Tag Archives: Quilt math

Math Is AWESOME!

Yes, awesome, inspiring feelings of awe; magnificent, amazing, stunning. Math rocks!

I truly enjoyed creating Untied, with its free-form, no math construction. Though I used a ruler for parts of it, it wasn’t because the outcome depended on the measurement. Instead, at those instances I was only interested in straight, and sometimes parallel, lines.

20160401_182836However, I love the challenge of a technically difficult quilt, too. For a change of pace, I began a brand new project. The inspiration for it is a piece of fabric I bought several years ago. It’s a fussy historical print that I’ve always loved. However, it’s fussy and historical, and the colors are just off, all of which have made it hard to use. If it is cut into small patches, the impact of the print disappears, but large pieces would require designing just for it. So I am.

I started by pulling from stash, which is how almost all my quilts begin. I pulled all the blues that could work with that print, which meant they had to have a tinge of green and a little grey. It’s the color I learned as French blue. As it turns out, I don’t have a lot of blue with that, and all of it is in scraps or small pieces, other than the inspiration print. We’ll see how far I get before needing to buy something.

Burgundy reds, creams, browns, and cheddar oranges also came out of stash, including from my scrap drawer. I’m trying to commit to using my scraps more effectively. They come in handy, as I’ll explain later.

My center block is 18″ finished. I knew I wanted to turn it on point twice. The method is exactly the same as used for the economy block setting. This is also called “square in a square.”

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And this is where the first round of harder quilt math comes in. (It’s not very hard, just a thing to learn, or store so you can look it up.) As noted in the linked post, when setting a square on point twice, the finished size of the resulting block is twice the dimensions of the initial finished square. For my 18″ block, I would end up with a 36″ center, once trimmed and finished. (See below for the calculation and reason why.) 

What a great way to quickly increase the size of a quilt!

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After trimming, I added a 1″ border all the way around, taking it to 38″ square. That’s an odd size.

Options for a 36″ Edge
Most quilt blocks are square, or are rectangular with length twice the width, like flying geese blocks. I typically divide a border length into a number of equal increments to find how many blocks could fit along the edge. For example, if I put a block border directly along the 36″ center, I would divide 36 in various ways to get possibilities. Let’s start with whole inches.
36/18 = 2. I could have 18 blocks measuring 2″.
36/12 = 3. I could have 12 blocks measuring 3″.
36/9 = 4. I could have 9 blocks measuring 4″.
36/6 = 6. I could have 6 blocks measuring 6″.
But remember we could also go to half-inches, such as
36/8 = 4.5. I could have 8 blocks measuring 4.5″.
36/24 = 1.5. I could have 24 blocks measuring 1.5″.

You can see there are actually infinite variations, though I like to stick to the ones that are easy to construct.

Options for a 38″ Edge
But I didn’t want to put a block border directly against the large center. I wanted the hard line of a narrow border before anything else, to contain the pale toile. That gave me 38″, which is harder to divide nicely than 36″.
38/19 = 2. I could have 19 blocks measuring 2″, but this was too narrow to work well with the proportions of the center. And 19 is a prime number, so I couldn’t subdivide it into whole numbers.
I could shift to fractionals, such as
38/8 = 4.75. Sure… This actually would work fine with something like HST or hourglasses.

My go-to blocks are half-square triangles and hourglasses. I don’t want this quilt to look like others I’ve made, so it’s good to try something different.

Then I had a thought, a math thought! What if I divide 38 by 1.414? (That actually was the first thought. Then I thought…) If I turn blocks on point rather than set them straight, I would need to know the number of blocks using the math for the diagonal.

Here’s the concept. The picture below shows a triangle that is 1″ on the vertical and horizontal sides. The diagonal measures 1.414″, which is the square root of 2″. (Check with your calculator if you don’t believe me.)

Sq_rt_of_2 Pythagorean Theorem
For a right triangle, the square of the length of the diagonal (hypotenuse) is equal to the sum of the squares of the other two sides. We often see this expressed as a² + b² = c². To find c, take the square root of c².

In the case to the left, a = 1; b = 1; c is the square root of 2, or 1.414.

In fact, the diagonal of every square is 1.414 times the length of the side. So
length x 1.414 = diagonal
1″ x 1.414 = 1.414″
2″ x 1.414 = 2.828″ or close to 2 7/8″
3″ x 1.414 = 4.242″ or close to 4 1/4″
4″ x 1.414 = 5.656″ or close to 5 5/8″
and so on.

That also means that if I know the diagonal of a square, I can find the length using
diagonal/1.414 = length. For example
6″/1.414 = 4.243″, or very close to 4 1/4″.
38″/1.414 = 26.875″. Hooray!!

Huh? Why is that good? 26.875 is very close to 27, which is a very easy number to use. For instance,
27/9 = 3.

I could use 9 blocks measuring 3″, turned on point. (There were other options, too, such as 6 blocks at 4.5″.)

3 x 1.414 = 4.242, the diagonal of a 3″ square, and the width of a 3″ block when turned on point.
4.242 x 9 = 38.178, or very close to 38″.

So if I use 9 3″ blocks turned on point, I’ll have the length needed for a 38″ border.

(And to review the concept in another way, let’s go back to the 18″ block. When I turn it on point twice, I’m doing this:
18″ x 1.414 = 25.452″
25.452″ x 1.414 = 36″.
This is the same as 18″ x 1.414² = 18″ x 2 = 36″.) 

I chose to use the historical print and other blue scraps as the 3″ finish squares. For the background fabric, a light background would give good value contrast for the blues so they stand out well. I picked a pink and tan print on pale cream. The pink works because the reds have a pink cast. Right now I’m still working completely from stash and scraps. I was able to cut most of the setting triangles from a couple of larger pieces, but for the last few I had to sew scraps together to cut patches.

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Can you find the seams where the setting fabric scraps had to be sewn together?

With the on-point border added, my current center is 46.5″ finished, another weird number. I plan to add an unpieced strip next, but I haven’t decided its width. I could add a strip to take it to 48″, 49″, 50″… So the next design decision, really, is the border after that, which will determine the width of the strip.

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I feel really fortunate to have the math skills as part of my craft toolbox. You can make beautiful quilts without knowing how to do any of this, but knowing increases the options open to me. Math is awesome!

Proportion, Part 3

Posts 1 and 2 on medallion quilt proportion addressed the center block. What size should the design be within the center block? What size should the block be as compared to the whole quilt? If you want it bigger, or just to appear bigger, how can you do that?

Who gets to decide the answers to these questions? You do. There are no rules for medallion quilt size, or for any of its components. (Believe me. I’ve looked.) And even if there were, I’d encourage you to break them whenever you want. You get to make your quilt your way.

The same is true for medallion borders. Their primary role is to support the center, the star of the show. But I’ve looked at historical medallions, and I’ve looked at contemporary show quilt medallions. I’ve read the only “rules” on this I can find (and I don’t necessarily agree.) Borders, and design components within borders, can be whatever size you want.

Let’s think about the various aspects of borders that affect proportions of the quilt:

1. How much of the total area of the quilt is borders, as opposed to the center (Part 1)
2. How big an individual border is, relative to other borders, the center block, or the whole quilt
3. How big the shapes within a border are, relative to that border, to other borders, to the center block, or the whole quilt
4. Visual weight characteristics, as discussed in Part 2.

Let’s start with how big an individual border is relative to the center block. Again, there are no hard rules here. But because the center block is intended to be the primary focal point, it makes sense that no border should be as wide as the center motif.

Indeed, Joen Wolfrom makes the argument, seemingly based on her personal preferences, that the total border width on a side should be less than half the width of the center. Here is advice she gave to entrants of the OEQC 2009: “If you are designing a medallion quilt … Generally, each side border should be no more than half the width of the medallion center. If the borders become much wider than the medallion center, the borders take over the design.” This is unnecessarily restrictive.

Okay, we’ve established that there is a “rule” for the width of borders as compared to the center, and we can ignore it. Let’s move on to the width of borders as compared to each other. Here again, there is no rule you need to follow, but there are some aesthetic principles that might be helpful.

The Golden Ratio is a geometric calculation that has been studied for at least 2,400 years. Some believe that it was used to develop ancient Greek architecture, not to mention innumerable other applications since. To simplify substantially, it is the ratio of 1.61 to 1. (huh?) And to make that applicable it suggests that two measures, to be aesthetically pleasing, should be scaled so one measure is 1.6 times greater than another.

For example, if a border is 3″ wide, the next border might be 3″ x 1.6 = 4.8″ wide. The next after that might be 4.8″ x 1.6 = 7.68″ wide. Here is an idea of what that looks like:

Proportion golden ratio

To approximate it in EQ7, this illustration uses a 15″ center block, 3″ first border, 4.75″ second border, and 7.5″ final border. The total quilt width is 45.5″, so the center is about a third of the total. To play devil’s advocate, what if we turned the order around? Does it look as good?

Proportion golden ratio 2

If I ignore the color/value use, I think yeah, it does look pretty good.

Okay, that is a rule that is worth keeping in mind. And there are other mathematical rules or progressions that could be tried, such as the Fibonacci sequence.

For practicality, though, designing pieced borders to fit the calculation could be rather torturous. My number one rule is to find the process rewarding, so I even if I use these mathematical progressions, I’ll certainly modify sizes to make design of individual borders easier.

That seems like a good segue to the next topic on proportion, the design of one border and how its shapes and sizes relate to the rest of the quilt.

Math Hater?

We all learn and think a little differently. Some people’s brains absorb images easily, while other people work in words. Still other people process numbers nimbly. But rarely do I ever hear of someone who hates pictures or hates words. They may say, “I learn more easily when I can see it done.” Or, “Pictures confuse me, but when it’s written out, I can get it.”

Frequently people from all kinds of backgrounds say, “I hate math,” or “I can’t do math.” It’s absolute, not just an expression of preference. Often they give an excuse, such as, “I’m an artist … a writer … a musician … just a mom … ” or “I had bad teachers in grade school” or “I never learned anything I would really use.” Often the math-aversion is expressed as a boast, rather than a regret.

Consider, many people are willing to say, “I can’t do math.” Few of those same people would proudly admit, “I can’t read.” Why do you think these lead to different responses?

Perhaps you wonder why this discussion belongs on a quilt blog. It belongs because quilting is full of math. Begin with buying fabric. How many yards do you need? Why do you need that many? Do you need to worry about shrinkage? (YES.) How many patches or strips can you cut from your purchase? How much will it cost? Do you know if the clerk rang up your purchase correctly?

Then move on to more complicated stuff, like when you’re designing your own projects. Maybe you have a bunch of blocks you’ve made or collected in swaps or purchased at an estate sale. How big of a quilt can you make from it? What if you use sashing? What if you turn them on point and use sashing? How many blocks do you need to make a queen-sized quilt? Or how about very complicated stuff, like how much should you charge for a quilt you are selling?

There are various ways to answer questions like these. Alternatives for the quilt size question include laying out all the blocks in various configurations and estimating how much disappears within seam allowances. Or you can draw it all out on graph paper or in quilt design software like EQ7. You can look it up in charts, if you happen to have the right resource and understand how to use it. Or you can do the math.

Very little of the math is actually hard to do. Almost all of it is very simple addition, subtraction, multiplication, and division, stuff we learned to do in grade school. There’s some geometry with shapes and areas, and a little algebra from time to time. But I’ve never had to write a geometry proof to quilt. And all the formulas are easy to find, and easy to calculate with your handy dandy electronic calculator (or computer or smart phone app.)

A few days ago I read a question from a quilter about how many blocks she would need, given a few assumptions. I wrote out the process she could use to calculate it. It’s a few steps, but by doing a few steps she can use the same process regardless of size of blocks she has, or size of quilt she makes. She can figure it out. SHE HAS THE POWER.

When you add basic quilting math to your set of quilting skills, you make yourself more powerful. You are less dependent on someone else’s patterns and instructions. You are freed to be more creative. You can figure out whether the remnant in your stash is enough for the patches you will cut, or if you’ll need to find something else. You can charge a fair price for your work when you sell your quilts.

Are you a math hater? What math quilting skills do you need to improve? Do you want to improve? Are there formulas or processes you would like to see here?

My previous blog posts with mathy stuff in them include:

Fraction Conversions
Economy Block ANY Size (With Cheat Sheet!)
How to Set a Block On Point
Design Process — Border Size Problems and Solutions
Tutorial: Straight-Grain Binding

Let me know how you see yourself in the math arena. If there is interest, I can build some tutorials on quilt math, and help you be more powerful.

Economy Block | Square in a Square

This morning the Quilt Alliance posted about Two Altheas and a Square Within a Square. The Altheas are American tennis champion Althea Gibson, and quilter Althea Orr Diament. Diament pieced and quilted the lovely quilt shown in the blog post. Please take a look.

I note this in particular because my all-time most viewed post is Economy Block ANY Size! (With Cheat Sheet). There must be some romance to this block that has made it so popular. Its graphic simplicity allows a sparkle as the primary block or an accent in a quilt.

And it’s easy to make using my instructions, though the trimming is a little fiddly.

Certainly there are many more ways to set it than side-by-side across the vast array of a quilt. If you didn’t look at the Quilt Alliance post yet, please do. The setting there is interesting and fresh. And my post showing seventeen ways to set economy blocks should spur a quilter’s thinking for more ideas.

Here is the baseball medallion that uses the block above.

Have you made a quilt using this block? Was it the primary block, an alternate, or an accent?

Fraction Conversions

Decimal to Fraction
.0625 = 1/16
.1250 = 1/8
.1875 = 3/16
.2500 = 1/4
.3125 = 5/16
.3750 = 3/8
.4375 = 7/16
.5000 = 1/2
.5625 = 9/16
.6250 = 5/8
.6875 = 11/16
.7500 = 3/4
.8125 = 13/16
.8750 = 7/8
.9375 = 15/16

Yardage — how many inches is
1/8 yard = 4.5″
1/4 yd = 9″
3/8 yd = 13.5″
1/2 yd = 18″
5/8 yd = 22.5″
3/4 yd = 27″
7/8 yd = 31.5″
1 yd = 36″