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.
Catbird Quilt Studio 
Ha! I’ve been exploring mathematical sequences by making strip sets from several of them, so I was entertained to see that you “eyeballed” the center-to-square proportion for your bullseye block rather than using the overworked Fibonacci sequence 😀
Not sure what to say to that. Thank you? 🙂 I read something last year (can’t begin to think of where I found it) about the golden ratio and how it’s a lot of hooey. While there are proportional relationships that we find pleasing, there are a LOT of them, and variations of them, and we don’t usually need to be exacting that way. Just as we find people with all kinds of faces attractive, we can find all kinds of quilt proportions attractive, too. (I’m just talking now, so bear with me.) So maybe the best reason to know the “rules” is to understand why things DON’T look right, rather than why they do. ??? I’ll have to think about that. Thanks as always, Mary!
When do we get to the fun stuff? Seriously, I know how important a good structure is to anything made, but I’m all about the reveal party.
Yeah, coming soon. I really had to do all this thinking before I could get to construction. In this case the math was too fully integrated into the design. And I’m GLAD I don’t have that with many of my quilts! 🙂
And I see how a feathered star works from the last diagram too. I guess I’m more visual than mathematical!
Right, I felt that way, too about the feathered star. Not that awful complicated when you can just SEE it.
I feel like you left us hanging without knowing where we are going from here or why any of these measurements were significant. Or maybe I am just tired! LOL! Somehow I think you will pull it all together for me with your next post.
Yes, totally left you hanging. There didn’t seem to be a good way to show more without showing A LOT more! 🙂 Thanks for hanging in there with me.