Phi and Fibonacci in Kepler and Golden Triangles

Gary Meisner

12 years ago

Creating a Triangle based on Phi (or Pythagoras meets Fibonacci):

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 = Phi^²

^{( 1+ 1.618… = 2.618… )}

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

Side	Length squared per above	Length, or square root	Length divided by phi so c = 1
a	1	1	1 / Phi
b	Phi	√ Phi	1 / √ Phi
c	Phi^²	Phi	1

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

The Pythagorean 3-4-5 triangle is the only right-angle triangle whose sides are in an arithmetic progression. 3 + 1 = 4, and 4 plus 1 = 5. The Kepler triangle is the only right-angle triangle whose side are in a geometric progression: The square root of phi times Φ = 1 and 1 times Φ = Φ.

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

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

The isosceles triangle above on the right with a base of 1 two equal sides of Phi is known as a Golden Triangle. These familiar triangles are found embodied in pentagrams and Penrose tiles.

Creating a 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:

Fibonacci Series				The Fibonacci Triangle
b’	a	c	b”	a²	b’xb”	a² + b’xb” = c²
0	1	1	2	1	0	1
1	1	2	3	1	3	4
1	2	3	5	4	5	9
2	3	5	8	9	16	25
3	5	8	13	25	39	64
5	8	13	21	64	105	169
8	13	21	34	169	272	441
13	21	34	55	441	715	1,156
21	34	55	89	1,156	1,869	3,025
34	55	89	144	3,025	4,896	7,921
55	89	144	233	7,921	12,815	20,736
89	144	233	377	20,736	33,553	54,289
144	233	377	610	54,289	87,840	142,129

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 Kepler Triangle above, with a ratio of the hypotenuse to the base of Phi, or 1.618…