Because F* _{n}*→ φ

*ⁿ*when

*n*→ ∞

Skip to content
# Tag: fibonacci

## φibonacci formula

## When matrices meet Fibonacci

## When Mondrian meets Pythagoras & Fibonacci

## Fibonacci’s Soup

## The Kepler Triangle, Phi and Pi

## Fibonacci Right Triangle

## Fibonacci Spiral Jigsaw Puzzle

## Circles and Golden Ratio

Because F* _{n}*→ φ

F_{0} = 1, F_{1} = 1, F* _{n}* = 1, F

Read more: https://mathworld.wolfram.com/FibonacciQ-Matrix.html

The side of medium white square / side of small black square = golden ratio, which is the limit of the ratio of consecutive Fibonacci numbers…

This geometric op art is available as prints and posters from our **online gallery.**

“Taste my recursive soup!”

– Fibonacci

A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.

Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.

Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.

Each piece of this puzzle is similar (the same shape at a different size). The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature (phyllotaxis), for instance on sunflowers. The puzzle features 8 spirals in one direction, and 13 in the other. You can build your own Fibonacci spiral puzzle by following John Edmark’s tutorial.

The last digit of the numbers in the Fibonacci Sequence are cyclic, they form a pattern that repeats after every 60th number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1.