The current "not quite Pi" blanket is progressing along in mindless knit stitches so I've been thinking about how to construct the second one.
Which leads to this:
Any n-sided regular polygon can be constructed out of n-isosceles triangles.
An isosceles triangle can be divided into two equal right angled triangles.
Right angled triangles can be constructed in knitting using short rows.
For example, a hexagon:
r is the radius of the polygon (how wide do you want it)
e is the edge of the isosceles triangle
a is the angle between the two equal sides of the triangle (and is equal to 360/n)
The triangle is then split into two right angled triangles thus:
R will be the cm value of the rows needed
s will be the cm value of stitches needed
How many stitches do you need though?
Enough to make s cm.
s is the acute of angle a, so s = cos(a/2) * r
how many rows will you need to work in total?
R is the obtuse of angle a, so R= sin(a/2) * r
so for a 40 cm radius, 6 sided polygon...
The individual triangles will have 34.6cm worth of stitches, and 20cm worth of rows.
Translating R and s into knit stitches will give a shape something like this (note this is not the actual number of stitches and rows!):
Using Rowan Alpaca Cotton (simply because I have a ball band to hand) which has a gauge of 30 stitches and 38 rows to 10cm. Each triangle would have 104 stitches and 76 rows.
I'll try and set up an excel spreadsheet that asks for n (how many sides), r (how wide the shape should be), stitch and row gauge, then spits out instructions on how to actually knit the triangles. More coffee required first however.