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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.

Fibonacci12_246k

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Quilter’s Design Board

design board blocksHow many of you have never played around with our Design Board? Did you know that there are 223 free patterns in three different sizes (6, 10 and 12 inches) and that a new pattern is added each month?

The blocks are grouped by how they are drafted such as 4-patch, 5-patch, 8-pointed star, etc. First, choose a block. You can print out templates for three different block sizes along with a template guide. Then the blocks can be put into a quilt and borders can be changed. There is a yardage calculator that gives the style numbers of the fabrics used in the block and also will determine how much fabric you need based on how many and what size blocks you want to use. It will even give you an estimate of the cost and put your fabrics directly into your shopping cart.

You can view the design in color or by the line drawing.  You can also change the border selection.
You can view the design in color or by the line drawing. You can also change the border selection.

 

This months’ block is Golden Tile. First and foremost, the block gets its name because it contains the Golden Proportions as was explained in a recent blog post. If the Golden Gauge Calipers are opened so that the smaller space fits on the shorter segment of the design, the larger opening fits on the longer segment.

calipers on Golden Tile 2 The design board is limited and is not meant to take the place of your graphics program but serves as a jumping off point. There are some wonderful software programs available which provide you amazing design possibilities. For blocks such as Golden Tile which are directional, you do not get the chance in the design board to see some of the other possible layout variations. If you have a graphics program that allows you to tile, rotate and flip blocks, experiment with different layouts. Here are some variations.

All blocks are oriented the same way.
All blocks are oriented the same way.
Four blocks are pinwheeled and that unit repeats.
Four blocks are pinwheeled and that unit repeats.
In any four block unit, opposite blocks are reversed.
In any four block unit, opposite blocks are reversed.

I hope you take the chance to play around with our Quilter’s Design Board and don’t forget to send us pictures of the quilts you make from it.

P.S. Golden Ratio by accident or design?

Dana, our staffer who did the layout for the blog sent it to me for approval. As soon as I saw her layout, I couldn’t help myself. I had to get out the calipers. So often when we are doing design or layout work, we select the proportions that are most pleasing to us and so many times it seems to fit the proportions of the golden ratio!

golden ratio by accident or design

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Make Way For Goslings

Luke & Gus on a rainy day
Luke & Gus on a rainy day with me in my office

Well, we almost had a disaster at the Beyers Monday morning. We have a pond on our property and every morning when I let out Gus and Luke (our two labs), we walk around the pond together. As we were coming along the dam, two adult geese suddenly flew from the sky, dive-bombed the dogs, then landed in the pond. Seconds later, another pair streaked up the dam and flew into the pond, all of them honking furiously. The dogs, of course, thought this great fun and went in after them.

Then, out of the corner of my eye,  I caught sight of a group of five precious little goslings on the slope of the dam. I knew if the dogs saw them the goslings would be completely vulnerable.  I yelled at Gus and Luke that it was time to eat and started running to the house. That got their attention and they jumped from the pond to follow me. (Well, they are labs, you know, and when you mention food…)

A few minutes later when I went to get the newspaper, I met my neighbor, Kim, who was carrying a camera with a huge lens. She said she was on her way to photograph some baby foxes she had seen. (Is my neighborhood starting to sound like that old TV show, “Wild Kingdom”?) I told her of the near miss with the goslings and invited her to come take a look. We could check to see if all the geese were okay.

goslings

When we arrived back at the pond, there were the goslings swimming along with their parents. Kim got some great photos and when I saw them, I couldn’t help thinking about the blog post two weeks ago which explained how to get colors out of a photograph. I couldn’t resist so here they are.

gosling colors
Colors drawn from gosling photo in Photoshop

The colors immediately reminded me of the version of Crayon Box that is made with the “tropical” set of Palette Pixie Strips. The only thing missing is the blue sky which was there but just didn’t make the photo.

crayon box_edited-1

I suppose I just can’t help myself from seeing color palettes, fabric designs and quilt possibilities all around me. I hope you are letting your world inspire you too.

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Spring is Here With Design Inspiration!

Luke, 3 months and Gus, 8 years
Luke, 3 months and Gus, 8 years

Spring is in full swing here in northern Virginia. How do I know this? Well, it could be the warmer temperatures or cherry blossoms blooming. But it’s not. For me, I know it’s spring because my 14-month-old dog, Luke, has been trying desperately to catch a frog since they have reemerged with the nicer weather.

We have always had dogs. I like having two. The older dog teaches the younger one and they keep each other company. Luke is our newest puppy. He came to cheer Gus up after we lost our Swissy, Gretchen. I have to say that Luke is one of the funniest dogs we have ever had. I will not be able to resist periodically sharing some of his and Gus’s antics.

However, for most of you, spring is about warmer temperatures and whatever it is you have blooming around you. I’m very fortunate that I live close to the Potomac River. I walk along the river three or four times a week. Monday of this week had to be the most gorgeous day of the year. Greeting me was a spectacular array of native flowers—bloodroot, toadshade, Dutchman’s breeches, toothwort.

wild flowers sm
Bloodroot, Toadshade, Dutchman’s breeches

The most amazing, though, were the Virginia bluebells which were at their peak. There was a carpet of bluebells stretching into the woods as far as one could see and reaching in the other direction to the banks of the river.

Bluebells woods and river smI came back fully inspired.

I often take colors from nature and use them in my quilt and fabric designing. Here are the colors of the Virginia bluebells.

colors of Virginia bluebells sm

I think these colors make a beautiful palette for a quilt. How did I get them from the image? It’s pretty simple if you have Photoshop.

  1. Open the image in Photoshop.
  2. Go to the top menu bar to Image/Mode/Indexed Color. Select the number of colors you want to see from the pop-up menu. You can go up to 250.
  3. Next go to Image/Mode/Color table. The chart with all the colors will come up.

bluebells in palette colors

Once you can clearly see the colors in the image, it is easy figure out your fabrics. I’ve used my Portable Palette which has swatches of all 150 of my Palette fabrics.

So take a walk and get inspired. Oh, and Luke? He’s still out there trying to catch a frog.

I know I could catch that frog if he'd just sit still!
I know I could catch that frog if he’d just sit still!