Shawl Calculations

I’ve just realized that I’m not anywhere close to finishing the shawl I’m knitting, so I’ve done the perfectly logical and rational thing and started a new one.

This one is using some yarn from the stash, so its size is limited by the available yarn, and I wanted to estimate how long it may be, so I weighted the ball of yarn at the beginning and then again after knitting 10 and 20 rows.

It’s a top-down crescent, with 6 increases every two rows (but these calculations should work for any uniform top-down shawl with a regular number of increases), so each block of 10 rows should use an aproximately fixed weight of yarn more than the previous block of 10 rows.

So, let w₀ be the weight of the first block of rows, w_r the (average) difference between two consecutive blocks and w_T the total weight of the shawl. Then the weight used by block i should be w_i = w₀ + w_r ⋅ i and the total weight of the shawl should be:

$$w_T = \sum_{i=0}^{N}w_i = w_0 + w_r ⋅ \frac{N ( N + 1)}{2}$$

where N is the number of blocks in the whole shawl.

This gives:

N² + N + 2/W_r ⋅ (w₀ − w_T)

and the only positive solution will be:

$$N = \frac{-1 - \sqrt(1 - \frac{8}{w_r} (w_0 - w_T))}{2}$$

or, in a few lines of python that can be easily copypasted (changing the values in ws and w_T, of course):

import math
import statistics

w_T = 200
ws = [2, 4, 6]
w_r = statistics.mean(map(lambda x: x[0] - x[1], zip(ws[1:], ws)))
(-1 + math.sqrt(1 - 8 / w_r * (ws[0] - w_T))) / 2

Which right now (using the actual measured values) tells me I will have about 140 rows in my shawl, but I’d really want to do a few more blocks of 10 rows and have more datapoints before I trust the numbers I’ve put in.

Which means that this shawl will also take forever.