Sunday, 9 January 2011

where's my polygon?

Sometimes ideas seem to collide. I've been listening to the bbc's history of mathematics podcast and thinking about how to knit blankets.

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.

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