Today I'm writing a little about geometry. Specifically, I want to show you what the r2 term really does to your yarn usage when you knit (or crochet) a center-out project.
If you have ever worked a center-out circle — or a center-out square, or even a neck-down triangular shawl — you may have noticed the project starts off quickly. Those first few rounds or rows just fly off the needles. And then the sprint turns into a run. And the run turns into a jog. And the jog turns into a crawl. And at the end, you find yourself spending an entire evening or more just binding off.
If you look at a circle, you can see that each round gets longer. The plodding tempo makes sense without getting into the math specifics. But what I want you to think about today is, "What does this mean for yarn usage?"
Let's say you want to knit a center out circle with ten individual skeins of yarn, each a slightly different color along a gradient but all the same yardage. What will happen to your stripes?
What happens is the stripes become narrower and narrower. (For those of you interested, there is a square root involved in radius/diameter progression. The numbers go 1, 1.4142, 1.732, 2, 2.236, 2.4494, 2.64575, 2.828427, 3, 3.162 as you track through the square roots of whole numbers.) If you keep knitting far enough, the gradient will "break," meaning it becomes so narrow that eventually a new color can't even complete one full round. For most people, this is not the effect envisioned.
Below is an example of the effect most people want.
The center circle is one unit in radius. The next circle out is three units radius. Then 5, 7, 9, 11, 13, 15, 17 and 19. The center circle and all the rings are the same width, just like a target bullseye. What does this mean for yarn usage?
The area of a circle = π r2
So the area of the center white circle is π 12 = 1π.
The next circle is π32 = 9π. But we want the area of the yellow-orange ring, not the area of a whole circle. That means we have to subtract the white center circle from the yellow-orange circle.
9π − 1π = 8π.
So if you used 1 meter of yarn to work the white center circle, you will need 8 meters of yellow-orange yarn to work the first ring. If you used 3 meters of yarn to work the white circle, you will need 24 meters (3 × 8 = 24) to work the yellow-orange ring.
What happens if we extrapolate the math for the whole figure?
π 12 = 1π
π 32 = 9π 9π − 1π = 8π
π 52 = 25π 25π − 9π = 16π
π 72 = 49π 49π − 25π = 24π
π 92 = 81π 81π − 49π = 32π
π 112 = 121π 121π − 81π = 40π
π 132 = 169π 169π − 121π = 48π
π 152 = 225π 225π − 169π = 56π
π 172 = 289π 289π − 225π = 64π
π 192 = 361π 361π − 289π = 72π
Interestingly, this same proportion works for squares, triangles, and half circles.
Area of a square = a2, where a is the length of a side.
If you draw out the squares, you get the same effect, with the center square being 1 unit on a side, the next square is 3 units on a side, the next 5 units on a side, and so on.
A neck-down triangular shawl is just a square rotated 45° and cut in half. A half-circle shawl is just a circle cut in half. It is the same proportions, since the extra "½" just carries through all the calculations.
How does this affect planning for a project?
If you are purchasing skeins (or making up kits), you will need much larger quantities of yarn for colors farther out. If you are doing your own hand-dying, you can wind off appropriate amounts to produce the desired effect. If you wind 10 meters for the center, then wind 80, 160, 240, 320 and so on for each progressive ring. If you wind a more random amount — say 7 yards — then the progression is 56, 112, 168, 224. You just multiply your beginning amount by 8, 16, 24, 32, and so on.
If you have purchased a gradient yarn or kit, be aware of which type of project will best suit the yarn. If the color amounts are graduated in length, a center out circle, square, or neck-down triangle will work well. If the color amounts are the same, however, consider a scarf, blanket, socks, sweater in the round, or some other pattern where the rows or rounds are close to the same size throughout the project.
Tomorrow: more tips of planning center-out projects when you have a limited amount of yarn.
If you have ever worked a center-out circle — or a center-out square, or even a neck-down triangular shawl — you may have noticed the project starts off quickly. Those first few rounds or rows just fly off the needles. And then the sprint turns into a run. And the run turns into a jog. And the jog turns into a crawl. And at the end, you find yourself spending an entire evening or more just binding off.
If you look at a circle, you can see that each round gets longer. The plodding tempo makes sense without getting into the math specifics. But what I want you to think about today is, "What does this mean for yarn usage?"
Let's say you want to knit a center out circle with ten individual skeins of yarn, each a slightly different color along a gradient but all the same yardage. What will happen to your stripes?
What happens is the stripes become narrower and narrower. (For those of you interested, there is a square root involved in radius/diameter progression. The numbers go 1, 1.4142, 1.732, 2, 2.236, 2.4494, 2.64575, 2.828427, 3, 3.162 as you track through the square roots of whole numbers.) If you keep knitting far enough, the gradient will "break," meaning it becomes so narrow that eventually a new color can't even complete one full round. For most people, this is not the effect envisioned.
Below is an example of the effect most people want.
The center circle is one unit in radius. The next circle out is three units radius. Then 5, 7, 9, 11, 13, 15, 17 and 19. The center circle and all the rings are the same width, just like a target bullseye. What does this mean for yarn usage?
The area of a circle = π r2
So the area of the center white circle is π 12 = 1π.
The next circle is π32 = 9π. But we want the area of the yellow-orange ring, not the area of a whole circle. That means we have to subtract the white center circle from the yellow-orange circle.
9π − 1π = 8π.
So if you used 1 meter of yarn to work the white center circle, you will need 8 meters of yellow-orange yarn to work the first ring. If you used 3 meters of yarn to work the white circle, you will need 24 meters (3 × 8 = 24) to work the yellow-orange ring.
What happens if we extrapolate the math for the whole figure?
π 12 = 1π
π 32 = 9π 9π − 1π = 8π
π 52 = 25π 25π − 9π = 16π
π 72 = 49π 49π − 25π = 24π
π 92 = 81π 81π − 49π = 32π
π 112 = 121π 121π − 81π = 40π
π 132 = 169π 169π − 121π = 48π
π 152 = 225π 225π − 169π = 56π
π 172 = 289π 289π − 225π = 64π
π 192 = 361π 361π − 289π = 72π
Interestingly, this same proportion works for squares, triangles, and half circles.
Area of a square = a2, where a is the length of a side.
If you draw out the squares, you get the same effect, with the center square being 1 unit on a side, the next square is 3 units on a side, the next 5 units on a side, and so on.
A neck-down triangular shawl is just a square rotated 45° and cut in half. A half-circle shawl is just a circle cut in half. It is the same proportions, since the extra "½" just carries through all the calculations.
How does this affect planning for a project?
If you are purchasing skeins (or making up kits), you will need much larger quantities of yarn for colors farther out. If you are doing your own hand-dying, you can wind off appropriate amounts to produce the desired effect. If you wind 10 meters for the center, then wind 80, 160, 240, 320 and so on for each progressive ring. If you wind a more random amount — say 7 yards — then the progression is 56, 112, 168, 224. You just multiply your beginning amount by 8, 16, 24, 32, and so on.
If you have purchased a gradient yarn or kit, be aware of which type of project will best suit the yarn. If the color amounts are graduated in length, a center out circle, square, or neck-down triangle will work well. If the color amounts are the same, however, consider a scarf, blanket, socks, sweater in the round, or some other pattern where the rows or rounds are close to the same size throughout the project.
Tomorrow: more tips of planning center-out projects when you have a limited amount of yarn.
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