Half a rectangle is a triangle

Lately I've been obsessed with shapes. No surprise there, I'm kind of a math geek, and shapes are the way that math and knitting get together and have a good time.

The most trivial non-trivial topology is the Möbius strip and we've already learned how to knit those. You can take the connection between knitting and mathematics much, much further, but, if you're only beginning, like me, there is a lot to be said about trivial shapes.

The easiest way to knit a triangle is to knit a half square or rectangle. The idea is to start with a certain number of stitches and as you go along keep on doing a regular amount of decreases. Or, if you prefer, you can do it the other way: start with only a few stitches and as you go along keep increasing the stitches regularly. Here is an example to make it more definite:

1. Cast-on 3 stitches.
2. Row 1: knit one, increase one (make one, or any other increase you prefer) and knit to the end of the row
3. Keep repeating row 1 until you've got the size you want and then bind-off.

This makes a garter stitch isosceles right-angled triangle. You can make it taller or shorter, narrowing or widening the 90° angle, by decreasing or increasing the amount of increases. For example, do two increases in each row and make a very wide triangle, or increase only once in every other row and make a very sharp one.

Don't like the regular shape of the isosceles, increase more to one side than the other and make a triangle with all different sides.

Note that the recipe above uses garter stitch. This is very important because you must take into account the ratio between the number of rows and the number of stitches per length, when predicting the shape, change the recipe to stockinette or any other stitch and you'll probably find that the triangle is no longer right-angled.

You can get a slightly different effect if you only increase to one side. Here is a recipe for a shallow shawl. Use variegated yarn with slow colour change for effect.

1. Cast-on 3 stitches.
2. Odd rows: knit straight.
3. Even rows:  knit one, increase one and knit to the end of the row.
4. Keep on working until you've got the size you want and then bind-off.

If you use a yarn-over to do the increase, you'll get a pretty eyelet side. And of course you can end with a row of eyelets if you'd like to have two eyelet sides. To be precise, on the row before the last do the following:

1. knit one, (yarn-over, knit two together) and repeat until you have one stitch left, and end with a yarn-over, knit one.