The hold is in the corners

A part resting against a contact may move away from it and may not move into it. A constraint that only pushes turned that into a test: a set of contacts holds a part when their rows, each the contact’s direction of push with the moment it makes about a chosen point, surround the origin. Four in the plane is the fewest that can, and the test gives a number as well as a verdict — the margin, the distance from the origin to the boundary of the rows’ convex hull, which is nought exactly when something escapes and grows with how far a hold is from letting go.

The same essay recorded the one planar part no contacts hold, and the test that decides it had once reported it held, which is how the disc came to be the case every route is checked against. Every normal to a circle passes through its centre, so every contact on a disc makes no moment at all, and a turn about the centre is resisted by nothing. A regular polygon is held by four. A regular polygon with a hundred sides is, to the eye, a disc.

So the hold is lost somewhere between the square and the circle, and the question is how. It could fall off a cliff at some number of sides, it could fade as the square of the edge length, as the area a polygon differs from its circle does, or it could fade as the edge itself. Which one it is decides how round a part can be before a fixture needs friction, a flat or a pin to hold it.

What a contact on a flat edge can resist #

A contact on an edge pushes along the edge’s inward normal. That line misses the polygon’s centre by the contact’s distance from the edge’s midpoint, and that distance is its moment arm: how much turning it can resist for each unit of push. At the midpoint the arm is nought and the contact is as useless against rotation as any contact on a disc. At either end of the edge the arm is largest, half the edge — and for a regular polygon of circumradius 1, half the edge is sin(π/n).

That already shows where the hold lives. On an octagon held by its best four contacts, move one contact from its corner towards the middle of its edge and the margin falls in a straight line, 0.284 to 0.158 at the midpoint. Carry on to the far corner and the arm returns to full length pointing the wrong way. The four arms no longer alternate in sign, and the margin reaches nothing. A contact is worth what its arm is worth, and the arm is set by where it sits on a flat edge.

The margin is measured with the polygon’s circumradius as the length that turns moments into the same units as forces, which a cone has no size showed is a choice every margin carries. A polygon twice the size, with the length doubled, has the same margin, so every number below is a property of the shape.

What the margin weighs #

Each contact’s row has three entries: its moment about the centre, divided by the circumradius, and the two components of its push. Scaled to unit length, a row on a nearly round part is almost entirely push, with a small moment entry of about the arm. Four such rows lie close to a circle in the plane of pushes, two of them lifted slightly above that plane and two dropped slightly below it.

The origin is inside their hull only if the rows surround it in all three directions at once. In the two directions of push, four rows spread round the circle surround it comfortably. In the third direction, the moment, the rows reach only as far as their arms, so the nearest face of the hull is at a distance set by the arms. That is why the margin tracks the arm so closely: on a part with small arms, turning is the direction in which the hold is thinnest, and a margin measures a hold at its thinnest.

On a square the picture is different, because the arms are not small. A contact at a corner of a square of circumradius 1 has an arm of 0.707, as large as its push is long, and the rows sit well away from the plane of pushes. The best margin there is exactly one third, and it is limited as much by how the four directions of push are spread as by the arms. The small polygons are nowhere near the half-edge law’s limit: the square’s margin is 47% of its half-edge, against 74% on the octagon.

Why the octagon is the picture #

The figure is the octagon’s best four. The thick line from the centre to each contact’s line of push is that contact’s arm, 0.383 each, and the four arms point in four directions a quarter-turn apart. The contacts sit at the start of one edge, the end of the edge a quarter-turn round, the start of the opposite edge and the end of the fourth. Their pushes point in four directions a quarter-turn apart as well, so the arrangement is as even in the plane of pushes as it is in its moments.

The margin is 0.284, 74% of the arm. It is not the whole arm because the rows are normalised: a row whose moment entry is 0.383 and whose push is 1 has unit length only after dividing both by 1.07, and the facets of the hull lean between the four rows rather than lying flat above them. As the arms shrink both effects vanish together, which is the $1/n^2$ in the law below.

Searching for the best four #

The question has an exact form. Over every placement of four contacts on a regular n-gon, what is the largest margin, and where are the contacts?

A placement is four edges and four positions along them, a space of dimension four with a discrete part. The search has two halves. The first is exhaustive over the placements with every contact at an end of an edge: there are 2n such positions, one at each end of each edge with that edge’s normal, and fixing the first by the polygon’s rotational symmetry leaves every choice of the other three. That is 455 placements for the octagon and 16,215 for the twenty-four-gon, each given its margin exactly. The best of them is then refined: each contact moved along its own edge by steps halving from an eighth of the edge to 10⁻⁷, and onto neighbouring edges, for as long as the margin improves. The second half is 24 random interior placements, refined in the same way, which ask whether anything better exists away from the corners.

No interior start beats the refined corner placement on any polygon. From six sides up, the refinement never moves a contact off its corner at all. The best four contacts sit at edge ends, and they take a particular shape.

On the square and every even polygon the four contacts take alternate ends of four edges a quarter-turn apart: the start of one edge, the end of the edge a quarter-turn round, the start of the opposite edge, the end of the last. Two push one way about the centre and two the other, and they push in four directions spread evenly round the plane. On every polygon whose number of sides is a multiple of four, the search’s best equals that arrangement’s margin to 10⁻⁹. It was not assumed; it is what the search returned.

Two polygons that break the pattern #

The triangle and the pentagon do not put their contacts at corners. On each of them the refinement walks two of the four contacts in from the ends of their edges to near the middles — 0.48 and 0.52 of the way along on the triangle, 0.49 and 0.51 on the pentagon — and leaves the other two at corners.

The reason is that the arm is not the only thing a contact supplies. Its push also has a direction, and four directions have to surround every translation as well as every turn. A triangle has only three edge directions, so four contacts on it must put two on one edge or accept directions that are poorly spread; the pentagon’s five directions are spread at 72°, and the quarter-turn arrangement has no pair of edges near 90° apart to use. On those two shapes the best compromise gives up some arm to get better directions. From six sides on there are always edges close enough to a quarter-turn apart that the corners win outright.

Odd polygons pay for this at every size the search covers. The heptagon holds at 0.206 against the hexagon’s 0.293 and the octagon’s 0.284; the nine-gon at 0.190 against 0.284 and 0.248; the eleven-gon at 0.183 against 0.248 and 0.223. An odd polygon has no edge exactly a quarter-turn from another, and its best contacts settle on edges a little more or a little less than a quarter-turn apart, which spreads their directions less evenly.

The half-edge, less a square #

Set every best margin against the half-edge sin(π/n), and the law appears.

From eight sides on, every margin is below the half-edge and closing on it: 74.2% at eight sides, 91.4% at sixteen, 95.9% at twenty-four, and for the quarter-turn arrangement 97.7% at thirty-two and 99.4% at sixty-four. The margin does not fall as the square of the edge. It falls as the edge itself, and at large n it simply is the half-edge — the arm of a contact at a corner.

How fast it closes on the half-edge is a second law, and it can be read off by multiplying the shortfall by $n^2$.

The product climbs from 16.5 at eight sides to 21.9 at sixteen and 23.9 at thirty-two, and then stops: 24.48 at sixty-four, 24.63 at 128, 24.66 at 256. So the margin is sin(π/n) times one less about 24.7/$n^2$. That constant sits within a tenth of a per cent of $5\pi^2/2$, which suggests the correction has a closed form in the angle between neighbouring edges; the measurement is the claim here and the closed form is not.

Why it is the edge and not its square #

The difference between a polygon and its circle is an area of order $1/n^2$, and it would have been natural to expect a hold that depends on how far a shape is from round to fade the same way. It does not, because a hold does not depend on how far the shape is from round. It depends on the largest moment arm a flat surface can give a contact, and that is a length along the surface: half an edge, of order 1/n.

A disc has no flat and gives no arm. A polygon’s flats are what hold it, and the part of each flat that holds is its ends. A contact at the middle of an edge contributes no resistance to turning at all; the octagon’s four best contacts, moved to the middles of their edges, hold nothing, because every one of them then pushes through the centre exactly as a contact on a disc does.

So a fixture holding a nearly round part by frictionless contacts is holding it with its corners, and the hold is proportional to the length of the flat a contact can reach the end of. A part with a hundred facets and a circumradius of 50 mm has a half-edge of 1.57 mm, and its hold margin with four ideal contacts is 0.031 — a ninth of the octagon’s. On the octagon, moving all four contacts a tenth of the way in from their corners already costs 12% of the margin, from 0.284 to 0.250; on the hundred-sided part a tenth of the way is 0.16 mm.

The comparison with a part that is smooth but not round is worth making. An ellipse has no flats and no corners, and held in a pocket its own size it has no first-order margin at all, yet it cannot turn: the penetration grows as the square of the angle. A regular polygon is the opposite case. It has a first-order margin, carried by its corners, and that margin shrinks to nothing as the corners flatten into a circle.

What it means for a fixture #

Put the contacts at the ends of flats. The intuition that pads belong well inside a face is right for a surface that bends or wears, and wrong for resisting rotation of a round-ish part. A contact near a corner is also the one whose position matters most, so the tolerance budget that decides which contact to make accurately should be read with the arm in mind: on a polygon with small flats the whole hold is in a fraction of a millimetre at each end.

Alternate the ends. Two contacts at the same end of their edges resist the same turn. The quarter-turn arrangement takes the start of one edge and the end of the next, so that half the contacts resist each sense of rotation.

Prefer an even number of flats. Up to eleven sides, an odd polygon holds less than either of its even neighbours with its best four contacts.

Expect the triangle and the pentagon to disagree. On shapes with too few, or badly spread, edge directions, the best placement is a compromise between arm and direction, and it puts some contacts near the middles of edges.

What the search does not settle #

That the corner arrangement is optimal everywhere. The corner placements are searched exhaustively, and refinement and random interior starts find nothing better on any polygon up to twenty-four sides. That is strong evidence and not a proof: a better interior placement on a large polygon would have to lie in a basin no random start fell into. Beyond twenty-four sides the figures use the quarter-turn arrangement and do not search.

Where the part ends up. A margin says whether a part is held and by how much. It does not say where it sits when contacts have clearance, which is a different question with a size of its own, and a polygon’s corners matter to that one too.

Contacts exactly at corners. A contact at the end of an edge is given the edge’s normal. A real pad at a corner meets a vertex, where the normal is undefined and the contact can slip onto the next face. How far from the corner a physical contact must sit, and what that costs, is the arm lost: a contact a tenth of the way in keeps nine tenths of it.

More than four contacts. Five or six contacts can spread their directions and arms more evenly, and a fixture with spare contacts survives one failing to touch. Whether the half-edge law survives more contacts, or its constant only changes, is not measured.

Friction. A contact with friction can push at an angle to the normal, so a contact at the middle of a flat regains an arm equal to the friction coefficient times the circumradius. That is of order one, not 1/n, and it is why real round parts are held at all.

Still open: the facet a friction cone repairs #

With friction coefficient μ, a contact can push anywhere within an angle arctan μ of its normal, and its row is no longer a single vector but a small cone of them. On a disc that restores a moment arm of μ at every contact, and a polygon then has two sources of arm: the flat’s half-edge and the friction’s μ.

The distinct argument there would be to find where those cross: the number of sides at which friction contributes more to the hold than the corners do, for a given μ, computed with the same margin and the same search over four contacts whose rows are replaced by the edges of their friction cones. The expectation from the arithmetic is a crossing near n ≈ π/μ — a hundred sides at μ = 0.03, ten at μ = 0.3 — and a measurement would say whether the hold is ever the sum of the two, or only ever the larger.