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:

Proof

Just for fun: The Fibonacci Hexagon.

Fibonacci Hexagon

We can remove the rhombuses and reassemble the
trapezoids to get a star.

Fibonacci 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

Just Lines