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.
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.
I love it! Just the kind of minutiae my husband and I are always wondering about. And I’m a big believer that no knowledge is wasted 😃
🙂 I know it’s not for everyone, but then, what is? Glad you enjoyed it.
I actually followed all of that and understood the why and how! It is funny what subjects can come up when traveling and noticing the surroundings. I was actually thinking about this same subject just last week, but hadn’t gotten to figuring out the math. Glad you and Jim did it for me. Thank you. I was wanting to apply it to a baby quilt for a different look than the normal 1/3 size of the block for the corners of snowballing.
By the way, your block is beautiful. What did you think of the paper piecing? Was the center of the block a fuzzy cut to make the square?
Hope your weather gets warmer soon!
Your comment made me smile so thank you for that! Thanks, too, for the nice comments on the block. Toby Lischko, who did our workshop, also did a lecture on using symmetries in your fabric designs to create interesting looks. So I did use fussy cutting to capture the look that I got. I used mirrors to identify what section I’d pick. You might have fun looking at this post about using mirrors. https://catbirdquilts.wordpress.com/2018/08/26/mirror-mirror/
The paper piecing was surprisingly easy. I had done a little before. Her method is to use freezer paper to secure the shape as you’re stitching it. I would have a hard time explaining it. She does teach for Craftsy and she has really good reviews, so you might check her out.
When confronted with problems like yours I tend to resort to gridded paper, pencil and scissors. I have to construct what I’m dealing with, and then I take measurements off my pieces. I began but never finished an online course called Math for Quilters.
Graph paper is one of the best tools ever. 🙂
Great post, Melanie. I appreciate your taking time to work out the calculations. And I hadn’t thought about snowball blocks not being true octagons. (Guess they are just polygons….which makes me think of making a bird-themed quilt named Polly Gone….)
Thanks, Nann. A snowball block *is* an octagon, but not a “regular” one, where regular means that the line lengths and angles are all the same. So a square is a regular quadrangle, but a rectangle is only “regular” if it is square. 🙂
I’m afraid I’m with Kate and the other Kerry on this–you kind of lost me! Maybe the nicest thing about quilting is that it works for those who love and can manage the math but also for the rest of us!
Yes, that’s absolutely one of the best things about quilting. 🙂
I have the idea of finishing a corner of a quilt that is snowballed and could not figure out this math. I found if I gave one dimension to the engineers I work with they would to the pathagorem therom very quickly, and then I do not have to overheat my No Math Allowed quilting brain. LOL, I will have to apply this and see if it will compute with no one else invovled.
Give it a try, and if you have questions, let me know! Of course if you don’t need the regular octagon, you can make it much simpler than I did. Thanks for reading and commenting.
You know me and quilt math. I got about three lines down and started doing a high pitched whine of terror. It’s magnificent, but *look* at it: algebra, geometry, converting decimals to base 16…. When things start getting too hairy, I generally get out a piece of paper and draw what I want, and then cut and paste bits till I have something that fits. Your way is definitely cleaner and quicker, but my way avoids the whole cowering in the corner saying “no no no no no” bit! 🙂
And shockingly both ways work! 😉
Whoosh over my head and my eyes glazed over! Bit like when my husband talks finances to me, so don’t feel it’s personal! LOL! Love the New York Beauty with the snowball – ingenious!
Thanks, Kerry. And I’ll freely admit to my eyes glazing over at some things, too. I don’t take it personally!
Excellent math lesson!
Thanks, Chela. And now I have it in writing and will be able to find it again!
I saved it too!
You’d better keep that math nerd Jim contented. “He’s handy” to very loosely paraphrase Red Green.
I try to keep him happy!
If math classes were this practical we’d have a nation of Einstein’s!
It certainly helps to see the practicality or the context of something, doesn’t it? Thanks, Jim!
I am going to save this as it is bound to be helpful at some point or another. Your block is lovely.
I’m back in math class. I’m sitting here, going “Huh?” You’d have to walk me through this over again, very slowly. Haha. It’s me, not your post, which is very thorough. I like how your medallion worked out though!
If you ever want to make a regular octagon for a quilt block and want help, please let me know. Then we can both refer back to this post. 🙂 Thanks for taking a look!
Oh my goodness – I never tied together the similarities between the octagon Stop sign and a Snowball block! I love this post with the math and design concepts/process. You are such a smart cookie Melanie!
Kind of funny to follow up the Liberated baby quilt with a completely nerdy, mathy post, huh?