l3

AI Drawing Prompt Reference

Draw a geometric proof diagram showing a regular dodecagon and a square connected by rotations and reflections. The figure should include vertices of the dodecagon, equilateral and isosceles triangle configurations, and constructions linking them via symmetry and congruence. Describe the logical sequence: start with the dodecagon, rotate to form the square, then reflect and connect points to complete the proof.
The prompt above was compiled from the original drawing commands and is for reference only.