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The Golden Ratio: Phi, 1.618

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You are here: Home / Geometry / Phi and Fibonacci in Kepler and Golden Triangles

Phi and Fibonacci in Kepler and Golden Triangles

May 13, 2012 by Gary Meisner 21 Comments

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 = Phi2

( 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 Phi2 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.

Golden Triangle based on Phi (1.618 0339 ...), or Golden Ratio relationships

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:

Golden Triangles based on the Golden Ratio with Phi (1.618 0339 ...) to 1 relationships

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.

 

Pentragram showing golden triangles based on phi (1.618 0339 ...), the golden ratio

Penrose tiles showing golden triangles based on phi (1.618 0339 ...), the golden ratio

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

Fibonacci triangles based on relationship from Marty Stange

 

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…

Filed Under: Geometry

Comments

  1. Ken Landin says

    March 14, 2013 at 5:22 pm

    What a brilliant analysis by Stange. I loved your beautifully done illustrations, and how clear your descriptions were.
    Anyone interested in phi should look up how using the Fibonacci numbers can create a series of rectangles that approach the phi ratio.

    Reply
  2. N K Srinivasan says

    May 19, 2013 at 3:14 pm

    A very useful presentation and quite interesting!

    Reply
  3. PANAGIOTIS STEFANIDES says

    June 23, 2014 at 3:44 pm

    CORRECT REPETITION
    ————————
    A very interesting topic.

    My decoding Plato’s Timaeus “MOST BEAUTIFUL TRIANGLE” shows that Kepler / Magirus Triangle is a similar triangle, “not the same” and” not as beautiful” , but constituent to that of Plato’s:
    http://www.stefanides.gr/Html/quadrature.htm
    This orthogonal scalene triangle has all its sides in ratio T and scalene angle ArcTan [ T ] , T=SQRT[Phi].
    Its hypotenuse is T^3, its bigger side is T^2 and its smaller is T^1.

    { http://www.stefanides.gr/Html/gmr.htm
    Shows the ruller and compass structure of T [ = sqrt(Phi) ] and Θ [ = ArcTan(T) ] ,
    via known construction of Phi [ = T^2].
    It is based on a fourth order equation : T^4 – T^2 -1 =0 }
    For quadrature of circle, [ as proved imposibility by transcendentals ] is obtained , however, by the irrational value of 4/[SQRT(Phi)] , following the equation : [ Πσχ ]^4 + { [ Πσχ ]^2}*4^2 – 4^4 =0
    Ref: http://www.stefanides.gr/Html/piquad.htm
    And an AutoCad drawing:
    http://www.stefanides.gr/pdf/D=5,083FOUR_1.pdf

    © Copyright 1999 P. C. Stefanides.

    Regards,

    Panagiotis Stefanides

    Reply
    • PANAGIOTIS STEFANIDES says

      June 14, 2021 at 5:13 am

      REF:
      http://www.stefanides.gr/pdf/BOOK_1997.pdf

      GEOMETRIC CONCEPTS IN PLATO
      Panagiotis Stefanides

      http://www.stefanides.gr

      Reply
      • Dennis ONeill says

        September 6, 2021 at 1:57 pm

        Wow, very cool, thanks for posting. Check out Ken Wheeler’s “Pythagoras, Plato and the Golden Ratio” available online free as a PDF.

        Reply
  4. Iuliana says

    April 15, 2016 at 10:06 am

    Awesome explanation. Thank you!!

    Reply
  5. pritam pagla says

    July 25, 2016 at 11:12 pm

    Really started loving golden ratio

    Reply
  6. AstroTime says

    February 19, 2019 at 12:14 pm

    One more amazing Golden Triangle exists, although unknown long time.
    It is described and represented here:
    http://eye-of-revelation.org/PDF/THEORY-SriYantra.pdf

    © 2003 The Watch Publisher

    Reply
    • Antonio Alessi says

      April 25, 2019 at 12:14 pm

      Completed further descriptions of the Great Golden Triangle,
      as well as the Golden Section-s in the Vitruvian Man by Leonardo da Vinci updated here:

      https://golden-ratio.eye-of-revelation.org/

      © 2019 The Watch Publisher

      Reply
  7. Kathy says

    October 6, 2019 at 1:03 pm

    Because the powers of Phi, unlike other numbers, are in Phi ratio to one another AND are in an additive sequence where any member is the sum of the prior two, it naturally follows that
    Right triangle sides a,b,c where c is hypontenuse. IF a = phi^n and c= phi^(n+1)
    THEN b must equal phi^ ((n+2)/2)

    Reply
  8. Giuseppe Fiorentino says

    March 13, 2020 at 7:15 am

    Beautiful work. A real joy. Thank You.

    Reply
  9. Mr. M says

    July 2, 2022 at 2:22 am

    “No three successive numbers in the Fibonacci series can be used to create a right triangle.” ?
    What about the sqrt of (3 successive) Fibonacci-numbers; not a Keppler-triangle, like 5 : 8 : sqrt(3×13) ?
    The proportions between the sides of a sqrt3 : sqrt5 sqrt8 -triangle looks just as right ánd wrong to me, as I would simply use fractions to get a Keppler triangle; with c being sqrt(8/3), the base 1 and a hight of sqrt (5/3.) The more perfect with higher F-numbers

    Reply
    • Gary B Meisner says

      July 9, 2022 at 4:41 am

      Yes, but the sqrt3, sqrt5 and sqrt8 are NOT Fibonacci numbers. They are square roots of Fibonacci numbers, which is not the same thing. Good analysis though!

      Reply
      • Mr. M says

        July 12, 2022 at 2:41 am

        A right triangle in the Euclidean plane is a Kepler triangle if and only if it is similar to a triangle with side lengths: 1 , sqrtPhi, Phi.
        1 Aside, Fibonacci-numbers are (of course) not similar to those dimensions, nor are their roots. In that sense single Lucas nr. can be use to generate Phythagorian Kepler-triangles with similar dimensions The higher the number the closer.
        123^0 : 123^1/20 : 123^1/10

        Reply
  10. Mr. M says

    July 22, 2022 at 9:17 am

    I should have used the roots of Fibonacci-products: (2×13),(3×13),(5×13) for all sides, to make it an even better one.

    Reply
  11. Mr. M says

    July 26, 2022 at 4:25 am

    I am sort of missing the link between the square sum of the 3 consecutive natural numbers i.e. 9 + 16 = 25 and the sum of the 3 consecutive Fibonacci numbers: 3 + 5 = 8 as 3 times those numbers gives: 9 + 15 = 24 which reveales the similarity a little bit more. as there is only 1 less on either side. Maybe it is all to obvious for math enthousiasts; it really was not to me.

    Reply
  12. Mr. M says

    August 27, 2022 at 9:44 am

    ps I notice(d) that it is not typically Fibonacci:
    as 7, 8, 15, 23 works too, or 12² + 11 x 35 = 23².

    Reply
    • Gary B Meisner says

      August 29, 2022 at 9:49 pm

      There are likely many combinations that will work. The article is just showing how Fibonacci sequence numbers consistently follow this pattern, which is not meant to imply that only those numbers work.

      Reply
      • Mr. M says

        August 30, 2022 at 6:56 am

        Somewhat less mathemagical
        (b-a)(a+b) = b²- a²
        (5-3)(5+3) = 5²-3²=4²
        5² = 4² + 3²

        Reply
  13. Mr M says

    November 8, 2022 at 5:23 am

    144 is the only Fib.square nr. that can be related to 3x 2,3,5,8 (6,9,15,24) or (15-9)(15+9) and a 9,12,15 Pythagoras triangle (9²+12²=15²) but also to a Pythagoras 5²+12²=13² which gives a nice equation:
    15²-9²=13²-5²
    225-81=169-25
    250=160+90

    Reply
  14. Mr. M says

    December 9, 2022 at 8:47 am

    I have found a different approach
    (a+b)²-(b-a)²=4(ab)
    (a²+b²)²-(b²-a²)²=4(ab)²
    (1²+2²)²-(2²-1²)²=4(1×2)²
    5²=4²+3²
    13²=12²+5²
    34²=30²+16²
    (3²+5²)²=4(3×5)²-(5²-3²)²

    https://math-journal.blogspot.com/2012/02/fibonacci-meets-pythagoras.html?m=1

    Reply

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