Entrelacs abound in knitting at the moment. In thinking about perhaps composing my thoughts for a post on the subject, I searched online and found some very good sources on this subject. One is found at howtoknitasweater.com. The author, Cheryl Brunette also shares an article on lace . Here are 2 of my teaching samples, executed on 4.5 mm machine
front view
a rearview with ends woven in
…
Removing a circle from the center of our pie yields the “donut”. The purpose and size of the “donut hole” can vary from the size needed to apply a central motif whether in knit, crochet, or other forms, to one large enough to allow for insertion of a “back piece” that can be anything from a “square” to one that included a bit of shoulder, armhole, even neck shaping and an optional curve at the waist/hip area. 
Additional rows of plain knitting in the “donut” itself alter the final forms. Seaming can occur where preferred; the direction of pattern repeats if in use further influence choice of seam placement.
If miter shapes are created in the knitting method, the corners of the triangles will want to “poke out”. This can be a purposeful design feature. If they are not wanted one way to soften them is to have stitches for at least an inch at the outer circumference of the circle knitting with no shaping in that area, adding a border, going the spiral route. Swatching helps determine preference in creating personal designs. Small scale paper collages sorting out geometric shapes and joins can inspire the large form variations.
Vests and sweaters built on circular shapes offer some challenges. Shawls and shoulder wraps are much more forgiving, but garments, particularly if sleeves are added, can provide sizing and fitting challenges. Making a muslin in disposable knit yardage prior to the actual knitting allows for trial placement of armholes and testing of overall measurements prior to charting out garment and sleeve shaping. Slits for the armholes can easily be taped or stitched closed to suit, and in turn, trial cut in a different location. Trimming or adding borders to circumference allows for visualizing size grading. This process helps spare the knitter regrets upon completion of the piece.
Drawing large circles is easy and accurate with a “yardstick compass”. Trammel points are available online and at many woodworking supply stores, etc. They convert any standard size yardstick for drawing arcs and circles up to 72 inches dia. Use a longer stick the same width and thickness as a yardstick and draw circles as large as you like. They are usually made of aluminum except for the steel point, measure about 3-1/2 inches in length.
Some beginning guidelines for drafting:
1. Use your bust measurement as the circle’s diameter and draw the corresponding shape. Two or more strips of freezer paper may be used as the drawing surface, temporarily fused onto the knit yardage, becoming the paper “pattern” for the piece and stabilizing the knit for cutting.
2. Measure your back from arm to arm to determine how far apart to place armholes or obtain this measurement from any well-fitting favorite.
3. Measure armhole depth from the top of the shoulder to 2-3 inches below the armpit.
4. Draw lines for armholes and center horizontally within the body of the circle, shoulder measurement apart. Commercially written patterns are bountiful online and in magazines and tend to center the armholes vertically as well. I prefer them shifted up for a less bulky “collar”, and for placement of sleeves with raglan or traditional caps. Binding off a few stitches at the base and casting them on at the top of the slit create a slightly shaped for easing in the sleeve top.
5. Cut “armholes”, remove freezer paper if used, try on for fit, adjust as needed. 6. Back to more math!
I made a series of long-sleeved circular sweaters for sale in 2008. Discovered problems with photos in my photo library (new computer). These are an attempt at “restored” shots of one of the first such sweaters. The yarn was fine Italian mohair.
close up
Holding/ short-rowing is used to knit each wedge. The basic rule for holding is followed: stitches are brought in to work on the carriage side, and into hold opposite the carriage to avoid floats. The greater the number of sections, for either the full circle or a donut, the smoother the curves at the outer circumference, as can be imagined in the form below, which divided into more sections than those 5 in our original calculation.
The pie slices” begin and end on open stitches, ultimately requiring a join where the radii meet, using whatever method is preferred by the knitter. Assuming the knit carriage is on left and set for hold in the diagrams below if all stitches are brought into hold except 1 on the carriage side (or number required by calculations for the individual piece), 2 rows are knit, and action is repeated until all needles are in work, the following shape starts to fill in and a miter with pointed edges is created.
If all the needles are in work, and one begins to bring them into hold opposite the carriage, one begins to fill in the shape creating the form below, and a spiral is created, with circumference edges more rounded.
Like shapes may be stacked sequentially. If shaping at the top of the initial triangle wedge is reversed, however, the following begins to occur.
Knit rows in between the triangles begin to create larger holes in the center of the pie, and a donut occurs. The donut center can be varied, the knit rows between triangles increased to suit. The illustration below shows some of the variables. The plain knit rows are another factor in smoothing that outer circumference.
Math is not always fun and is downright dreaded by some. One instance in knitting wherein basic calculations are required is in obtaining stitch and row gauges. I have known one hand knitter who would purchase yarn (not necessarily the one used in pattern), knit happily away, and try the finished product on everyone she knew until she found an accommodating body shape and size. If a large number of family and friends did not oblige, sweaters were stored until such a correct body appeared. Predictable results require careful measurements and some basic formula calculations.
Using home knitting machines to produce circular forms one resorts to breaking down the round object into pie wedges, which in turn are knit as triangles with straight line outer edges. The outer final circumference curve is controlled in a number of ways, one is by creating a far greater number of pie slices. For this exercise, I will work with 5 segments. 
There are some math constants. One example: to find the circumference of a circle its diameter is multiplied by pi = 3.14. If the diameter of our knit is 44 inches, its circumference will measure 44 X 3.14 = 138.16 inches. Using the rule of 5 or less than 5, this measurement is rounded to 138 inches.
The radius becomes the width of the pie wedge. In this instance, it would measure 22 inches. Let us assume our gauge is 4 stitches and 6 rows per inch. The radius is converted to stitches: 22 X 4 = 88 sts. The circumference becomes rows: 138 X 6 = 828 rs. If subdivided into 5 slices, each slice would be composed of 166 rows.
To knit the pie slice, short rows are used; since they happen every 2 rows, our row number for outer edge is divided by 2, yielding the total of now 83, which in this exercise I will round off to the even # 84.
24 stitch repeat
60 stitch repeat
more than the recommended 16 sections for full circle would be required in this particular yarn gauge
This post was written originally in 2011, in one of my lace “phases”, with test swatches knit on a 910 using a mylar sheet or on my 892E Andare Brother punchcard machine. Early on many of the posts were used to record what I had made, sometimes providing the repeats, but not usually step by step instructions. The shares were intended for anyone with previous experience who might want to create their own version of such fabrics. Now in 2020, my swatches are usually knit with download cables img2track on a 930 or less frequently on a 910 using ayab, via an iMac. In revisiting those early posts I sometimes find myself wondering about the content and how their fabrics were achieved. At the time, as now, I tried to respect copyrights. Since then many machine knitting publications are now downloadable for free online and I began to include information from them, and when possible, links to the source.
There is an excellent online resource for the Bond Machine. Techniques are applicable to other KM models for those who enjoy hand techniques. The round lace tablecloth series provides a number of “doily” charts.
The slip setting in both directions is used on the KH for automatic shaping, end needle selection is canceled. It is critical that carriages be off the machine and on the lace extension rails while the alternate carriage is in use as they both engage the timing belt. The latter can be broken if pulled in opposing directions at the same time.
Comparing a pattern on 2 machines: one of the critical differences when using 2 carriages to select patterns, is that on electronic machines such as the 910 or later models, with each carriage pass the design repeat advances one row. With Brother punchcards, the first pass of the second carriage does not advance the card as it makes its first “trip” from the opposite side, the previous preselection and function are repeated.
When operating the lace carriage on a punchcard machine, one of the critical differences is that the preselection row from left to right made with the lace carriage is performed with the punchcard not locked as usual, but rather, set to advance normally. From: 
Any circular knit that is shaped with sideways holding segments will need to be seamed when completed. Taking that into account, at least one row is knit with “doily” yarn from left to right after several rows are knit in waste-yarn before the transfers to create open stitches are begun. It is a personal choice and preference whether upon completion of the required number of segments to join by rehanging open stitches and binding off on the machine or grafting (the method I prefer) with the work off the machine after several rows of waste yarn and dropping the work off the KM.
When 2 carriages are selecting needles from opposite sides, the “rule” as provided in the manual is broken, the card starts in the locked position on one side, is released when it reaches the opposite one, as is standard in knitting other stitch types. In this chart, patterning begins with the locked reader and the LC used for the initial preselection row 
Beginmomg with the LC on the left, the end needle selection is canceled. The two positions in LCs for both punchcard and electronic KMs
Back in 2002 exchanges with a fellow member of an Australian Yahoo Group, OzMKers, led to her final edit of this half punchcard repeat for the test doily. All transfers are in the same direction, to the right
A 24 stitch sample, knit in cotton yarn. The center of the circle needs to be managed as opposed to simply gathering it in order to keep the finished shape flat. 
It is possible for the lace carriage to transfer while at the same time preselecting every needle to be knit by the next pass of the knit carriage from the opposite side. An expanded view of part of the repeat: 
In order to get the repeat working properly, I found I needed to edit out one of the punched holes at the start of each repeat, revised card 
The original and the amended start of each segment are shown below, I skipped the extra knit rows numbered 21-23 on the original punchcard with the intention of eliminating extra knit rows at the very center, making the circumference at the closure of the doily smaller. I prefer to start with knit rows, in which case the first preselection row would take place from the right. Note that lace transfer marks are directly on top of the solid lines representing knit stitches 
If drawing on the back of mylar for use on the 910, either image may be drawn as is, with the number 1 pattern case “A” selector reverse lever to up position.
Repeat design principles are shared in creating edgings, ruffles, and more.
The amended 24 stitch repeat with all transfers to the right knit on the 930 after a few rows misstart. The download is automatically mirrored. The end needle cancellation setting on the 910 LC for the pattern to work properly with the carriage in use. 
In turn, this repeat as given was used in 2020 for a pattern test on my 930 for all transfers to the left. Note: here the knit stitch black pixels are followed by a blank row, and the transfer marks occur on design row 3, directly below the next series of all black pixels 
Depending on the machine model, the .png for the repeat may, in turn, need to be mirrored horizontally.
The preselection row for my test swatch is from right to left, using the knit carriage (KC). All black pixels will preselect needles. When the left side is reached, the KC is set to slip in both directions and will knit all preselected needles on its way back to the right. Subsequently, the slip setting will be performing the function of bringing stitches in and out of hold opposite the carriage. Each carriage pass advances the repeat one row. The lace carriage preselects for transfers from left to right, transfers to left, and preselects for the next knit row as it returns to the left and onto its rail. 
A 40-stitch adaptation from the Bond site to try
A 60-stitch repeat also inspired by the Bond post knit on the 910 using a mylar. Gauge still matters, more than the recommended 16 sections to form a full circle would be required using this particular yarn 
The electronic repeat as drawn on the mylar, 
the 60X66 png 
In the post on lace edgings automated with slip stitch on Brother machines written in 2020, I shared a modified version of a punchcard published by Susanna Lewis in “A Machine Knitter’s Guide to Creating Fabrics” (1981), the ultimate resource for punchcard knitting for knitters with any amount of experience. The chart on p 223 was modified by me since I like to start my lace edgings on the widest number of stitches in the pattern. The punchcard repeat on the left below is shown after being reorganized to start with a knit row on the full 24-stitch width. On the right, rows of black pixels are added, for a version of the full repeat to be used on electronic machines. The slip stitches are substitutes for manual holding techniques, used to knit rows that are in turn form increases and decreases to alter the outside edge of the trim. The lace pattern resulting from the transfers changes along with their direction as the outside edge does. This image is for the desired effect on the purl side of the fabric. Again, depending on the electronic model or the download program, it may in turn need to be mirrored. 
Subsequent posts on knit edgings:
Lace edgings on Brother machines- automated with slip stitch
Lace edgings on Brother machines- automated with slip stitch 2
From automated lace edging patterns to doilies and ruffles
Blocking is one of those knitting preferences that can arouse strong pro/con arguments, and goes the range from casual to nearly compulsive with wires, pins, and assorted tools used to achieve desired results. My shawls continue to sell well: the photos below illustrate part of the process and 2 of the most recent in their family. All shaping and joining are achieved through the knitting process; the shawls are reversible, may be worn and draped in a variety of ways.
before steaming and pressing
detail shot after steaming/pressing
one way to wear, purl side facing out
Layers of knit copper and brass wire petals, assorted beads, leaves, form another no wilt corsage suspended on a 20 inch knit copper tube “chain”. The closure is magnetic. The flower and leaves measure 5.5 inches at their widest axis.
In anticipation of the fiber invitational in Lowell and the annual Art to Wear show at Cambridge Artists’ Cooperative, the copper wire is resurfacing in new neck pieces. Some of the presently completed work may be seen below. Materials often come with their own stories. I purchased my nearly invisible wires from an elderly gentleman at a RI yard sale years ago for only a few dollars. He in turn had worked using them on TV and radio tubes in the “old days”. The 32 gauge version was obtained with the assistance of a brother-in-law-electrician. A beaded piece is “in the works”, and then there is all that colored telephone wire and a shoebox full of acupuncture needle cases periodically calling out to me… I tend to work freeform, without sketching, piecing elements and working out designs and problems as I go. There are elements of surprise for me as well in each piece I create.
The beginning of a new/different direction
ladder lace and short rowed ruffles, modified commercial toggle closure
The finished collar tuck lace and ruffle collar; 4 strands of nearly invisible wire used throughout