A heel turns a right angle because of the loop

A sock is a tube, and a foot is not. Somewhere between the leg and the sole the tube has to turn through something close to a right angle, and it has to do it without a seam, because a seam under the heel is a blister.

The commonest way of doing it is the oldest. Half the needles — the ones at the back — are knitted back and forth on their own while the instep half waits. Each row stops one stitch short of the last, alternately at each end, until a third of the heel’s stitches are left working in the middle; then each row goes one stitch further, until all of them are working again. Knitting resumes in the round and the tube now points along the foot.

Why a third? Every pattern book says a third and none says why. The answer turns out to be a constant that has already been measured twice for other reasons, and the same arithmetic says something less comfortable about what the construction actually makes.

The heel is a trapezoid of courses #

Count what the short rows add, column by column. A stitch d places in from the heel’s edge is worked in about the first 2d short rows of each half — it drops out as the turning point passes it — and every stitch within the middle third is worked in every one of them. So each column gains a number of extra courses that rises by four for every stitch in from the edge and levels off across the middle.

On 64 needles the heel is 32 stitches, the turn leaves 11, there are 21 short rows each way, and the middle columns gain 42 courses. The two outermost columns gain almost nothing, and the 32 instep columns gain nothing at all.

That shape is not an approximation of the construction; it is the construction, and it is counted here row by row rather than read off a formula — each short row is worked, the columns it passes over are credited, and the trapezoid is then a check on the count. An odd number of short rows puts one side a row ahead of the other, which a smooth formula would hide and the count does not.

Why the back line is the thing to measure #

A tube bent through an angle θ about its inside edge is longer along its outside than along its inside, by the angle times the tube’s diameter. That is the whole geometry of a bend: the inside line is the hinge and every other line round the tube is further from it.

A sock’s inside line is the front of the ankle, where the instep meets the leg, and its outside line runs down the back of the heel. The short rows have made the back line 42 courses longer than the front and left the front exactly as it was. So the heel has done, along one line, exactly what a bend does.

The angle it has turned is therefore the back line’s extra length over the tube’s diameter, and both are counts of stitches: 2(H − m) course spacings of extra length, and a diameter of N wale spacings over π.

The needle count cancels, and so does the yarn #

Write it out and almost everything leaves. The extra length is 2(H − m) courses. The diameter is 2H wales over π. Their ratio is

θ = π (1 − m/H) · (course spacing / wale spacing)

and the needle count has gone, because the heel’s stitches and the tube’s circumference are both half of it. The loop length has gone, because a knit’s course and wale spacings are both the loop length over a constant and it cancels from their ratio. The yarn, the fibre and the gauge have gone for the same reason.

What is left is the share of the heel left at the turn, and one number: the loop’s own aspect, Munden’s $k_r$, the ratio of a wale’s width to a course’s height — 1.2791 in the fully relaxed state on Munden’s constants.

That is the same cancellation that made a fashioned edge’s angles the same in every yarn and at every gauge, and it happens for the same reason. Any shape made by adding or removing whole courses against whole wales is an arctangent or a ratio of those two counts, and the loop’s aspect is the only thing that converts one into the other.

A third gives a right angle, near enough #

Put the trade’s third in:

$$\theta = \frac{2\pi/3}{k_r} = \frac{2.0944}{1.2791} = 1.6376~\text{radians} = 93.8^\circ.$$

The oldest rule in sock knitting turns the heel through a right angle to within four degrees, and nothing in the rule mentions an angle. The practice was arrived at by fitting heels, not by computing them. What fitting found is that the share to leave is about one minus half the loop’s aspect — 0.360 in the relaxed state — and the nearest simple fraction to that is a third.

The relaxation states move it a little, because they move the aspect a little: 96.0 degrees on dry-relaxed constants, 92.8 wet-relaxed, 93.8 fully relaxed. A knit relaxes for as long as it is allowed, and a sock is allowed to more than most garments. A sock is knitted, worn and washed through all three, so the heel’s angle wanders by three degrees over its life, and at every point of the wander it is within six degrees of square.

Where the rounding shows #

The stitches left at the turn must be a whole number, and a third of the heel usually is not. So the share left is a third only when the heel’s stitches divide by three, and otherwise it is the nearest whole stitch — which moves the angle.

Across fourteen ordinary needle counts the turn runs from 91.5 degrees at 40 needles to 96.0 at 44, a spread of 4.5 degrees, and it lands exactly on 93.8 at 48, 60, 72, 84 and 96. That is the whole of the needle count’s influence: a sock on 44 needles leaves 7 of 22 heel stitches, which is under a third, and its heel turns a little further; one on 64 leaves 11 of 32, which is over, and turns a little less.

So a knitter who adjusts the rule by a stitch for a particular foot is making an adjustment of about two degrees per stitch on a small sock and one on a large one. That is the resolution of the construction, and it is the same kind of resolution a fashioned edge has: whole stitches against whole courses, with nothing in between.

The surprising part is which constant it is #

The loop’s aspect has turned up before, twice, and neither time in a heel.

It was the constant that made every fashioned angle universal — the reason a pullover’s side seam stands at the same eighteen possible angles in every yarn ever spun. And it was the constant that a tube’s taper cancelled out of, leaving the loop length as the only free quantity. In both places it was a nuisance to be carried through.

Here it is the whole answer. A heel is a right angle because a knitted loop is about four-thirds as wide as it is tall, and a third of the heel left at the turn is the fraction that converts two thirds of π in courses into ninety degrees of turn. Had the loop been square, a third would turn the heel through 120 degrees and the rule would be a half; had it been twice as wide as tall, a third would give 60, and even working the heel down to a single stitch would only just reach a right angle.

That is a claim a reader can check without a calculator. A sock knitted in a structure whose loops have a different aspect should need a different share at the turn for the same heel, and the arithmetic above says which way: squarer loops, more stitches left; flatter loops, fewer.

What a heel does not do #

The back line is one line. A bend is a property of every line round the tube, and the check that a short-row heel is a bend is whether every other line has the extra length a bend would give it.

It does not. A bend hinged at the front of the ankle wants extra length that grows smoothly round the tube, as one minus the cosine of the angle from the hinge — nothing at the front, half the maximum at the sides, the full amount down the back. The heel supplies the full amount down the back and nothing at the sides, because the heel’s edge columns are worked in no short row at all. At the heel’s edges a right-angle bend wants 21 extra courses and the heel has none.

The shortfall can be put as one number. A bend needs, in all, the tube’s circumference times its radius times the angle in extra fabric — a quantity the whole heel and instep would have to supply between them. The heel supplies its trapezoid, and the ratio is

(1 + m/H) / 2

which is two thirds at the trade’s rule. It does not depend on the loop’s aspect at all, because both the fabric supplied and the fabric needed are areas of the same cells. A short-row heel supplies two thirds of a right angle’s fabric and the missing third is on its sides.

The missing third is where the other heel puts its gusset #

That last sentence names a place, and knitters will recognise it.

The other classic heel — a flap worked straight down the back, a turn, and then stitches picked up along the flap’s sides and decreased away over the next few rounds — adds fabric in exactly two places: down the back of the heel, as the flap, and on each side of the foot, as the triangular gussets. The gussets are precisely the region the curve above shows as missing.

So the two constructions are not two styles of the same thing. One makes the back line of a bend and stops; the other makes the back line and the sides. That is why a short-row heel is known to fit a shallow foot and to be tight over a high instep — a high instep is a foot that needs the missing sides — and why knitters so often pick up an extra stitch or two at its corners, which puts a little fabric back exactly where the curve says it is missing.

The practice has known this for a long time, as a matter of fit. What the arithmetic adds is the size of it: a third of the fabric, located on the sides, and fixed by the share left at the turn rather than by anything about the foot.

Leaving more buys fabric and costs angle #

The share supplied rises with the share left, so the obvious repair is to leave more stitches in the middle. It does not work, and the reason is that the same number sets both.

Leave half the heel instead of a third and the trapezoid fills out: the share of a bend’s fabric rises to 75 per cent. But the back line now has only half the heel’s width in short rows each way, and the turn falls to 70 degrees. Leave a fifth and the turn rises to 113 degrees while the share falls to 60 per cent.

No choice of the one number gives both a right angle and a whole bend, because the construction has one free parameter and two things to do. That is the same shape of limit the tube’s taper ran into: one control moving two outputs along a fixed curve, so that getting one right fixes the other. The heel flap’s gussets are a second control, which is exactly why they exist.

A dart that costs no cloth #

Set beside a woven garment, the heel is doing something a woven cloth cannot.

A woven cloth shaped to a curved body needs darts, because its shear runs out: a dart takes a wedge of cloth away and closes the gap with a seam, and the cloth removed is waste. A short-row heel is a dart run the other way. It adds a wedge of courses where more length is needed instead of removing a wedge where less is, and it adds it seamlessly, because the extra courses are knitted in place.

So knitting has two ways to make a three-dimensional shape out of a flat structure and weaving has one. The fashioned edge removes wales, which is the knitted dart; the short row adds courses, which is the dart inverted. The two are quantised in different units — a fashioned edge in whole wales at whole courses, a short row in whole stitches at a turn — and both are converted into angles by the same constant.

The model named #

Every number here comes from one model and it is worth stating plainly what it is.

The tube is a cylinder of relaxed plain knit whose wale and course spacings are the loop length over Munden’s two constants, 4.3 and 5.5 in the fully relaxed state, from this collection’s loop geometry. The heel is a count: each short row is worked, the columns it passes over are credited with a course, and the columns’ totals are the heel. The angle is the back line’s extra length over the tube’s diameter, which is the definition of a bend’s angle about its inside edge. The fabric a bend needs is the extra length a bend gives every line round the tube, integrated over the circumference.

The count and the trapezoid are required to agree: the worked rows fall short of the continuous trapezoid by under one course a column, which is the stitch each short row leaves at its turn, and the two are required to agree that closely. The share supplied is required to sit within one part in the heel’s stitch count of (1 + m/H)/2 on every needle count in the sweep. And the back line of the bend and of the heel are required to agree exactly, because that is how the angle was defined — a check that the definition was implemented rather than assumed.

What was counted #

Fourteen needle counts, from 40 to 96 in steps a pattern book would print: each heel worked row by row at the nearest whole stitch to a third, its turn computed and its share of a bend’s fabric computed. Three relaxed states of the loop constants, at the exact third. And a sweep of the share left from 0.15 to 0.60 in each state, for the straight lines in the turn figure.

Nothing was fitted. The loop constants are Munden’s published ones, used unchanged, and the only choice made here is the definition of the angle — along the back line, where the heel and a bend agree — which is stated and not hidden inside a least-squares fit that would quote a smaller angle and a less honest one.

What the picture cannot show #

A heel is not a bent tube, and the drawings treat it as one. A real short-row heel is a cup: the missing third on the sides is taken up by the fabric stretching and the heel pocket bulging, and the shape it actually takes on a foot is a membrane problem with the wearer inside it. Nothing here computes that shape. What is computed is the extra length each line round the tube has, and the claim is only that a bend of the back-line angle would need more on the sides than the heel gives.

The wrap is left out. Each turn in a short row is wrapped or otherwise secured to stop a hole, and the wrap’s own yarn and tension are real and small. They add nothing to the count of courses and they are where the holes at a short-row heel’s corners come from, which is a separate matter from the missing fabric and is not modelled.

And the constants are plain jersey’s. A heel worked in a slip-stitch or a twisted-stitch pattern, or in a rib, has loops of a different aspect, constants nobody has published for most structures, and a different share to leave for the same turn. The direction of the change is predicted; its size is not.

A turn and a taper are one arithmetic #

A seamless tube’s taper and a sock’s heel are the same calculation made along two different axes. The taper changes a tube’s circumference, which is wales; the heel changes a tube’s length on one side, which is courses. Each is a count of whole loops in one direction divided by a count in the other, and the loop’s aspect is the exchange rate between them.

That is what makes the aspect worth carrying as a constant in its own right rather than as a quotient of two others. It is the only number in plain knitting that converts a count of courses into a count of wales, and every three-dimensional shape a knitter makes without cutting is built out of that conversion.

Who found it, and when #

Short-row heels are old enough that their origin is not recorded; the wrapped short row as a named technique is twentieth-century, and the rule of working to a third is in knitting manuals of the nineteenth. The flap-and-gusset heel is as old, and the observation that it fits a high instep better is craft knowledge with no author.

Munden’s constants are from 1959 and his aspect ratio is published with them. This collection’s own loop geometry carries them and has already found that the relaxed shape of a plain loop is one curve for every yarn.

Putting the aspect into the heel is what is new here, and it is one division. It says why a third, and it says that a third gives the back line and two thirds of the fabric — which turns a piece of fitting lore about high insteps into a fraction that can be read off a pattern.

Still open: whether a knitted disc can close #

The heel is a curvature made along one line. The same arithmetic asked of a whole surface is a harder question and a more revealing one.

A flat disc can be knitted two ways: outward in rounds, with increases spaced round the circle, or sideways in wedges of short rows, each wedge a trapezoid like this heel’s pointed at the centre. Either way the disc is flat only if its circumference grows by exactly 2π per unit of radius, and either way the growth is a count of loops in one direction over a count in the other. So a knitted disc is flat only at particular values of the loop’s aspect, and the two constructions need it on opposite sides of the relaxed value — which would mean that the same yarn, knitted into the same size of circle, lies flat one way and cups the other. Whether the numbers bear that out, and by how much a wash moves each disc, is the next calculation.