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Missing Square Dissection

Missing Square Dissection
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2013-04-22 04:35
PolygonTransformationGeometric Construction
This example demonstrates a geometric dissection puzzle where a seemingly perfect square is split into pieces that, when rearranged, leave a missing area. The core math concepts involve geometric transformations, area conservation, and rotation. It uses GeoGebra commands like Intersect, Point, Polygon, Midpoint, OrthogonalLine, Rotate, Circle, Segment, and LineBisector to construct the puzzle. Suitable for high school mathematics to explore geometric transformations and area invariance.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → poly1, a_1, b_2, c_3, d_3, C, D
  4. Midpoint(D, B) → E
  5. OrthogonalLine(E, xAxis) → e_5
  6. Rotate(e_5, a, E) → e'
  7. Intersect(e', a_1) → H
  8. OrthogonalLine(E, e') → f
  9. Intersect(f, b_2) → I
  10. Polygon(H, B, I, E) → poly3, h_1, b_1, i, e_2
  11. Intersect(f, d_3) → G
  12. Polygon(A, H, E, G) → poly2, a_2, h, e_1, g
  13. Circle(A, e_1) → p
  14. Intersect(p, a_2, 1) → J
  15. Circle(J, e_2) → r
  16. Intersect(r, xAxis, 2) → L
  17. Circle(L, (2 * e_2)) → k_1
  18. OrthogonalLine(L, xAxis) → j_1
  19. Intersect(k_1, j_1, 2) → N
  20. OrthogonalLine(N, j_1) → l_1
  21. Intersect(l_1, yAxis) → O
  22. Segment(E, O) → s_1
  23. LineBisector(s_1) → c_2
  24. Circle(O, e_2) → p_1
  25. Intersect(p_1, l_1, 1) → P
  26. Segment(G, P) → t_1
  27. LineBisector(t_1) → d_2
  28. Intersect(c_2, d_2) → R
  29. Segment(E, N) → m_1
  30. LineBisector(m_1) → r_1
  31. Segment(I, P) → n_1
  32. LineBisector(n_1) → q_1
  33. Intersect(q_1, r_1) → Q
  34. Segment(E, L) → s
  35. LineBisector(s) → t
  36. Segment(I, J) → n
  37. LineBisector(n) → q
  38. Intersect(q, t) → M
  39. Intersect(e', c_3) → F
  40. Polygon(I, C, F, E) → poly4, i_1, c_1, f_1, e_3
  41. Polygon(F, D, G, E) → poly5, f_2, d_1, g_1, e_4
  42. Segment(A, E) → j
  43. LineBisector(j) → k
  44. Segment(G, J) → l
  45. LineBisector(l) → m
  46. Intersect(k, m) → K
  47. Rotate(A, b, K) → A'
  48. Rotate(H, b, K) → H'
  49. Rotate(E, b, K) → E'
  50. Rotate(G, b, K) → G'
  51. Polygon(A', H', E', G') → poly2', a', h', e'_1, g'
  52. Rotate(B, c, M) → B'
  53. Rotate(I, c, M) → I'
  54. Rotate(E, c, M) → E'_1
  55. Rotate(H, c, M) → H'_1
  56. Polygon(H'_1, B', I', E'_1) → poly3', h'_1, b', i', e'_2
  57. Rotate(C, d, Q) → C'
  58. Rotate(F, d, Q) → F'
  59. Rotate(E, d, Q) → E'_2
  60. Rotate(I, d, Q) → I'_1
  61. Polygon(I'_1, C', F', E'_2) → poly4', i'_1, c', f', e'_3
  62. Rotate(F, e, R) → F'_1
  63. Rotate(G, e, R) → G'_1
  64. Rotate(E, e, R) → E'_3
  65. Rotate(D, e, R) → D'
  66. Polygon(F'_1, D', G'_1, E'_3) → poly5', f'_1, d', g'_1, e'_4

AI Drawing Prompt Reference

Draw a seemingly perfect square split into pieces that, when rearranged, leave a missing area. Show key geometric transformations like rotation and symmetry to emphasize area conservation. Start with a square and coordinate axes, then construct several irregular shapes using rotations, midpoints, and perpendicular lines. Rearrange the pieces to highlight the missing area.
The prompt above was compiled from the original drawing commands and is for reference only.