# 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.

# Cheshire Cat?

Now you see her, now you don’t? Yes, that’s me! Two weeks ago I opted out of my own 30-day challenge, stopping at 20 days in a row of blogging. And since then was Thanksgiving, with several days of family fun. Besides that, I’ve been trying to finish some projects — don’t know if you know this, but the end of 2017 is rolling right up! (And not a moment too soon, huh? Heckuva a year, and not all in the good way…)

Right now I have three quilts in progress. Two are from my medallion class. Christmas Is Coming! needs binding attached.

The other is the bear’s paw quilt, which is still on the frame. Well, in fact, it is AGAIN on the frame. It is again on the frame because I’m having thread tension trouble on my longarm. Remember this?

This was the worst of it, but not the last of it. After struggling with getting the tension settings improved, I decided to take the quilt off the frame and putting a testing sandwich on. Prior to taking it off, I basted all the way around the edge, and also used great big stitches to baste through the body of the quilt. The basting stabilized the piece, so layers would stay put for returning to the frame later.

My test sandwich got covered with stitching. I managed to get the top and bottom tension adjusted well, but still had intermittent messy looping areas on the back. When the tension is BAD and there is looping, it’s because the tension is bad. When the tension is GOOD and there’s looping, it’s often because the thread is catching somewhere, like a rough spot in the thread path. I’ve done this long enough to know some of the places to look. (I have a long list of them, if you are interested.) I worked through all those things that I could. While it improved, it still wasn’t as good as it should be.

I called the company and spoke with the head technician. He agreed I’d done all the right things and asked me to bring the machine to the factory. Fortunately, that’s only about a half hour away from me, so I took it the same day. After two hours working with a technician, the best we could come up with was replacing an inexpensive part, the last thread guide above the eye of the needle.

Before returning to my bear’s paw project, I wanted to test it again on something small. Son’s fiancee likes seasonal decorating and I hoped to make a table runner for her for Christmas. I figured that would be a good project to test quilt. After all, if there was looping on the back, it wouldn’t matter. When used as intended, no one would see the back!

I tend to make things more, rather than less complicated. So I had to fight my instincts and make this a simple project. I made three puss-in-the-corner blocks with fussy-cut centers. (Yeah, I couldn’t go all the way to simple!) I set them on point and framed them with a border. I had a piece of appropriate fabric for the back. So I loaded it all up and quilted it.

(There is more to the story of how the quilting went and what happened next. That part of the story will come later this week.)

The last step on the table runner was the binding. I had just the right amount of just the right fabric. However, when it came to attaching it, I wasn’t sure how! All my quilts prior to this have had squared 90° corners. This also had 45° angles. Have you used them before, or other angles than 90°?

This morning I watched a video tutorial that explained how.

Here is my finished table runner.

The table runner has the edge of red binding. Underneath it is another quilt on the table.

Taking a couple of extra minutes to fussy cut the centers made these simple blocks look fancy.

Can you see the figure-8 Christmas tree stitched into this setting triangle? Another easy way to make an easy project look more intricate.

I’m off and running again. Thanks for reading! I always read comments and try to respond promptly. If I don’t get back to you soon, it’s because I’m offline. See you soon!

# Two Ways To Make Flying Geese

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. (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. (See the tip below for trimming your flying geese units.) 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. The photo below is a little murky. The small squares are of dark blue, with wrong side up.

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’ll use this example, with flying geese units with finished measure 3″ x 6″. To make FOUR units this size:
Stitch-and-flip requires 4 (units) x 3.5″ x 6.5″ = 91 square inches of the base fabric, and
4 (units) x 2 (per unit) x 3.5″ x 3.5″ = 98 square inches of the sky fabric.

Four-at-a-time requires 7.25″ x 7.25″ = 52.5625 square inches of the base fabric, and
4 x 3.875″ x 3.875″ = 60.0625 square inches of the sky fabric.

For this size example, stitch-and-flip requires substantially more of each fabric, compared to the four-at-a-time method. While that might not seem like much, if you need a lot of geese, it adds up quickly. 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.

A trimming tip: if your geese are slightly too big and need to be trimmed, make sure you leave the point, or “beak,” intact. Trimming at the bottom, along the “wings,” will be less noticeable.

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

# Lessons: Unpieced Borders (aka “Strip” or “Slab” Borders)

One of the biggest stumbling blocks making medallion quilts is getting borders to fit. Pieced borders provide their own special challenges. But plenty of quilters, of any kind of quilt, have trouble with unpieced borders.

When I started quilting, almost every block quilt or strip quilt had an unpieced border, or perhaps two. Books on quilting gave recommendations to make the width proportionally pleasing to the center of the top. Too wide and it would look like you’re trying to make the quilt bigger. Too narrow and it wouldn’t have enough visual impact.

Quilt police chimed in with edicts to join lengths together at an angle, or with a perpendicular seam, depending on their own preferences.

Patterns continually called for cutting border strips across the width of fabric.

Most sources recommended measuring the quilt center in three places (the same direction) and averaging those three. Somehow, that would magically make your border fit.

And occasionally I saw instructions to mark the center of the border strip, the center of the top, and perhaps at the quarters, and pin those marks. Usually it was recommended to pin every few inches in between.

Whatever these instructions did, they did not, generally, make borders fit better. I’m in a longarm quilting group in Facebook, and badly fitting borders might be the number one complaint quilters have about their customers’ work. They share photos of the worst cases. Once a top is loaded on the frame, it’s easy to see the ripples and waves of excess border fabric. That will not quilt out!

How does this happen? A lot of ways. The most common, I expect, is that people cut a long strip of fabric, probably across the grain, lay it on without measuring or pinning, and sew. As they sew, they smooth the border out, continuing to stretch it farther and farther as compared to the quilt piece below it. The feed dogs pull slightly more against that bottom layer, making the problem even worse. It’s almost like gathering a skirt by having more fabric on one layer than on another, and easing it in. Do you do this? I have!! What a mess!

Medallion quilts often use unpieced strips in the interior, as well as outer borders. Rippling inner borders make it nearly impossible to correctly fit the next borders to them. You can sew it on, but the distortions will make a flat, squared top impossible. The flatter the top, the more easily it can be quilted and the better it will look finished.

There are easy ways to make borders fit better. Here are a few tips.

• Square the center’s corners before attaching a border. Splayed corners will multiply if they aren’t fixed. (The “center” is everything that is already assembled into something that will be bordered. If you are adding multiple borders, each new border becomes part of the center once it is attached.) Use your largest square ruler to check the center’s corners. If the center’s edges don’t align to the square, you can either trim them to square, or adjust seam allowances perpendicular to the edge to improve the shape. OR if the problem is minor, attach the strip border and then trim it to square.
• Cut border lengths along the selvage if possible. The grain is more stable than on width-of-fabric, meaning you’ll get less stretch and distortion.
• If you need to join lengths, use a perpendicular seam. It is easier to align the pieces correctly this way. You won’t have a bias seam to stretch. And the seam is shorter than if joined on the diagonal, so any mismatch in the print extends for a smaller length.
• Determine the correct length of the strip border and cut it to size. (More on that below.)
• Pin. A lot. A lot of pins. Smooth the center its full length and find the middle of its edge. Mark that point with a pin. Find the middle of the border strip and mark that point with a pin. Match the middle-point pins. Remove the pins (each through only one layer) and pin the two layers, center and border, together at that point. Smooth the border strip along the center’s edge until it reaches each corner. Pin the corners. I pin near each corner twice, about a half inch apart. It keeps the layers from shifting at the start and end of sewing. If you’ve measured and cut your border correctly, and if your center isn’t too out of square, the two pieces should fit well together. Pin about every 2″, easing with more pins where needed. (Why pin so much? The pins allow you to ease the layers together where they don’t fit exactly. And they help support the weight of the layers so they don’t shift, which makes sure your seam allowance maintains its width. The bigger the center is, the more weight and the more closely you need to pin.)
• If you have corner blocks, they will be on the third and fourth strips of the border set. Begin your pinning by matching the seams of the corner blocks to the first and second strips of the border set. Continue to pin as above.
• Secure your long seams by backstitching at both ends.
• Use your walking foot (even-feed foot) if it helps keep the layers from shifting, giving a smoother seam.

How to determine the correct length of the strip border
When I began quilting, I relied heavily on a few online sources of information (and back then, there were only a few!) One of them was Bonnie Hunter of Quiltville.com. (She still has great stuff on her site. Take a look around, especially at her Tips & Techniques.) Bonnie has a whole page just on border hints, and this is where I learned to cut and attach border strips.

According to Bonnie, the best way to assure your border will fit and your quilt top will lie flat is to use one measure (for each direction) across the center of the quilt top. She says:

Some people take several measurements across the quilt and average that measurement for borders. (hear me gasping in fright here!) I *NEVER* “average” when measuring for borders because they can still flare, and where they are going to flare the worst is at the center of the quilt sides…That’s why the CENTER measurement is the one to go for. If the ‘averaged’ measurement is longer than the quilt CENTER measurement, you are GOING to have a flared border. If the ‘averaged’ measurement is smaller than quilt  center measurement, you are going to have borders that are too tight for your quilt center, and the center of your quilt is going to balloon out. Just use the center measurement and your quilt will lie flat!

How to get that centerline measurement? Should you hold the quilt top in your lap and move the measuring tape across it a few inches at a time? (Can you see my eyes rolling?!?) No.

1. Lay the quilt top out flat, preferably on the floor. If you don’t have enough room to spread it out, you can bunch up or fold in the sides. But the center must be spread out flat in a straight line, without twisting. Smooth it out without stretching, just to flatten it to the floor.
2. Cut two border strips for that direction and stack them on top of each other. Cut one end perpendicular to the length.
3. Lay them across the center. Start with the cut end flush with the edge of the quilt top. Smooth the strips out so they are flat against the center. Don’t stretch them!
4. Mark the other end of the top strip using a straight edge and pen or pencil.
5. Cut the strips on the marked line.
6. If you will have corner blocks, repeat with the other two strips in the other direction prior to sewing the first two strips on. If you won’t have corner blocks, sew the first two strips on, and then repeat.

Click on any photo below to open the gallery.

I’ve applied hundreds, maybe thousands of borders using this method. My quilt tops are almost always square and flat. Thanks to Bonnie Hunter for the lesson!

# 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.

*~*~*~*~*

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? 🙂