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Animated Area Paradox: Area Differences Under Rotation and Partition

Animated Area Paradox: Area Differences Under Rotation and Partition
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2020-09-08 19:45
PolygonVectorTransformationGeometric Construction
This example uses an animation to demonstrate an “area paradox” phenomenon: when a configuration of regions is rotated and partitioned, the displayed area difference (or an equivalent quantity) can behave in a way that challenges naive intuition. The central math ideas include polygon area relationships, coordinate-geometry construction via intersection points, and geometry transformations—rotation and translation—implemented through vectors and line segments to precisely define the evolving boundaries. From the construction summary, the diagram is built with Polygon to create the base regions; Rotate to change their orientation over time; Intersect to compute the key intersection points that determine how the regions are split; and Vector plus Segment to represent directions and boundary pieces. Translate is used to align or shift configurations for comparison across frames. A structured set of parameter objects (e.g., a, b, c, d and q1, q2, e, f, g, h, t along with starred variants) drives the dynamic construction and the animated area comparisons. The activity is well-suited for high school through early undergraduate use in dynamic geometry, coordinate geometry, and conceptual instruction about why apparent area conclusions may depend on the exact construction and partitioning rules.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → q1, a, b, c, d
  2. Polygon(E, F, G, H) → q2, e, f, g, h
  3. Rotate(E, (-((90 * t°))), I) → E'
  4. Rotate(F, (-((90 * t°))), I) → F'
  5. Rotate(G, (-((90 * t°))), I) → G'
  6. Rotate(H, (-((90 * t°))), I) → H'
  7. Polygon(E', F', G', H') → q2', e', f', g', h'
  8. Polygon(J, K, L, M) → q3, j, k, l, m
  9. Intersect(xAxis, yAxis) → N
  10. Vector(N, O) → u
  11. Translate(J, v) → J'
  12. Translate(K, v) → K'
  13. Translate(L, v) → L'
  14. Translate(M, v) → M'
  15. Polygon(J', K', L', M') → q3', j', k', l', m'
  16. Polygon(Q, R, S, T) → q4, q, r, s, t_1
  17. Rotate(Q, (-((90 * t°))), U) → Q'
  18. Rotate(R, (-((90 * t°))), U) → R'
  19. Rotate(S, (-((90 * t°))), U) → S'
  20. Rotate(T, (-((90 * t°))), U) → T'
  21. Polygon(Q', R', S', T') → q4', q', r', s', t'
  22. Polygon(V, W, Z) → t1, z_1, v_1, w
  23. Vector(A_1, B_1) → i
  24. Translate(V, n) → V'
  25. Translate(W, n) → W'
  26. Translate(Z, n) → Z'
  27. Polygon(V', W', Z') → t1', z', v', w'
  28. Polygon(C_1, D_1, E_1) → t2, e_1, c_1, d_1
  29. Vector(F_1, G_1) → p
  30. Translate(C_1, u_1) → H_1
  31. Translate(D_1, u_1) → I_1
  32. Translate(E_1, u_1) → J_1
  33. Polygon(H_1, I_1, J_1) → t2', j_1, h_1, i_1
  34. Intersect(a, d) → K_1
  35. Segment(A, E) → f_1
  36. Segment(E, C) → g_1
  37. Segment(C, Q) → k_1
  38. Segment(Q, A) → l_1
  39. Segment(H', E') → m_1
  40. Segment(E', F') → n_1
  41. Segment(G', F') → p_1
  42. Segment(G', H') → q_1
  43. Segment(V', Z') → r_1
  44. Segment(W', Z') → s_1
  45. Segment(V', W') → a_1
  46. Segment(J_1, H_1) → b_1
  47. Segment(H_1, I_1) → f_2
  48. Segment(I_1, J_1) → g_2
  49. Segment(T', Q') → h_2
  50. Segment(Q', R') → i_2
  51. Segment(R', S') → j_2
  52. Segment(S', T') → k_2
  53. Segment(M', L') → l_2
  54. Segment(K', L') → m_2
  55. Segment(K', J') → n_2
  56. Segment(J', M') → p_2
  57. Segment(A, N_1) → q_2
  58. Segment(N_1, M_1) → r_2
  59. Segment(M_1, L_1) → s_2
  60. Segment(L_1, A) → t_2

AI Drawing Prompt Reference

This is a dynamic geometric figure illustrating how the area difference of a region changes under rotation and partitioning. The figure includes initial polygons, rotated polygons, translated polygons, and partition lines. The construction order is: first draw initial polygons, then rotate them to change orientation, then translate to align configurations, and finally draw partition lines to define boundaries. The figure aims to illustrate the area paradox phenomenon, showing how rotation and partitioning affect area calculations.
The prompt above was compiled from the original drawing commands and is for reference only.