Gaia asked how I achieved an invisible join between the Capecho pentagons. I promised to explain. (I am only making an abbreviated circle of six pentagons instead of the entire shrug so you will need to tweak these modifications for your project.)
Look at the schematic below. The pentagons are numbered in the order I plan to knit them. Number six will join five and one to complete the circle. The black and blue lines look very similar in the thumbnail below; click on the image to see the true colors.On one needle (black lines), I cast on 12*28 stitches using a long-tail cast-on (LT CO). On another needle (blue lines), I LT CO 6*28 stitches. On double pointed needles (red lines), I used a provisional crochet cast-on for 28 stitches 6 times. I learned the technique from Lucy Neatby's Spun Yarn Newsletter #13.
I've been very busy lately so the only picture I took of the process is spread out over two chairs in my immunologist's waiting room. I am starting pentagon #5. I will pick out the crochet chain on pentagon #4, place those stitches on a needle, and knit (or purl) those 28 stitches, knit 2*28 stitches off the bamboo needle (black lines in schematic), knit 28 provisional crochet stitches off the double-pointed needle (red lines), and then 28 stitches off the green needle (blue line).
Remember when I obsessed over needle sizes to get the desired pentagon size? I needed to begin on row 5 of the Capecho chart so that the cables flowed smoothly between pentagons. If I started on another row, I would have had to adjust the cabling maneuver rows. That's not too difficult if you are knitting your pentagons in the round as I am.
I am not entirely happy with this technique. The LT CO stitches stretch between pentagons and the trailing multiple needles are a pain.
Earlier, I tried using a long-tail cast-on for 2 sides, then knitting a crochet cast-on by alternating between the two yarn ends, then LT casting on for another side, dropping the short tail and then knitting up the last crochet cast on. If this sounds confusing, that is because it is.
Anyway, that attempt was fine until I got to the neighboring pentagon and tried to pick up the provisional stitches. All those extra loops from the two yarn tails were a nightmare. I gave up and decided to cast-on all outer edge stitches at once.
There were several other frustrations. When picking up the provisional loops in ribbing, the crochet chain does not come out cleanly. Furthermore, there is a half-stitch jog at the joins if you look closely.
If only I had read cmeknit's grafting in rib tutorial earlier! She doesn't seem to have a half-stitch discontinuity at the join. (Update: I posted my own pictorial tutorial for grafting in rib pattern.)
Design your own knitted pentagon
You can knit a regular pentagon in any stitch pattern as long as you know your row and stitch gauge. Take a look at Wolfram Mathworld's pentagon page. Do you see the regular pentagon with labels R (circumradius), r (inradius), and a/2 (side)? Decide your side length and multiply that by your stitch gauge for the number of stitches to cast-on per side; let's call that number n. (Read Sizing Knitted Pentagons for help.) You want to cast on 5*n stitches and reduce stitches until you end up with 5 or 10 stitches at the center of the pentagon.
Calculate the length of your inradius, or the little r in the Wolfram pentagon schematic. Type sqrt(25+10*sqrt(5))/10 into your Google search bar and you see that r = 0.688*a. Multiply r by your row gauge to find out how many rows (call this RC, your row count) you need to knit till you reach the center of your pentagon.
If you start with n stitches and need to decrease to 2 stitches, then you need to decrease once every (n-2)/RC rows.
Suppose you want to knit a stockinette pentagon with 3 inch sides with a 5x7 gauge. You need to cast on 15 stitches and decrease over 14.5 rows. Let's round it to 14 rows. That means you decrease at the beginning (or end) of each side on every row for 14 rows. Then cut off your yarn with a longish tail, thread it through a needle and draw it through your remaining loops twice to tighten up your hole. Weave in the end. That's all there is to it.
Don't be afraid of the math. Just cast-on and play. It's only knitting and yarn is very forgiving.
3 minutes ago