Accommodating Curves

Turning Circles into Squares

Something has been bothering me for a while now.... the mathematics behind crochet. I noticed in a forum last week that someone was asking about circles in squares- How can they find a way of doing them that enables them to design their own Afghan without using someone else's pattern, and using different sizes throughout. I did some searching on the internet and couldn't find ANYTHING that related to simple mathematics that could help us to design patterns.

It got me to thinking.... what IS the mathematics behind turning circles into squares easily? Could many different sizes be used in the same Afghan?

I've been stewing on this all week and guess what?!!!

The answer is YES!!!!!

We are not talking about hyperbolic crochet or polar angles here -  the mathematics is actually pretty rudimentary. I mean... all you need to know are your 4 times tables and the definition of "median".

FIRST you need to know to begin making circles into squares is ... the "law of circles" .

You can now make your circle as large as you like.... BUT - the stitch count MUST be divisible by four (4).

Image from Whipup

Double Crochet Circles are an easy example because your first row is a multiple of 4:
3x 4 = 12 

A square has 4 corners, to make a square from a circle with a stitch count of 12, a corner will be made in every 3rd stitch.

To keep the work flat, we still need to increase each round by 12 stitches, so we need to increase by 3 stitches in each corner. Easy right?

To make a 1 row DC circle into a square: 
When your circle is complete,  [ch3, dc, ch1, 2x dc] into same stitch (first corner made), dc into next 2 stitches, [2xdc, ch1, 2xdc] into next stitch (corner made), dc into next 2 stitches, corner, dc into next 2 stitches, corner, dc into next 2 st, slst to join.

Can you see the maths happening? 

To make a 2 row DC circle into a square: 
When your circle is complete, [ch3, dc, ch1, 2x  dc] into same stitch (first corner made), dc, hdc, sc, hdc, dc, *corner, dc, hdc, sc, hdc, dc*, repeat 3 times, slst to join.

Here is where it starts to get slightly more involved.... notice how there are now alternative stitches in the "squared" row? this is because in bigger circles, the widest part of the circle needs to be accommodated to achieve a square.

The MEDIAN of the stitches BETWEEN each corner will also be the widest part of the circle, so that is where the shortest stitch needs to be placed.

In the example above, the corner is made into every 6th stitch. The stitch count BETWEEN the corners is 5.

1, 2, 3, 4, 5

The Median is 3. This is where the shortest stitch is placed (Single Crochet). Notice that we have placed a HDC either side of the SC? This helps to accommodate the curve of the circle.

Are you still with me? Lets keep going.....

Making a 3 row DC circle into a square: 
 [ch3, dc, ch1, 2x  dc] into same stitch (first corner made), dc, hdc into next 2 st, sc into next 2 st, hdc into next 2 st, dc, *corner, dc, hdc into next 2 st, sc into next 2 st, hdc into next 2 st, dc*, repeat 3 times, slst to join.

In this example, the corner is made into every 9th stitch. There are 8 stitches between each corner. This means we have a pair of 'middle numbers'.

1, 2, 3, 4, 5, 6, 7, 8

This is why there are 2 SC flanked by 2 HDC.

So there you have it. The basic maths behind turning a circle into a square.

Now to answer the question about joining them in different ways:
THEORETICALLY (I haven't tried it, I avoid working in motifs wherever possible) these 3 sizes alone SHOULD be able to be joined in a few ways:

1 row circle/square made into a 4 patch block could be joined to a 3 row circle/square.

1 row circle/square made into a 9 patch block can be joined to 2 row circle/square 4 patch block.

Summary:
Make your circle.
Divide your stitch count by 4.
Make a corner into the (divisible number) stitch of your circle
Accommodate the curves by using shorter stitches in the middle of the space between 2 corners, and longer stitches towards the corners.


Post Script: 
The Larger your circle, the taller your corners will need to be.
Alternatively, you could add a 2nd row to your square round, still using accommodating stitches, to help keep your edge straight.

I love seeing your creations! Feel free to post pictures of your work on my Facebook page.
Feel free to message me with questions or suggestions.

Happy Crafting!