Phi and Fibonacci in Golden Triangles

Pythagoras meets Fibonacci

Creating a Golden Triangle based on Phi

Pythagoras discovered that a right triangle with sides of length a and b and a hypotenuse of length c has the following relationship:

a² + b² = c²

A foundational equality of phi has a similar structure:

1 + Phi = Phi²

^{( 1+ 1.618... = 2.618... )}

By taking the square root of each term in this equality, we have the dimensions of a Golden Triangle, a right triangle based on this phi equality, where:

This triangle is illustrated below.  It has an angle of 51.83° (or 51°50'), which has a cosine of 0.618 or phi.

Although difficult to prove due to deterioration through the ages, this angle is believed by some to have been used by the Egyptians in the construction of the Great Pyramid of Cheops.

Other Golden Triangles can be created with a Phi (1.618 0339 ...) to 1 relationship of the base and sides of triangles:

These familiar golden triangles are found embodied in pentagrams and Penrose tiles.

Creating a Golden Triangle based on Fibonacci numbers

No three successive numbers in the Fibonacci series can be used to create a right triangle.  Marty Stange, however, contributed the following relationship in January 2007:  Every successive series of four Fibonacci numbers can be used to create a right triangle, with the base and hypotenuse being determined by the second and third numbers, and the other side being the square root of the product of the first and fourth numbers.  The table below shows how this relationship works:

Thus for the illustration highlighted in gold, Stange's Treatise on Fibonacci Triangles reveals that a triangle with sides of 5 and the square root of 39 (e.g., 3 x 13) will produce a right triangle with a hypotenuse of 8.

As greater numbers in the series are used, the triangle approaches the proportions of the phi-based golden triangle above, with a ratio of the hypotenuse to the base of Phi, or 1.618...