Free Patterns, Geekery

Guaranteed to be a Pointless Endeavour

Get it?? Pointless??? Cuz it’s a circle.

Ah, puns.

SONY DSC

Anyway, here is the latest shape-themed pattern. This one is a circle made from single crochet stitches in spiral rounds.

SONY DSC

The largest circle I’ve made using this pattern is 18 rows (which is a decent sized circle), but I’ve calculated the number of single crochet increases needed in each row all the way up to 70 rows. I’m not sure what you’d do with a crochet circle that big … but that’s completely up to you.Crochet Ideal Circle front

NOW THE MATHY BIT:

In this simplified diagram, each circle represents a row and the spacing from one row to the next (called x) is the same as the height of one crochet stitch.

circle diagram

For this to work, I’m assuming every single crochet stitch is as tall as it is wide and that every single crochet stitch is identical.

The circumference of a circle is equal to twice its radius times π (‘pi’, which is equal to 3.14159…). So the circumference of the first circle is C = 2πr = 2π(x), which means that you’ll need 6.28 stitches. (This rounds to 6 stitches.)

Hey, we usually start circles and spheres off with 6 single crochet stitches!

For the second circle, C = 2πr = 2π(2x) = 4πx, which means that you’ll need 12.57 stitches. This rounds to 13 stitches, so in this circle, you need to increase by 7 Sc.

For the third circle, C = 2πr = 2π(3x) = 6πx, which means you’ll need 18.85 stitches. This rounds to 19 stitches – an increase of 6 Sc compared to row two.

Noticing a pattern? Every row needs 2π more stitches than the row before. Usually, when we make balls or circles in the round, the rule of thumb is to increase by six stitches each row, which makes sense since 2π (6.28319…) rounds to six. Ever notice that this sort of a circle results in edges that curl up? That’s because by rounding down to six increases in each row, you’re not adding enough stitches to make a perfect, flat circle!

Frequently Asked Questions (that I just made up)

Hey, this pattern is in spiral rounds, not circles! What if I want closed circles for each row instead of spirals?

Good observation and question.

You can totally use this pattern to make perfect little concentric circles – just crochet in rows instead of in the round. After row 1, slip stitch to the very first Sc. Chain 1 to begin the next row (this starting chain 1 stitch becomes the first stitch of the next row). Follow the pattern around and finish by slip stitching to that starting chain 1 stitch. Depending on how you want your circle to look, you could even turn the piece after your starting chain 1 stitch on each row so that the stitches alternate from facing back to facing front.

If you do it this way, you will have a little seam running up your circle.

So, since this pattern will technically make a spiral, is it still mathematically correct?

Yes it is! Ready for more math? (That was rhetorical!)

Here are the parametric equations for a spiral that models what we are making.

x(t)

y(t)

And here’s a graph of that spiral. The spacing between spiral rotations is constant: it’s one unit. That one unit represents one single crochet stitch. This model doesn’t represent the first row of our circle/spiral very well, but we just won’t look at that part too closely.

archimedean spiral

I would like to find the distance (called the arc length) around one full revolution of this spiral in order to determine how many stitches I will need. That means, I need to integrate. In math speak, one full revolution is 2π radians (360 degrees), and since I want to make this work for any round of my spiral, I will go from 2(n-1)π to 2nπ, where n is any round you want.

line integral general

Here’s a simplification (using the parametric equations above) of the square root bit:

simplification pre integration

Here is the integration step, which I’d like to evaluate from t = 2(n-1)π to 2nπ.

integration.png

So evaluating the above equation gives me this generalization for any round, n:

answer

This is a bit different than when we modelled our crochet circle/spiral as a series of concentric circles (see above), but keep in mind that the inner region of this spiral isn’t a great model for what we are crocheting. What we’re really interested in is how many stitches you need to increase from one row to the next in order to maintain this nice, flat spiral shape.

So, if I take the difference in the total stitches required from one row (n-1) to the next (n), heres what we get:

generalization

We still need to increase by 2π stitches from one row to the next! Same as the circle! Woohoo!

In summary, it’s the little things that get you through the day.

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2 thoughts on “Guaranteed to be a Pointless Endeavour”

  1. Seeing math applied to crochet makes me INCREDIBLY HAPPY. Love that you did the parametric proof for a spiral in addition to concentric circles.

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