Fibonacci Numbers in Equilateral Triangle

In an equilateral piece of a triangular lattice, color the top triangle and one below. Then continue coloring isosceles trapezoids that are obtained by cutting off rhombi with sides on two colored shapes.

Then, the sides of successive rhombi form a Fibonacci sequence (1,1,2,3,5,8,...) and the top, sides and base of each trapezoid are three consecutive Fibonacci numbers.


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Fibonacci Numbers in Equilateral Triangle


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References

  1. Brian J. McCartin, MYSTERIES OF THE EQUILATERAL TRIANGLE, First published 2010, p. 68
  2. Hans R. Walser, Proof Without Words: Fibonacci Trapezoids, Mathematics Magazine, Volume 84, Number 4, October 2011, pp. 295-295(1)

Fibonacci Numbers

  1. Ceva's Theorem: A Matter of Appreciation
  2. When the Counting Gets Tough, the Tough Count on Mathematics
  3. I. Sharygin's Problem of Criminal Ministers
  4. Single Pile Games
  5. Take-Away Games
  6. Number 8 Is Interesting
  7. Curry's Paradox
  8. A Problem in Checker-Jumping
  9. Fibonacci's Quickies
  10. Fibonacci Numbers in Equilateral Triangle
  11. Binet's Formula by Inducion
  12. Binet's Formula via Generating Functions
  13. Generating Functions from Recurrences
  14. Cassini's Identity
  15. Fibonacci Idendtities with Matrices
  16. GCD of Fibonacci Numbers
  17. Binet's Formula with Cosines
  18. Lame's Theorem - First Application of Fibonacci Numbers

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