Whenever I look at the completed projects section of my Ideal Sphere page on Ravelry, I love how people have incorporated it into a huuuuge variety of projects! For a while now, I’ve wanted to go back and make more basic shapes for people to use as a spring board. And now I am! Yay!
This is a crochet square. Now, if you crochet, you have almost certainly made a granny square before. (I think it’s one of the first things I ever crocheted while I was still learning.) This one is a little different.
This square is made in single crochet and is worked in rows (from the inside to the outside). I’ve also made sure that it has the correct number of stitches in each row to make it a mathematically ‘ideal’ square. Because reasons. (Scroll to the end of this post for more geeky details.)
You may be saying, ‘Emily, I already know how to make a square with single crochet: you make it bottom to top one row at a time, like a washcloth.’
Yes, that is also a way to make a crochet square/rectangle (and one that I’ve used many a-time). But, I’m not always a huge fan of the jaggedy edges. Sometimes I like the neat-and-tidy quality of crocheting in rows.
Without further ado: here’s the pattern. There are written instructions as well as a stitch diagram. You can stop at any row you like to make a smaller square, or – once you get the hang of the pattern – continue crocheting to make a ginormous square. 
NOW FOR THE MATH:
If you are completely uninterested in geometry, feel free to skip this bit. But, if you’re interested in why I’m calling this an ‘ideal’ square, read on…
First: a few assumptions.
I’m assuming that every single crochet stitch is as tall as it is wide. This is an ok approximation considering that the stitches smoosh together quite a bit at the corners. In reality, single crochet stitches are about 10% wider than they are tall, but this will vary and depend on your yarn choice, your hook size and your crocheting tension.
I’m assuming that every single stitch is the same size. Scrunching will happen in places. Whatevs.
Second: the geometry.
Each square represents a single row. The first square has side lengths equal to twice the dimension of a single crochet stitch. (Let’s call the height or width of a single crochet stitch x for simplicity.) Therefore, the perimeter of the inner square is:
P1 = 2x + 2x + 2x + 2x = 8x
So the first row needs to be 8 stitches.
Each side length of the second row is now 4x, so the perimeter of the second square is:
P2 = 4x + 4x + 4x + 4x = 16x
So this row needs to be 16 stitches. Notice that the pattern goes up by 8 stitches each time, since you are adding two stitches to every side of the square.
For any row, N, the number of stitches required is:
PN = 2Nx + 2Nx + 2Nx + 2Nx = 8Nx
So any following row needs to have < 8 times the row number > stitches.
Ms Premise-Conclusion 
I’m thinking this square is actually crocheted in rounds rather than rows since it’s never turned and worked back the other way.
Hi there, Actually these are worked in rows since you will need to turn after each new row. You could also do it rounds (without turning), which I think looks a little nicer, however, I found that this makes the diagonals curve if you do more than about 10 rows.
Cheers!
Emily
Reblogged this on vikkihaffenden and commented:
I love to see such detailed technical knowledge shared so generously – thank you