Making a sphere is something that comes up a lot in the realm of crocheting stuff. But how do you make a really good sphere?
This is something that I pondered one day. After a very brief and half-hearted go of it with google, I decided to answer the question myself… by throwing some math at it. How do you create a sphere when you are working in a row-by-row system (like crochet) where all you can alter is the number of stitches per row?
Well I assumed that each row is equal to a fixed polar angle , Δθ, within the sphere. At each θ, the circumference of the circle that is traced out within the sphere is proportional to sin(θ) (this was just a quickie derivation). Below, is a plot of what the circumference (y-axis) looks like for each θ (x-axis). So, to completely recreate this, I would want the number of stitches in each row to scale to fit this plot:
So far, so good. In the past, I have only ever used one method to make spheres. It’s fairly common because it’s really easy and, once you stuff it, it makes for a decent sphere. But… it’s not ideal. This method has you increase by the same number of stitches every row until you hit the middle bit, where you work rows even for a little while before you start decreasing by the same amount for each row.
Problem: this is what this method looks like when plotted against the ideal case (the blue dots represent the old method):
It’s just not the greatest. So what I did was to figure out what the ideal number of stitches per row would be, given the sin(theta) dependence. Result: I can make an ideal sphere for however number of rows I want. It’s not as clear as the tried and true method and it’s not intuitive, but it makes a nice looking sphere.
Here is a pdf file where I’ve written up the pattern for ideal spheres of different sizes. I haven’t tried all the different sizes out myself (since I would up to my eyeballs in spheres) so they *should* all work out nicely. Let me know if they don’t.
p.s. Math rules!
UPDATE: Here is a link to a newer post where I include the pattern for some even bigger spheres! Embiggen!
***03.25.10 Update: An error has been spotted on the 10 Row Sphere, rows 5 and 7. (Oops – sorry all!) So this mistake has been fixed and the link has been updated.
***03.27.10 Update: Oops#2 located on 16 Row Sphere and is now fixified.