A seventh leg has no dead yaw of its own

A seventh leg must lean its own way mapped where a seven-legged Gough platform fails near the yaw at which its six legs fail together. The six legs run between a base circle of radius 2.2 and a platform circle of radius 1.1 in three pairs. Seen from above, each leg’s platform attachment sits some way round the axis from its base attachment, an angle called its lean. At the dead yaw all six lean 82.5° or 97.5°, two angles with the same sine, so their moments about the vertical are equal and together they cannot resist one twist. A seventh leg fails with them only if its lean has that same sine. The best seventh leg, from the placement map of a seventh leg opens the dead yaw, leans 285°, the other way round the axis, and keeps its seven-leg singular set out of the dead yaw’s neighbourhood up to eight degrees of tilt.

The essay ended on a doubt about yaw. Every leg’s lean changes by the same angle as the platform turns. Perhaps at some other yaw the seventh leg’s lean comes round to match the six legs’ common lean, and the seven then have a dead yaw of their own, somewhere else in the turn.

They do not, and the reason is short enough to state before any measurement. Finding where the best leg’s singular set does lie takes longer, and the answer is that it hugs the six legs’ dead yaws at tilts of fifteen degrees and more.

Seven can only fail where six do #

A platform’s legs are lines, and each contributes one row of six line coordinates to a matrix. Six legs are singular when their 6 × 6 matrix loses rank. Seven legs are singular when their 7 × 6 matrix loses rank, which means its rank falls below six. Then every set of six of its rows is dependent, and every 6 × 6 minor vanishes. One of those minors is the six original legs’ own determinant.

So the seven legs’ singular set lies inside the six legs’. At any pose where the six legs are not singular, their six rows already span every twist, and a seventh row cannot take that away. Whatever the seventh leg’s lean does as the platform turns, the seven can only fail at a yaw where the six already fail.

That turns the question around. The thing to find is where the six legs are singular over a whole turn of yaw. A seventh leg can have a dead yaw of its own only at one of those yaws.

The six legs’ second dead yaw #

The six legs’ dead yaw comes from their moments about the vertical being equal, which is how tilted near the dead yaw explained it. Turn the platform by an angle and each leg’s lean grows by that angle. Half the legs then lean $82.5° + \psi$ and half $97.5° + \psi$, and their moments are proportional to the sines of those. The sines are equal when the two angles add to 180° or to 540°, which happens at a turn of 0° and again at 180°. Half a turn from the dead yaw the six legs lean 262.5° and 277.5°, whose sines are again equal, both −0.991, and the platform has a second dead yaw.

The argument predicts two singular yaws and does not exclude others, since the closed form holds only at a centred position. So the measurement scans the whole turn. At each of 122 positions spread through the workspace, x and y within ±0.8 and heights of 1.8, 2.4 and 3.2, the six legs’ determinant is computed every half degree of yaw, level, and each sign change is bisected.

Every one of the 122 positions has exactly two singular yaws, at 0° and 180° from the dead yaw, to $10^{-13}$ of a degree. A sign change can miss a zero the determinant only touches, so the six legs’ least singular value is also checked across the turn away from those two. It stays above 0.0036 everywhere more than three degrees from them. Level, the six legs fail at the two dead yaws and nowhere else.

The second dead yaw repeats the first #

At either dead yaw the six legs leave one twist free, a screw about the vertical axis. A seventh leg fails with them, at every level position, exactly when it does no work on that twist. At the first dead yaw that happens when its lean is 82.5° or 97.5°. At the second the six legs’ leans and the seventh’s have all moved by 180°, which flips the sign of every sine. The seventh leg again fails only when its moment matches the six legs’, at the same two leans as before.

The measurement confirms both halves. Leans 82.5° and 97.5° do no work at either dead yaw, to $10^{-9}$. At every other lean round the axis the work at the second dead yaw has the opposite sign to the first. The sizes differ a little, $-0.287$ against $0.254$ at a lean of 0°, because the legs’ lengths at a given position are not the same half a turn apart, but they never pass through nought.

For the best leg, the work across the level workspace runs from $-0.587$ to $-0.373$ at the first dead yaw and from $0.373$ to $0.560$ at the second. It is never nought, and at home the seven legs’ least singular value over the whole turn is 0.0287.

Putting the two results together: the seven legs could fail level only at the six legs’ two dead yaws, and at both the best leg does work on the free twist. The seventh leg has no dead yaw of its own. The yaw question the earlier essay asked has a negative answer that holds for any seventh leg between the same two circles that does not already lean as the six do.

Where the singular set does lie #

The count still says the seven legs’ singular set exists. Two conditions in the six-dimensional space of poses leave a four-dimensional set. The earlier essay looked for it within eight degrees of tilt of the dead yaw and found none for the best leg. It has to be somewhere among the six legs’ singular poses, and those are a five-dimensional set. Level they are just the two dead yaws, but tilted they leave them and spread. The dead yaw opens into a wedge measured the first stage of that spreading: each tilted position becomes singular at its own yaw, displaced from the dead yaw in proportion to the tilt and the position’s offset, so that within a few degrees of tilt the singular poses fill a wedge of yaw around the dead yaw. The seven legs’ singular set lives on that wedge and on whatever grows out of it at larger tilts, so the search has to follow the wedge upward, past the eight degrees where the earlier map stopped.

To find it, the search fixes a position and an axis to tilt about. Each pose is then a point in a plane, with yaw across and tilt up, and the six legs’ singular poses form curves in that plane. The seven legs are singular at points on those curves where the seventh leg does no work. Those points are the common zeros of two determinants: the six legs’ own, and the six with one leg replaced by the seventh. When the six are singular, the second determinant is the seventh leg’s work on the free twist multiplied by a cofactor of the replaced leg, so it vanishes where the seven fail.

It also vanishes where the cofactor alone does, and those zeros are false. Each candidate is found from a grid, one degree of tilt by three of yaw up to 40° of tilt, refined by Newton’s method to $10^{-11}$, and then checked by a second route that shares nothing with the determinants: the seven legs’ least singular value, computed directly. A true seven-leg singular pose has it at nought. Of 711 candidates at nine positions and eight axes, 99 pass at below $10^{-7}$ and 612 are the replaced leg’s cofactor, rejected.

At this position the six legs’ singular set is two vertical lines at the bottom of the plane, the two dead yaws, which bend away as the tilt grows. New curves join them above about 15°. The seven legs’ singular poses sit on those curves, the lowest at a tilt of 15.3° and a yaw of 177.4° from the dead yaw, almost exactly half a turn.

Fifteen degrees, near the dead yaws #

The same search at nine positions, from the centre to the corners at all three heights, and eight tilt axes gives the picture across the workspace.

The lowest tilt anywhere is 14.9°, at a low corner of the workspace tilting about the 315° axis. By axis the lowest are 18.6° about 0° and 45°, 16.2° about 90°, 15.4° about 135°, 15.3° about 180°, 18.7° about 225° and 14.9° about 315°. Tilting about 270° brings in nothing at any of the nine positions up to 40°. Every one of the first appearances lies within 26° of yaw of a dead yaw, at the first or the second. At higher tilts the poses spread further round, up to 88° from either dead yaw, but by then the platform is tilted by more than any ordinary task asks.

So the seven legs’ singular set is not somewhere else in the turn of yaw. It stays near the six legs’ dead yaws, where the six legs themselves fail, and the seventh leg pushes it up in tilt rather than across in yaw. The earlier essay’s eight degrees of clear tilt is a lower bound with room to spare: the true margin is nearly twice that.

That has a direct reading for a task. At the nine positions searched, a platform whose tilt stays within about 14° in every direction meets no seven-leg singularity at any yaw in the turn, and the level result holds at every one of the 122 positions. A task that needs more tilt than that near a dead yaw, or works far from the positions searched, has to check its own path. A task that tilts only about one axis may have more room still, since the 270° axis brought in nothing.

Leaning further buys tilt #

The same search, run at fewer positions for speed, compares other seventh legs.

A leg leaning 84°, 1.5° from half the six, fails with them at a tilt of 1.3°, which agrees with the earlier essay’s finding that it enters between one and two degrees. One leaning 105° fails at 3.8° and one leaning 90° at 6.1°. The earlier essay’s denser sampling found the 90° leg entering by 4°; fewer positions can only find a higher least, so these figures are upper bounds of the same kind as the best leg’s. Further round, 120° needs 7.6°, 150° needs 11.7°, 180° needs 13.9° and 230° needs 15.2°. The best leg, at 285°, needs 14.9° over the larger set of positions.

The order follows how far a leg’s lean is from the six legs’ own, but the curve flattens. Past about 180° every lean is far from the six legs’ in the sense that matters, with a moment about the vertical of the opposite sign, and the least tilt settles near fifteen degrees. What sets that ceiling is the six legs’ singular set, not the seventh leg. Above about 15° of tilt the six legs’ singular curves grow new branches at the positions searched, and on a branch long enough, any seventh leg’s work changes sign somewhere.

One placement of every placement found the same structure on a planar platform, where a fourth leg removed the three legs’ singular orientations unless it shared their similarity. Here the seventh leg removes the six legs’ dead yaws, level and gently tilted, unless it shares their lean. It cannot remove what the six legs’ singular set becomes at large tilts, because there the six legs fail in too many ways for one more line to be independent of all of them.

Two dead yaws are one design fact #

The second dead yaw changes one practical statement. The dead yaw is a design choice found that rotating the platform’s pairs against the base’s moves the dead yaw. It moves both dead yaws, since they are always half a turn apart. A platform whose working range of yaw is wide enough to reach half a turn from its dead yaw meets the six legs’ singular set twice. No path across the dead yaw misses it found that a six-legged path across the dead yaw must meet a singular pose. With a turn of half a revolution or more there are two dead yaws to cross.

The seventh leg handles both at once, because a leg that does work at the first dead yaw does work at the second. That is the sign reversal in the pair of curves above, and it makes the rule for placing the seventh leg unchanged by yaw. Lean it away from the six, and both of the six legs’ dead yaws are covered, level and up to about fifteen degrees of tilt.

A yaw that is singular everywhere introduced the dead yaw as one line of trigonometry about the six legs’ moments. The same line, read once more, gives the second dead yaw: equal sines come in pairs half a turn apart.

Two routes, and what each caught #

The answer rests on two separate computations agreeing.

The level scan found the six legs’ singular yaws by the sign of their determinant, and checked the stretches between by their least singular value, which cannot be fooled by a zero the determinant only touches. Neither found anything other than the two dead yaws at any of 122 positions.

The tilted search found seven-leg singular poses as common zeros of two determinants, and kept only those the seven legs’ least singular value confirms. The cofactor’s false zeros outnumber the true ones six to one, and without the second route the search would have reported 711 singular poses instead of 99. A search by the least singular value on a grid alone shows small values in the right places but cannot say whether they reach nought. Common zeros confirmed by the singular value can.

What the search leaves out #

Positions between the nine. The tilted search ran at nine positions, the centre and edges at mid-height, the centre at both other heights and two opposite corners. The least tilt varies by several degrees from one position to the next, and a position between them could come in lower than 14.9°. The level result, run at 122 positions, does not share this limit.

Other circles. Every seventh leg here runs between the six legs’ circles. The minors argument holds for any seventh leg, but the second dead yaw’s sign reversal was measured for legs between these two circles only.

Stiffness near the set. A pose fifteen degrees of tilt from the nearest seven-leg singularity can still be weakly held, and how fast the seven legs’ least singular value falls as the tilt approaches that set is not mapped.

Still open: what the six legs’ new branches are #

Above about fifteen degrees of tilt the six legs’ singular set grows branches that do not come from either dead yaw. Those branches, not the seventh leg, set the ceiling that every far-leaning seventh leg runs into.

The distinct argument would be those branches named. It would trace the six legs’ singular curves in the plane of yaw and tilt from the dead yaws upward, find where new branches are born and what kind of singularity each is, and ask whether they belong to the paired platform’s geometry or would appear on any Gough platform tilted that far. If they are this platform’s, a different pairing could push them higher and give a seventh leg more room. If they are any Gough platform’s, about fifteen degrees is the most a single redundant leg can buy near a dead yaw.