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Regular Pentagon to a Square Dissection

Regular Pentagon to a Square Dissection
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2018-05-08 02:15
PolygonVectorTransformationGeometric Construction
① This example demonstrates how to dissect a regular pentagon and rearrange the pieces via translation to form a square of equal area, illustrating an area-preserving transformation. ② Core mathematical concepts include polygon dissection, area conservation, translation transformations, and geometric construction; it relates the side lengths of a regular pentagon and a square, combining geometric intuition with algebraic reasoning. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Point, Polygon, Segment, Intersect, OrthogonalLine, Midpoint, Circle, Line, If, Vector, and Translate are used, creating multiple auxiliary points, lines, circles, and vectors. The dissection pieces are translated to form the square, involving objects like A, B, poly1, f, g, h, i, j, C, D, E, and many auxiliary lines and circles. ④ Suitable for middle/high school geometry, math competitions, or teacher training on geometric transformations, useful for exploring equal-area dissections and dynamic geometry demonstrations.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 5) → poly1, f, g, h, i, j, C, D, E
  4. Segment(E, C) → k
  5. Segment(D, A) → l
  6. Intersect(k, l) → F
  7. OrthogonalLine(D, f) → m
  8. Intersect(m, f) → G
  9. Segment(D, G) → n
  10. Midpoint(n) → H
  11. OrthogonalLine(C, f) → p
  12. Circle(C, Segment[H, G]) → c
  13. Intersect(c, p, 1) → I
  14. OrthogonalLine(I, p) → q
  15. Intersect(q, j) → J
  16. OrthogonalLine(J, q) → r
  17. Segment(A, C) → s
  18. Line(J, s) → t
  19. Intersect(t, k) → K
  20. Intersect(q, g) → L
  21. OrthogonalLine(L, s) → a
  22. Intersect(t, a) → M
  23. Polygon(E, D, F) → t1, f_1, e, d
  24. Polygon(F, D, C) → t2, c_1, f_2, d_1
  25. Polygon(K, M, L, C) → q1, k_1, m_1, l_1, c_2
  26. Polygon(L, M, J, A, B) → poly2, l_2, m_2, j_1, a_1, b
  27. Circle(K, Segment[J, L]) → e_1
  28. Intersect(e_1, k, 1) → N
  29. OrthogonalLine(N, t) → g_1
  30. Intersect(t, g_1) → O
  31. Polygon(N, O, K) → t3, k_2, n_1, o
  32. Polygon(E, N, O, J) → q2, e_2, n_2, o_1, j_2
  33. OrthogonalLine(D, yAxis) → h_1
  34. Point(h_1) → P
  35. If(0 ≤ u < 1, Dilate[M, 1 - u, P], P) → Q
  36. Vector(M, Q) → v
  37. Translate(L, v) → L'
  38. Translate(M, v) → M'
  39. Translate(J, v) → J'
  40. Translate(A, v) → A'
  41. Translate(B, v) → B'
  42. Polygon(L', M', J', A', B') → poly2', l', m', j', a', b'
  43. If(2 ≤ u < 3, Dilate[C, 3 - u, B'], B') → R
  44. Vector(C, R) → w
  45. Translate(F, w) → F'
  46. Translate(D, w) → D'
  47. Translate(C, w) → C'
  48. Polygon(F', D', C') → t2', c', f', d'
  49. If(2 ≤ u < 3, Polygon[Rotate[F', (((u - 2) * 2) * π / 10), R], Rotate[D', (((u - 2) * 2) * π / 10), R], Rotate[C', (((u - 2) * 2) * π / 10), R]], Polygon[Rotate[F', (((3 - 2) * 2) * π / 10), R], Rotate[D', (((3 - 2) * 2) * π / 10), R], Rotate[C', (((3 - 2) * 2) * π / 10), R]]) → t4
  50. If(2 ≤ u < 3, Rotate[F', (((u - 2) * 2) * π / 10), R], Rotate[F', (((3 - 2) * 2) * π / 10), R]) → S
  51. If(4 ≤ u < 5, Dilate[L, 5 - u, L'], L') → T
  52. Vector(L, T) → u_1
  53. If(4 ≤ u < 5, Rotate[Polygon[Translate[K, u_1], Translate[M, u_1], Translate[L, u_1], Translate[C, u_1]], ((u - 4) * π), T], Rotate[Polygon[Translate[K, u_1], Translate[M, u_1], Translate[L, u_1], Translate[C, u_1]], π, T]) → q3
  54. If(4 ≤ u < 5, Rotate[Translate[K, u_1], ((u - 4) * π), T], Rotate[Translate[K, u_1], π, T]) → U
  55. If(6 ≤ u < 7, Dilate[E, 7 - u, S], S) → V
  56. Vector(E, V) → v_1
  57. If(6 ≤ u < 7, Polygon[Translate[E, v_1], Translate[D, v_1], Translate[F, v_1]], Polygon[Translate[E, v_1], Translate[D, v_1], Translate[F, v_1]]) → t5
  58. If(8 ≤ u < 9, Dilate[K, 9 - u, U], U) → W
  59. Vector(K, W) → w_1
  60. If(8 ≤ u < 9, Polygon[Translate[N, w_1], Translate[O, w_1], Translate[K, w_1]], Polygon[Translate[N, w_1], Translate[O, w_1], Translate[K, w_1]]) → t6
  61. If(10 ≤ u ≤ 11, Dilate[J, 11 - u, J'], J') → Z
  62. Vector(J, Z) → b_1
  63. If(10 ≤ u ≤ 11, Rotate[Polygon[Translate[E, b_1], Translate[N, b_1], Translate[O, b_1], Translate[J, b_1]], ((u - 10) * π), Z], Rotate[Polygon[Translate[E, b_1], Translate[N, b_1], Translate[O, b_1], Translate[J, b_1]], ((11 - 10) * π), Z]) → q4

AI Drawing Prompt Reference

Draw a regular pentagon and demonstrate how to dissect it into pieces and rearrange them via translation to form a square of equal area. Start by constructing the pentagon, then identify key points and draw auxiliary lines, circles, and vectors. Dissect the pentagon into several parts, and show how translating these pieces forms a new square, ensuring area conservation throughout the transformation process.
The prompt above was compiled from the original drawing commands and is for reference only.