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Octagon to Square Dissection

Octagon to Square Dissection
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2019-10-15 15:42
PolygonAngleTransformationGeometric ConstructionLocus
This example demonstrates a geometric dissection that transforms a regular octagon into a square of equal area, illustrating area conservation and similarity transformations. Core mathematical concepts include polygon area, translation, rotation, similarity, and special points such as midpoints and intersections. Construction uses GeoGebra commands: Polygon, Line, Midpoint, Circle, Intersect, OrthogonalLine, Angle, and If, to build auxiliary lines, midpoints, and perpendiculars, then reassemble pieces via translation and rotation. Suitable for middle and high school geometry classes, this interactive model helps students explore area-preserving dissections, transformation geometry, and problem-solving in geometric construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 8) → poly1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  2. Line(E, F) → k_1
  3. Midpoint(g) → M
  4. Midpoint(i) → J
  5. Circle(J, Segment[A, B]) → c
  6. Intersect(c, j, 1) → K
  7. Midpoint(m) → I
  8. Line(I, K) → n
  9. OrthogonalLine(M, n) → r
  10. Intersect(n, r) → N
  11. Line(J, n) → p
  12. Intersect(p, r) → O
  13. Polygon(N, O, 4) → poly2, s, t, a, b, P, Q
  14. Line(P, Q) → r_1
  15. Angle(k_1, r_1) → α
  16. If(2 ≤ u < 3, Translate[poly2, Vector[N, Dilate[N, 3 - u, F]]], Translate[poly2, Vector[N, F]]) → q1
  17. Midpoint(k) → L
  18. OrthogonalLine(L, p) → q
  19. Polygon(P, L, F, E, J) → poly3, p_1, l_1, f_1, e, j_1
  20. Polygon(O, M, C, D, J) → poly4, o, m_1, c_1, d, j_2
  21. Polygon(M, N, I, A, B) → poly5, m_2, n_1, i_1, a_1, b_1
  22. Polygon(I, H, G, L, Q) → poly6, i_2, h_1, g_1, l_2, q_1
  23. If(0 ≤ u < 1, Polygon[Rotate[O, (u * π), J], Rotate[M, (u * π), J], Rotate[C, (u * π), J], Rotate[D, (u * π), J], Rotate[J, (u * π), J]], Polygon[Rotate[O, π, J], Rotate[M, π, J], Rotate[C, π, J], Rotate[D, π, J], Rotate[J, π, J]]) → poly7
  24. If(0 ≤ u < 1, Polygon[Rotate[I, ((-u) * π), L], Rotate[H, ((-u) * π), L], Rotate[G, ((-u) * π), L], Rotate[L, ((-u) * π), L], Rotate[Q, ((-u) * π), L]], Polygon[Rotate[I, π, L], Rotate[H, π, L], Rotate[G, π, L], Rotate[L, π, L], Rotate[Q, π, L]]) → poly8
  25. If(0 ≤ u < 1, Rotate[I, ((-u) * π), L], Rotate[I, π, L]) → R
  26. If(0 ≤ u < 1, Polygon[Rotate[M, ((-u) * π), L], Rotate[N, ((-u) * π), L], Rotate[I, ((-u) * π), L], Rotate[A, ((-u) * π), L], Rotate[B, ((-u) * π), L]], Polygon[Rotate[M, π, L], Rotate[N, π, L], Rotate[I, π, L], Rotate[A, π, L], Rotate[B, π, L]]) → poly9
  27. If(3 ≤ u, Rotate[q1, ((u - 3) * α), F], Rotate[q1, α, F]) → q2
  28. If(0 ≤ u < 1, Rotate[poly9, ((-u) * π), R], Rotate[poly9, π, R]) → poly10

AI Drawing Prompt Reference

Draw a regular octagon and construct an inscribed square of equal area. Use midpoints of sides, diagonal intersections, and perpendicular auxiliary lines to divide the octagon into several polygonal pieces. Then translate and rotate these pieces to reassemble them into a square, demonstrating area-preserving dissection. Include visible construction lines, intersection points, midpoints, and angle markers to show the transformation process.
The prompt above was compiled from the original drawing commands and is for reference only.