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Fibonacci and the Golden Ratio

A few blog posts ago, when I talked about the Golden Ratio, (1 to 1.618 or .618 to 1) there were several questions about how the golden ratio relates to the Fibonacci number sequence.

Leonardo Fibonacci was an Italian mathematician (c. 1170-1250) who devised a number sequence where the relationship of one number to the next or previous one provided perfect proportions. Mathematicians and artisans have been using this number sequence ever since. Some quilters use these numbers to plan proportion for their designs.

Da Vinci - the strips and the interior as well as the borders follow the Golden Ratio proportions
Da Vinci – the strips and the interior as well as the borders follow the Golden Ratio proportions

Fibonacci’s number sequence goes like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.

Can you see how the numbers are determined? Here’s how the sequence works. Start by adding our first two numbers: 0+1=1. Go to the second and third numbers, 1+1=2, then 1+2=3 and so on. Each successive number is the sum of the previous two numbers.  You can select any number in the sequence. It is always the sum of the previous two numbers.  For example 21 is obtained by adding 8 and 13.

But in actual fact, this is virtually the same as the Golden Ratio.  As the numbers get higher the relationship between two adjacent ones approximates the golden ratio.  In fact from the 10th number on, you will get a value of almost 1.618 or .618 every time!

The rectangles and spirals shown here, illustrate exactly how the Golden Ratio relates to the Fibonacci sequence of numbers.

fibonacci golden ratio comparison both

Fibonacci Spiral:
Fibonacci begins with two squares, (1,1,) another is added the size of the width of the two (2) and another is added the width of the 1 and 2 (3). As more squares are added the ratio of the last two comes closer each time to the Golden Proportion (1.618 or .618). Put quarter circles in each of the squares to get the Fibonacci Spiral.

Golden Spiral:
The Golden Spiral begins with a square and a rectangle is added whose width is .618 of the first square. Another square is added that is the width of the first square and rectangle (1.618) This proportion continues so that all the relationships are either .618 or 1.618. Once again the spiral is achieved when quarter circles are drawn in each of the squares.

Comparison of the two spirals:
An overlay of the two spirals shows that at the beginning they do not match up but as Fibonacci’s numbers grow the two spirals are virtually the same. The Golden Gauge Calipers show that the spiral is in perfect Golden Ratio proportions, 1 to 1.618!

fibonacci golden ratio comparison1

All of this fascinates me. And I discovered that you can do the same type of number sequence starting with a different number. For example, we can call this one “Jinny’s Sequence”.

3, 3, 6, 9, 15, 24, 39, 63, 102, 165, etc.

Once again, by the time you get to the 10th number, and divide the 10th by the 9th you get very close to the Golden Ratio….1.6176

It seems to come out this way no matter which number you start with. So you may be asking yourself, do quilters really use this? My quilt, DaVinci was something of an ode to the proportion with the strip widths determined by this mathematical ratio.  I am a huge fan of the work of Caryl Bryer Fallert, who has created an entire Fibonacci series of quilts. Why don’t you give it a try?

If you find all this fascinating check out the previous blog posts on the Golden Ratio.


15 thoughts on “Fibonacci and the Golden Ratio

  1. Thank you ~ math has always been so difficult for me to wrap my brain around – that was so easy to understand – I taught my kids with cooking for simple fractions – but this was over the top ^_^

  2. Fascinating!

  3. It is in quilting that math has finally made any sense to me at all. The love of quilting began with the exposure of the very first Country Living Magazine which in 1984 was very traditional and Early American . The quilt was part of the style which I wanted and quilts were not available to purchase back then unless you knew a quilter and commissioned one. Math never made sense to me because math by itself has very little to relate it to. Quilts make math relatable..Like measuring ingredients to make bread, math for quilting making it come to life!! Thank you Jinny You made a connection for my mind to grasp . Isabel

  4. Been a chemist I use the “Fibonacci” at all time and intuitively apply his sequence on my quilts (yeah, I am a man who sew!). From now it will be more a accurated on my hobby!….
    (Great tips!!)

  5. Fascinating! I need to study this more.

  6. This fascinates me also. I will be ordering the calipers Monday. Love the 60′ ruler and templates as it makes cutting painless and much more accurate. I use the templates for many other things also. The new 45′ will also be on my order.
    But that is not what I wanted to ask. Is the article on Fibonacci in any of your books? I am not “number friendly” so I know I will want to refer back later.

    1. This is not in any of my books but it will always be here in the blog if you want to reference the article later.

  7. I have long been a fan of Fibonacci, but I did not know there was any such thing as the Golden Gauge Calipers. Look for my order!!!! I am excited!!!

  8. This is amazing!! Fascinating–16 yrs of math & I’ve never heard of it. Love it! Thanks.

  9. Gorgeous, can’t wait to have time worked into busy schedule to do this!

  10. I knew about this formula but never thought to use it in a quilt. Thank you so much for the info! Your quilts are absolutely stunning!

  11. Who would ever think one would learn such advanced mathematical concepts just doing simple old quilting. I am amazed that I, who couldn’t pass 2nd year algebra, found this very interesting and comprehensible.
    Thank you for such an interesting article.

  12. Using these proportions to apply multiple borders starting with a 2″ what would come next? I am thinking that I might go with a 2″ + 3″ + 5.” I have been trying to figure this out for a quilt that I am working on. I put a 2″ border on according to the pattern but I only had enough fabric that I love for the next to border make it 3.” So, I decided to add a third border and nothing looked right. Then I tried 5″ and that seems pretty good. Is it because of fibonacci’s sequence?

    1. Yes, see Jinny’s past blogs on the golden ratio. The Fibonacci’s sequence is almost the golden ratio which is 1 to 1.628. So if it is 2″ the next border would be 2 x 1.618 or 3.236. I would round it up to 3 1/4. Then the next border would be 3.25 x 1.618 or 5.25 and so on.

  13. Precioso.

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