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Proof without words with Fibonacci numbers

Proof without words with Fibonacci numbers
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2020-01-03 21:55
TrianglePolygonSequenceInequalityGeometric Construction
① This GeoGebra example demonstrates a visual proof without words for the Fibonacci identity involving sums of squares, using a geometric tiling of squares and triangles with side lengths equal to consecutive Fibonacci numbers. ② Core mathematical concepts include the Fibonacci sequence, sum-of-squares identities, geometric dissection, and the interplay between number theory and geometry. ③ The construction uses GeoGebra commands such as Sequence to generate Fibonacci numbers, Polygon to create squares, and Midpoint/Segment to partition triangles, with objects like liste1 through liste9 managing the polygonal sequence, resulting in a spiral-like arrangement that visually confirms the identity. ④ Suitable for high school to early university students, this example is ideal for teaching sequences, geometric proofs, and mathematical reasoning, and can serve as an engaging resource for exploring algebraic identities through visual patterns.

GeoGebra Commands & Parameters

  1. Sequence(Sequence[C + (i, (-j)), i, 0, 1], j, 0, 1) → liste2
  2. Sequence(Sequence[B + (i + 1, (-j)), i, 0, 2], j, 0, 2) → liste3
  3. Sequence(Sequence[D + (i, (-j)), i, 0, 4], j, 0, 4) → liste4
  4. Sequence(Sequence[E + (i + 1, (-j)), i, 0, 7], j, 0, 7) → liste5
  5. Sequence(Sequence[F + (i, (-j)), i, 0, 12], j, 0, 12) → liste6
  6. Sequence(Sequence[G + (i + 1, (-j)), i, 0, 20], j, 0, 20) → liste7
  7. Sequence(Sequence[H + (i, (-j)), i, 0, 33], j, 0, 33) → liste8
  8. Sequence(Sequence[I + (i + 1, (-j)), i, 0, 54], j, 0, 54) → liste9
  9. Polygon(A + (-0.5, -0.5), B + (0.5, -0.5), B + (0.5, -2.5), A - (0.5, 2.5)) → poly1, b, c, d, e
  10. Polygon(B + (0.5, 0.5), B + (3.5, 0.5), B + (3.5, -2.5), B + (0.5, -2.5)) → poly2, f, g, h, i
  11. Polygon(D + (-0.5, 0.5), D + (4.5, 0.5), D + (4.5, -4.5), D + (-0.5, -4.5)) → poly3, j, k, l, m
  12. Polygon(E + (0.5, 0.5), E + (8.5, 0.5), E + (8.5, -7.5), E + (0.5, -7.5)) → poly4, n, p, q, r
  13. Polygon(F + (-0.5, 0.5), F + (12.5, 0.5), F + (12.5, -12.5), F + (-0.5, -12.5)) → poly5, s, t, a_1, b_1
  14. Polygon(G + (0.5, 0.5), G + (21.5, 0.5), G + (21.5, -20.5), G + (0.5, -20.5)) → poly6, c_1, d_1, e_1, f_1
  15. Polygon(H + (-0.5, 0.5), H + (33.5, 0.5), H + (33.5, -33.5), H + (-0.5, -33.5)) → poly7, g_1, h_1, i_1, j_1
  16. Polygon(I + (0.5, 0.5), I + (55.5, 0.5), I + (55.5, -54.5), I + (0.5, -54.5)) → poly8, k_1, l_1, m_1, n_1
  17. Polygon(B + (-0.5, -0.5), B + (-0.5, 0.5), B + (0.5, 0.5), B + (-0.5, 0.5)) → poly9, p_1, q_1, r_1, s_1
  18. Polygon(A + (-0.5, -0.5), A + (-0.5, 0.5), A + (0.5, 0.5), A + (-0.5, 0.5)) → poly10, t_1, a_2, b_2, c_2
  19. Midpoint(f) → J
  20. Midpoint(n) → K
  21. Midpoint(c_1) → L
  22. Midpoint(r_1) → M
  23. Midpoint(e) → N
  24. Midpoint(m) → O
  25. Midpoint(b_1) → P
  26. Segment(A + (-0.5, -0.5), A + (0.5, -0.5)) → d_2
  27. Segment(B + (0.5, 0.5), B + (0.5, -0.5)) → e_2
  28. Segment(B + (-0.5, -0.5), B + (0.5, -0.5)) → f_2
  29. Midpoint(a_2) → Q
  30. Segment(B + (-0.5, 0.5), B + (0.5, 0.5)) → g_2
  31. Segment(A + (0.5, 0.5), A + (0.5, -0.5)) → h_2

AI Drawing Prompt Reference

Create a geometric illustration showing a visual proof of the Fibonacci identity involving sums of squares, composed of squares and triangles with side lengths equal to consecutive Fibonacci numbers. Start with the smallest square and expand outward, each square divided into two triangles, forming a spiral pattern. Use varying sizes of squares and triangles to demonstrate the geometric dissection and visually confirm the mathematical identity.
The prompt above was compiled from the original drawing commands and is for reference only.