Tag Archives: Quilt math

Turning a Block On Point Twice

Friday Jim and I drove down the Mississippi River from La Crosse, WI. We were returning from a two week trip to see our son, who lives in Washington state. With 2,000 miles behind us on the train, it felt great to switch to our own car.

In Prairie Du Chien, WI, we stopped for lunch. On one wall of the diner hung a quilt with a patriotic theme. It was a medallion quilt, centered by a stylized American flag. The flag block was turned on point twice, emphasizing its importance and creating a nice, large center.

I liked the setting, and especially liked that a non-square rectangle was turned that way. It’s a setting I haven’t used myself.

I’ve written plenty about turning large blocks on point to center a quilt. In one post I described the types of blocks suited for an on-point setting, if it is only turned once. In another I showed how to do that, with the math needed to cut your setting triangles large enough. I’ve also written about turning small blocks twice, creating an “economy block.”

But I’ve never written about turning a larger rectangular block twice. Here are some cool things I learned about it.

*~*~*~*~*

The Part I Already Knew
If you turn a square block twice, you’ll double its dimensions. Consider an example of a 15” block. Turn it twice with an exact (not over-large) setting, and you will create a block that is 30” wide. Using the math for diagonals,
15” x 1.414 = 21.21”.

Now turn it again:
21.21” x 1.414 = 30”.

This block setting is often called an “economy block.” It’s an especially effective way of highlighting a small centerpiece, such as a special or fussy-cut piece of fabric.

Economy block setting – a square turned on point twice.

The Part I Didn’t Think About Much, But Probably Knew, Too
As it turns out, you can do this with non-square rectangles, too.

Non-square rectangle, squared first with unequal triangles, and again with equally-sized ones.

The Part I Didn’t Know, And Figured Out Last Night
The size relationship for both types of blocks can be generalized, and is far easier than multiplying by 1.414. If the length of the inside shape is A, and the width of the inside shape is B, the distance across the diagonal of the interior square is A+B. That means the length of the exterior square is A+B.

In the economy block example above, the interior square is 15”.
The resulting block is 30” square, or 15” + 15”.

In the second example, if the interior blue rectangle is 12” x 18”,
the resulting block is 30” square, or 12” + 18”.

The next time you want to frame a rectangle with setting triangles, remember how easy it is to determine the finished size. Length plus width of the interior rectangle (square or not!) is the width of the resulting square.

Ain’t math fun? 🙂

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Four Flying Geese, Three French Hens, Two …

[I published this in April when I was making lots of geese. Well, I’m at the geese stage again! Since I needed to remind myself of the four-at-a-time method, I thought I’d remind you, too. Cheers!]

Fooled ya, huh? It’s actually “four calling birds…” The geese don’t come in until later, when there are six geese a-laying…

One of the reasons I love flying geese blocks is they can be made almost any size. Blocks that are on grid are less adaptable for sizes. For example, regular 9-patches are on a 3-grid (3 rows by 3 columns) and are most easily made in sizes that multiply easily by 3, such as 6″ finish with 2″ patches, or 7.5″ finish with 2.5″ patches. For a harder one, bear’s paws are on a 7-grid, so are most easily made in sizes like 7″, 10.5″, or 14″. But geese are ungridded. The flying geese I need finish at 8.25″ in length. That’s a weird size, but easy to make. (Half-square triangles and hourglass blocks also can be made any size.) 

There are a variety of ways to make geese, but I only use two of them. One is the stitch-and-flip method. With this, the base (goose) is cut the size of the finished block, plus 1/2″ each direction for seam allowances. For example, if you want a block that finishes at 3″ x 6″, cut the base as 3.5″ x 6.5″. The background (sky) is cut as two squares, both half the length of the finished block, plus seam allowances. Using the same size block, cut two pieces that are 3.5″ square. Pat Sloan shows how to put these together to make a perfectly sized flying geese unit.

The method works great. However, you do have waste triangles of fabric cut off. Some people make good use of them and convert them into half-square triangles for other purposes. I do not. For small flying geese, throwing away the waste doesn’t bother me much. For larger ones, it does.

The other method I use is the four-at-a-time method. Why choose this one? The process allows more efficient use of fabric, because there are no waste triangles. For me, the disadvantage is I have to be more careful of my seam allowance. To adjust for that, I check sizing on the first set I make. If I need to trim slightly, that means I need to use a slightly bigger seam allowance, perhaps only a thread width bigger. Even so, it’s a great way to make a lot of geese quickly and with no waste.

Here is a video that clearly explains the process, as well as a link to another set of instructions from Connecting Threads.

For each FOUR geese units, use 1 large square and 4 small squares.
Large square = finished length of unit + 1.25″
Small square = finished width of unit + .875″ (that is 7/8″)

Example: for four flying geese units finishing at 3″ x 6″, cut 1 large square (the geese) at 7 1/4″, or 7.25″. Cut 4 small squares (the sky) at 3 7/8″, or 3.875″.

Draw a diagonal line across the wrong side of each small square, corner to corner. Arrange two of them right sides together in diagonally opposite corners of the large square, with the drawn lines meeting in the middle. The small squares will overlap a little. Pin them in place.

FG 1
Stitch from corner to corner, a scant 1/4″ away from the drawn line. Then turn around and stitch the other direction on the other side of the drawn line.

Cut on the drawn line between the two stitching lines. The video shows using the rotary cutter and ruler, but scissors work fine.

FG 3
Press toward the sky squares. You end up with two pieces shaped sort of like a heart.

FG 4

On each of those pieces, pin another of the small squares with the drawn line running from the corner through the “cleavage” of the heart. Sew 1/4″ from both sides of the drawn line, as you did before. Cut apart on the drawn line, and press toward the sky triangles.

FG 5

What is the difference in fabric used for the two methods? I needed 32 flying geese units with finished measure 4.125″ x 8.25″. To make FOUR units this size:
Stitch-and-flip requires 4 (units) x 4.625″ x 8.75″ = 161.875 square inches of the base fabric, and
4 (units) x 2 (per unit) x 4.625″ x 4.625″ = 171.125 square inches of the sky fabric.

Four-at-a-time requires 9.5″ x 9.5″ = 90.25 square inches of the base fabric, and
4 x 5″ x 5″ = 100 square inches of the sky fabric.

For this size example, stitch-and-flip requires almost twice as much of each fabric, compared to the four-at-a-time method. For 32 of them, that’s almost a half yard difference for each fabric. I don’t always have that much more fabric available. Note that different sizes of flying geese will have different outcomes on this calculation, because of the proportion of the seam allowance compared to the rest of the unit.

One more alternative is to create the effect of flying geese using half-square triangles. Instead of 32 flying geese, I could have used 64 half-square triangles. I chose not to do this because I wanted the toile of the base fabric unseamed.

Do you use flying geese in your quilts? Do you have a favorite way of making them? Questions or comments?

Four Flying Geese, Three French Hens, Two …

Fooled ya, huh? It’s actually “four calling birds…” The geese don’t come in until later, when there are six geese a-laying…

One of the reasons I love flying geese blocks is they can be made almost any size. Blocks that are on grid are less adaptable for sizes. For example, regular 9-patches are on a 3-grid (3 rows by 3 columns) and are most easily made in sizes that multiply easily by 3, such as 6″ finish with 2″ patches, or 7.5″ finish with 2.5″ patches. For a harder one, bear’s paws are on a 7-grid, so are most easily made in sizes like 7″, 10.5″, or 14″. But geese are ungridded. The flying geese I need finish at 8.25″ in length. That’s a weird size, but easy to make. (Half-square triangles and hourglass blocks also can be made any size.) 

There are a variety of ways to make geese, but I only use two of them. One is the stitch-and-flip method. With this, the base (goose) is cut the size of the finished block, plus 1/2″ each direction for seam allowances. For example, if you want a block that finishes at 3″ x 6″, cut the base as 3.5″ x 6.5″. The background (sky) is cut as two squares, both half the length of the finished block, plus seam allowances. Using the same size block, cut two pieces that are 3.5″ square. Pat Sloan shows how to put these together to make a perfectly sized flying geese unit.

The method works great. However, you do have waste triangles of fabric cut off. Some people make good use of them and convert them into half-square triangles for other purposes. I do not. For small flying geese, throwing away the waste doesn’t bother me much. For larger ones, it does.

The other method I use is the four-at-a-time method. Why choose this one? The process allows more efficient use of fabric, because there are no waste triangles. For me, the disadvantage is I have to be more careful of my seam allowance. To adjust for that, I check sizing on the first set I make. If I need to trim slightly, that means I need to use a slightly bigger seam allowance, perhaps only a thread width bigger. Even so, it’s a great way to make a lot of geese quickly and with no waste.

Here is a video that clearly explains the process, as well as a link to another set of instructions from Connecting Threads.

For each FOUR geese units, use 1 large square and 4 small squares.
Large square = finished length of unit + 1.25″
Small square = finished width of unit + .875″ (that is 7/8″)

Example: for four flying geese units finishing at 3″ x 6″, cut 1 large square (the geese) at 7 1/4″, or 7.25″. Cut 4 small squares (the sky) at 3 7/8″, or 3.875″.

Draw a diagonal line across the wrong side of each small square, corner to corner. Arrange two of them right sides together in diagonally opposite corners of the large square, with the drawn lines meeting in the middle. The small squares will overlap a little. Pin them in place.

FG 1
Stitch from corner to corner, a scant 1/4″ away from the drawn line. Then turn around and stitch the other direction on the other side of the drawn line.

Cut on the drawn line between the two stitching lines. The video shows using the rotary cutter and ruler, but scissors work fine.

FG 3
Press toward the sky squares. You end up with two pieces shaped sort of like a heart.

FG 4

On each of those pieces, pin another of the small squares with the drawn line running from the corner through the “cleavage” of the heart. Sew 1/4″ from both sides of the drawn line, as you did before. Cut apart on the drawn line, and press toward the sky triangles.

FG 5

What is the difference in fabric used for the two methods? I needed 32 flying geese units with finished measure 4.125″ x 8.25″. To make FOUR units this size:
Stitch-and-flip requires 4 (units) x 4.625″ x 8.75″ = 161.875 square inches of the base fabric, and
4 (units) x 2 (per unit) x 4.625″ x 4.625″ = 171.125 square inches of the sky fabric.

Four-at-a-time requires 9.5″ x 9.5″ = 90.25 square inches of the base fabric, and
4 x 5″ x 5″ = 100 square inches of the sky fabric.

For this size example, stitch-and-flip requires almost twice as much of each fabric, compared to the four-at-a-time method. For 32 of them, that’s almost a half yard difference for each fabric. I don’t always have that much more fabric available. Note that different sizes of flying geese will have different outcomes on this calculation, because of the proportion of the seam allowance compared to the rest of the unit.

One more alternative is to create the effect of flying geese using half-square triangles. Instead of 32 flying geese, I could have used 64 half-square triangles. I chose not to do this because I wanted the toile of the base fabric unseamed.

Do you use flying geese in your quilts? Do you have a favorite way of making them? Questions or comments?

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.”

20160401_182819

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!

20160401_182927

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.

20160401_184553

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.

20160401_183045

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.