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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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
   • Getting a Scout out of Desert
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

Golden Ratio

1. Golden Ratio in Geometry
2. Golden Ratio in an Irregular Pentagon
3. Golden Ratio in a Irregular Pentagon II
4. Inflection Points of Fourth Degree Polynomials
5. Wythoff's Nim
6. Inscribing a regular pentagon in a circle - and proving it
7. Cosine of 36 degrees
8. Continued Fractions
9. Golden Window
10. Golden Ratio and the Egyptian Triangle
11. Golden Ratio by Compass Only
12. Golden Ratio with a Rusty Compass
13. From Equilateral Triangle and Square to Golden Ratio
14. Golden Ratio and Midpoints
15. Golden Section in Two Equilateral Triangles
16. Golden Section in Two Equilateral Triangles, II
17. Golden Ratio is Irrational

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