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Octagon to Square Dissection by Steve Phelps

Octagon to Square Dissection by Steve Phelps
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2019-09-16 17:21
PolygonGeometric Construction
This example demonstrates how to dissect an octagon and reassemble it into a square. Key mathematical concepts include properties of polygons and area calculations. The construction uses GeoGebra commands such as Polygon, Line, Midpoint, etc. Suitable for middle school and above students to learn about the area transformation between polygons and squares.

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

Start by drawing a regular octagon. Then, draw two of its diagonals and find their midpoints, drawing perpendicular lines from these midpoints. Use these perpendicular lines and midpoints to construct several triangles and quadrilaterals. Finally, through translation and rotation, reassemble these shapes into a square.
The prompt above was compiled from the original drawing commands and is for reference only.