Hans Walser, [20090715c]
Fibonacci in the Triangular Lattice
In a regular triangular lattice we draw on top a red regular unit triangle, underneath a yellow rhombus and beneath a second red triangle. Under the red triangle an other yellow rhombus and beneath a red isosceles trapezium. And now always under the red trapezium a yellow rhombus and under the yellow rhombus a red trapezium.
Filling the triangle with trapeziums and rhombuses
Now the sides of the rhombuses are the Fibonacci numbers. The top side of a trapezium, the two isosceles sides and the base are three consecutive Fibonacci numbers.
The proof is simple:
Just for fun: The Fibonacci Hexagon.
We can remove the rhombuses and reassemble the trapezoids to get a star.
And of course we can also reassemble the rhombuses to get another star.
Another Fibonacci Star
The Fibonacci Triangle again
Using the Fibonacci Triangle we can prove the identity:
We can transfer the Fibonacci triangle into a square lattice. Compare with the Matterhorn in the Swiss Alps.
In the Square Lattice. Matterhorn