Wind River Beauty, Math Part 2

Wind River Beauty is the  name I’ve given to my current project. I’m using the New York Beauty center (shown below) along with 45° diamond star arms. The overall look is of a Lone Star or Star of Bethlehem variation.

In Math Part 1 I noted that my Wind River Beauty quilt center is 17″. In fact, the block as designed by Toby Lischko finishes at 16″, which is a more typical, or perhaps more useful, size for a quilt block or medallion center. I first built it as a 16″ block according to her instructions, and then I rebuilt the orange surround to finish at the larger size. The 17″ block allowed two things. First, it made the proportions of the block better, once the corner treatment was added. And second, it made the math easier for the next stage.

Proportion

I previously wrote about proportion in three posts (here, here, and here.) In the first post, I covered the proportions of the center block design. In the first picture, the circle is tangent to the sides of the square. The size seems to crowd the square, and may seem “too big.” In the second picture, the circle floats in the square and seems “too little” within a sea of background. The one on the right, to my eyes, seems “just right.” It is still related to the edges of the square while not touching them.

For the Wind River Beauty (my name for the quilt,) once I decided to create an octagon of the background orange fabric, I knew it would need to be larger. A slightly bigger block makes more room for the corners, without crowding the inner circle. As shown below, the width of orange is approximately the same as the width of the teal circle. If the block were 1″ smaller, the orange width would be skimpy. Overall, the proportions are better with it bigger.  

More Math

Another reason the make the block bigger is some more quilt math. The diagonal of the square is 1.414 times the length of the side. (See the Pythagorean theorem review in the prior post for details.) The diagonal measure of a 17″ block is 17″ x 1.414 = 24.04″, or just over 24″. Half of the diagonal, or just to the very center of the block, is 12″, which is much easier to work with than the result from a 16″ block.

Look at the illustration below of an 8-pointed star. The very fine line draws a square, representing the center block of the quilt. The square is 17″ on each side. The diagonal all the way from corner to corner is 24″. Half of the diagonal, represented by the dotted segment B, is 12″. Each of the star points in orange and turquoise is a 45° diamond. All four sides of a diamond are the same length. A is the distance along the right edge of the turquoise point. A and B are the same length, 12″. With the 12″ length I could build out the star with relatively easy piecing.

As mentioned before, the design of the quilt is a variation of a Lone Star or Star of Bethlehem quilt. The sketch from EQ8 below gives a glimmer of how this will work. On this picture I drew two squares the same size. One is rotated 45° from the other, around the center point. You might be able to see that the octagon formed by the overlap is marked with a heavier line.

If you compare the illustration with my center block above, you can see that the corners of my center block, in purple, are like the corners beyond the octagon.

In the next post on this quilt, and past most of the math, I’ll show you next steps for construction.

Wind River Beauty, Math Part 1

In 2017 Jim and I drove thousands of miles in a number of different trips. When you’re in the car together that much, literally a few inches apart, it helps to have entertainment. Fortunately, we like to talk with one another, so the types of things one might muse about silently instead become topics for conversation. For example, after noticing a stop sign and considering how it looks like a snowball block, I asked, “If you start with a square and want to make a regular octagon from it, how do you calculate the length so each of the 8 sides is the same?” Huh?

Okay, look at the two illustrations below. The one on the left is a stop sign. It’s a regular octagon, meaning that all of the angles are equal and all of the lengths are equal. The diagonal segments of the octagon are the same lengths as the horizontal and vertical segments. I noted the dimensions as a for the vertical and horizontal segments, and c for the diagonal segments. As you can see, a = c. (Click the image to open the gallery and see larger.) The segment lengths are all the same. The dotted segment noted as b is not part of the octagon. If you extend the vertical and horizontal lines to create a square, b is the extension.

On the right is an illustration of a square, red & white snowball block. (This specific snowball block is designed to pair with something like a 9-patch block.) For the octagon (white, 8-sided shape,) the angles are all the same. However, the lengths of the octagon line segments are not the same. The diagonal segments of the octagon c are longer than the horizontal and vertical segments a. Why? For this particular block, each side of the square is cut in thirds; a = b. Going down the left side of the square, the top red segment b is equal in length to the center white segment a, which is equal to the lower red segment. The equal lengths make it easy to pair this block with a 9-patch. But the equal lengths of a and b mean the diagonals c are longer by a factor of 1.414. The general idea is the same for all snowball blocks, with the length of c the diagonal dependent on the length of the two triangle legs. See the primer on the Pythagorean theorem at the bottom of the post if you want to know more. 

So how do you take a square and make a regular octagon from it? I’m not bad at math but will be the first to admit I didn’t learn my geometry. Jim worked it out for me. What he found is

b = .707a
2b = 1.414a
==> side of the square = 2b + a = 2.414a

Why does it matter? At the time it was just curiosity, but I quickly found a project to apply it. In spring of 2018 I took a workshop with Toby Lischko on making New York Beauty blocks. She taught a simple way to use curved rulers and paper piecing to create these lovely, complex blocks. This was mine.

After I made it, I thought about how to use it to center a quilt. My design idea would work best if the center was a regular octagon.

With Jim’s formula in hand, knowing the size of the square, I solved for a and b, which let me know how big to cut the stitch-and-flip squares to make the corners.

I wanted a center block finishing at 17″. (Why 17″? That comes later, more math!) That means
side of the square = 2b + a = 2.414a
17 = 2.414a
a = 17/2.414 (just dividing both sides by the same number to solve for a)
a = 7.04, or just barely over 7″.

Since the finished side of the square is 17″ and a (the center segment) is 7″, the other two segments b are 5″ each. I cut my stitch and flip corners 5.5″ each. This is the result. The finished length of the diagonals (along the purple/orange seam) is the same as the finished length of the orange segment along the horizontal and vertical sides of the square. 

Pythagorean Theorem

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.)

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; a² = 1; b = 1; b² = 1; a² + b² = c² = 2; 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″.
This is also useful in the next step of the Wind River Beauty.

Agreed, you gotta be something of a math nerd to work through all this. I’m glad all my quilts don’t require this process, but it’s a wonderful tool to use for a few.

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? 🙂

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.

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.

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

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.

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.

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.

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

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.

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?