Quilting with Fibonacci

First, I want you to repeat after me:

“Numbers are our friends.  Numbers do not lie.”

Repeat this phrase as many times as necessary while reading this blog.

I am rather consistently amazed with quilters who don’t know how to “math” out their quilts or don’t understand how to.  I’m even more amazed at quilters who would rather not learn how to do quilt math and simply follow all the directions on a pattern.  Quilt math sets you free to alter patterns or design your own quilt.  And there’s nothing to really dread about this math. It’s addition, subtraction, multiplication, and division.  If you can balance your checkbook and come up with a workable household budget, you can easily conquer the math needed to change patterns or design your own quilt top.

Quilts and quilters generally have an uneasy working relationship with Algebra and Geometry.  Occasionally the fields of Algebra and Geometry will throw us quilters a formula we can use.  And what I find gratifying about these higher maths (especially geometry), is when the formula is introduced in the concept of concrete numbers, it makes a lot more sense than it did sweating out variables in Ms. Blalocks’ seventh period geometry class.  Which is why I also think Algebra should be taught in lockstep with chemistry, but that’s a different battle for a different day.  I have written a lot about the Golden Ratio (1.618) and quilting (Go here: ) and how we can use it to produce wonderfully balanced quilts, sashing, and borders.  Today I want to introduce another similar formula called Fibonacci.  But before we get into what exactly Fibonacci numbers are and how we use them in quilts, let’s talk a little bit about Fibonacci himself.

Fibonacci, also known as Leonardo BonacciLeonardo of Pisa, or Leonardo Bigollo Pisano (‘Leonardo the Traveller from Pisa’), was an Italian mathematician from the Republic of Pisa, considered to be “the most talented Western mathematician of the Middle Ages”. The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci (‘son of Bonacci’). However, even earlier, in 1506, a notary of the Holy Roman Empire, Perizolo mentions Leonardo as “Lionardo Fibonacci”.  Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.

Fibonacci was born around 1170 to Guglielmo, an Italian merchant, and customs official. Guglielmo directed a trading post in Bugia (Béjaïa) in modern-day Algeria, the capital of the Hammadi empire. Fibonacci travelled with him as a young boy, and it was in Bugia (Algeria) where he was educated that he learned about the Hindu–Arabic numeral system.

Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic. He soon realized the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation), which popularized Hindu–Arabic numerals in Europe.

Fibonacci is thought to have died between 1240 and 1250, in Pisa. (Wikipedia)

The key phrase from all this history is this:  Sequence of Numbers.  While Fibonacci was pretty darn keen about substituting Hindu-Arabic numbers for Roman numerals (because the Hindu-Arabic numbers we use today make computations so much easier – can you imagine three-digit addition with Roman numerals?), he was also fascinated with number sequences.  What made it even more fascinating was Fibonacci saw his number sequences reflected in nature, art, and architecture.  The way the number sequence works is though simple addition.

Begin with the number one.  Add the number before one (in this case, zero) and one together to come up with two.

1, 1, 2

Now add the 2 and the second one together to get 3.

1, 1, 2, 3

Keep adding the new number to the one immediately preceding it, and this is the Fibonacci Sequence:

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

So, how do these numbers work in day-to-day life?  The Fibonacci Sequence is seen in tree branches, as the sequence begins with the trunk and then works out and up in the branches.  If you’re a storm tracker, the swirling masses of hurricanes is a great example of the Fibonacci numbers at work.  The numbers in sequence, with one being the eye or center of the storm, expands in a tight formation of the Fibonacci numbers.  Pinecones, flower heads, galaxies, flower petals, nautilus shells, and humans all exist as great examples of the Fibonacci Sequence.  However, in nature, instead of these numbers lining up in horizontal or vertical row, most of the time they appear in a spiral sort of form like this:

The Fibonacci spiral or Fibonacci sequence is one of the mathematical formulas par excellence in terms of the proportion aurea or divine proportion. The number is repeated infinitely, we can find it both in the organic form of nature and in the galaxies of the universe itself. It consists of drawing circular arcs connecting the opposite corners of the squares adjusted to the succession values, by putting side squares 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on.

Which, if you look closely, can be found reflected in nature and architecture.

By now, if you’re still reading about the plethora of numbers I’ve thrown at you, I bet you’re wondering what does any of this have to do with quilting?  And that’s a fair question.  But before I answer, I would like to ask you to do something:  Think about a Log Cabin quilt block.  For my example in this blog, I am using this Log Cabin variation.

If you follow the progression of this block, in the lower right-hand corner are two squares of equal size.  For the sake of keeping things simple, let’s say both of these squares are 1-inch.  So, these are the first two numbers of the Fibonacci Sequence: 1, 1.  The patches adjacent to the left side and top are twice the size of the joined first two patches, making them 2-inches.  This is the third number of the Fibonacci Sequence:  1, 1, 2.  The next set of patches would be 3-inches (1, 1, 2, 3) and the sequence would continue on until the block was as large as you needed it to be.  It would also be balanced and pleasing to the eye because it used the Fibonacci Sequence during construction.

Fibonacci may also be observed in some applique pieces, especially flowers.  Note the spiral formation in the rose and the petals in the other flower.

Even star blocks reflect the Fibonacci Sequence.

So, at this point we know who Fibonacci was, his number sequence, and how to compute his number sequence.  Which brings us to the main topic of this blog: How do we use the Fibonacci Sequence in our quilting?  Is it anything like working with the Golden Ratio?  Let’s tackle the first question before the second.

One of the easiest quilts to make is a Rail Fence Quilt.

This quilt is made of blocks like this:

Which are simply strips cut the same length and width and sewn together.  There are two things I love about Rail Fence Quilts.  First, they are super-duper stash busters.  If you like to cut your leftover fabric into some kind of manageable stash-keeping system, Rail Fence Quilts is a great vehicle for this.  After you’ve constructed your quilt top, cut the remaining fabric into strips in the width of your choice (I usually cut mine 2-inches wide).  Store them somewhere until you have enough to begin piecing your blocks.  Sew the strips together lengthwise until the sewn together strips are as wide as you’d like the blocks, and then cut the strip apart into blocks. 

The second reason I like the Rail Fence Quilt is, despite its simplicity, there are some serious design variations you can throw out with these strippy blocks. 

But…let’s play with this block by throwing in the Fibonacci Sequence.  Going back to our initial sequence (1, 1, 2, 3, 5), what if we cut our strips the width of the Fibonacci Sequence?  How would that look in a block? It’s not hard to figure this out.  Remember, we always work with the finished size of the block, and then add a ½-inch seam allowance.  This would the actual width we cut the strips.

Strips One and Two: 1 + ½ = 1 ½-inches

Strip Two: 2 + ½ = 2 ½-inches

Strip Three: 3 + ½-inches = 3 ½-inches

At this point, our block would be 7-inches finished, and would look something like this:

And there are lots of fun ways to lay this Rail Fence Quilt out.

You can have even more fun by dividing the 7-inch block into two rectangles with coinciding Fibonacci numbers. 

Rectangle One could measure 3 x 7-inches finished and the second rectangle could measure 4 x 7-inches finished.  Within these two rectangles you could piece units within the Fibonacci Sequence and come up with something like this:

Depending on your color choices, you can get some mind-bending layout ideas. 

Please realize, too, you don’t have to stick to strippy blocks when Fibonacci is in play.  Let’s revisit our sequence again:  1, 1, 2, 3, 5.  As long as the units in your block (even if the units are pieced) measure these finished sizes, your good to go.  In other words, you could have a simple pieced block with the units measuring 1, 1, 2, 3, 5 like this:

Or you could take those units and piece them.  As long as the finished measurements of each unit come up to 1, 1, 2, 3, and 5, you’re still well within the Fibonacci Sequence. 

The Fibonacci Sequence doesn’t just pertain to pieced blocks.  Applique blocks also use them.  The circles in this appliqued quilt

Used the Fibonacci Sequence to determine what size they needed to be. And those applique blocks with flowers all over the place?  Many times the designers of those blocks used the Fibonacci Sequence to determine how big to make them.

Let me leave you with one last very practical example of how we quilters can use the Fibonacci Sequence.  Most of us have one of these:

Somewhere in our stash.  We saw the panel and we liked it.  Or in my case, I keep a few nursery panels around to make quick baby quilts for gifts.  The quickest and easiest way to deal with panels is to simply put borders around the panel.  An afternoon of quickly cut borders, sewing, and some simple quilting results in a nice baby shower gift.  We can use the Fibonacci Sequence to determine the borders’ sizes.  However, I can hear some of you right now, “One-inch borders are so narrow to sew and don’t show up well against my panel.”

That very well may be true, but allow me to also throw in this caveat – you can make the first border 2-inches wide (you simply add the 1, 1).  The second border would also be 2-inches wide.  The third, 3-inches wide, and the final one would be 5-inches wide.  You also don’t necessarily have to begin the sequence with one.  You could begin with two.  In this case, your sequence would look like this: 2, 3, 5, 8, 11.

Finally, let’s look at the last question:  Does the Fibonacci Sequence and the Golden Ratio have anything in common?  Truthfully, beyond both producing pleasing blocks, applique, or quilt size, no.  But it is amazing how close the numbers come without landing squarely on top of each other.  We use division with the Golden Ratio (1.618).  If we take the Fibonacci Sequence numbers and divide them by their preceding number, the answers look like this:

2/1=2, 3/2=1.5, 5/3=1.667, 8/5=1.6, 23/8=1.625, and 21/13=1.615.

So you can see that the final number (21/13) gets super close to the Golden Ratio.

The Fibonacci Sequence is another tool you can tuck away in your quilting toolbox and bring out when you want to alter a quilt pattern or design your own.  It also comes in pretty handy when you’re dealing with some orphan blocks in different sizes or a quilt panel.  And always remember, no matter how traumatic your high school math classes were, numbers are our friends.  They never lie.

Until next Week, remember the Difference is in the Details!

Love and Stitches,


6 replies on “Quilting with Fibonacci”

It’s been a long time since I thought of Fibonacci! Thanks for the interesting post! I’m going to mention it to the math instructors when I’m at school subbing tomorrow.

Fantastic! I am the math guru for the quilting team of my wife and me. Fibonacci is a favorite of mine so your article has given me a lot to think about and design with. Your golden ratio article is next.

I am so glad you enjoyed my Fibonacci blog! I hope you enjoy the Golden Ratio one. Quilting math isn’t hard. And numbers don’t lie!

Thanks for reading and taking the time to comment.


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