"Snowflakes are suggested by an all-white Spot Bronson weave." (Clotilde Barrett, from the original article, Weaver's Journal, Fall 1983.)
Here's the draft I promised, re-drawn since the old one was a bit fuzzy. I hope this one's clear enough to work from the image. Notice that Spot Bronson is threaded with every other thread on the same shaft. This means you will need lots of heddles on the one shaft, and may have to move some from other shafts (unless you have more than four and can use some other shafts to tie up identically and work along with it). The treadling is "as drawn in". I've marked the center pick of the pattern, from which point it reverses. You can use the threading sequence for the entire pattern. Repeat as many times as needed for the towel length of 38".
Now, about yarns: I don't have any 40/2 or 20/1 linen in my stash. When I looked to buy it, I found that would mean quite an outlay in dollars to do all the projects (most of them use these two yarns). So although linen towels sound absolutely scrumptious, I'm going to try using some yarn I have already. I have some 20/2 cotton that according to weaving books can be sett at 30 epi (the same as the article recommends for the 40/2 linen).
As an aside, in my search I could not find bleached linen in 40/2 or 20/1 anyway; I could only find half-bleached. If anyone knows a source, please let me know!
So have you decided to try a kitchen towel in Spot Bronson this month? What kind of yarn will you use?