The ball that remembers where it has been

Put a ball on a table, mark a point on it, and push it round a small closed square without ever twisting it about the vertical. It comes back to the same place on the table. The mark does not come back.

That is the deepest instance in this field of the thing every essay in it is about, and it is the one with the neatest answer: the angle the ball has turned through is the area of the square divided by the square of the ball’s radius.

What a ball on a plane is #

Five numbers say where a ball is: the contact point on the table, which is two, and the ball’s orientation, which is three. Rolling without sliding ties the contact point’s velocity to the ball’s spin,

$$(\dot x,\ \dot y) = r\,(\omega_y,\ -\omega_x),$$

which is two conditions and leaves three freedoms. The third is $\omega_z$ — the ball may be twisted about the vertical line through its contact point without any sliding anywhere, because the contact is a point and a point has no width to scrub.

So a ball on a table has three freedoms where a wheel has two, and its growth vector is $3, 5$: three directions available, one round of brackets, and the remaining two arrive together.

Now forbid the twist. That is not an artificial restriction: it is what happens to a ball held between two plates, rolled under a fingertip, or driven by two rollers pressed against it, and it is what makes a ball-and-plate mechanism a mechanism rather than a free body. With $\omega_z = 0$ the same ball has

• two controls rather than three,
• three constraint rows rather than two,
• and a growth vector of $2, 3, 5$.

One constraint added, one entry gained. Nothing about the geometry has changed, and the second round of brackets now produces two new directions at once rather than one — which is what makes this particular distribution the object it is.

The rotation, measured #

The ball is rolled round a square, with the rotation composed from exact exponentials of the increments rather than integrated as nine numbers. That keeps the orientation orthogonal to machine precision instead of drifting off the group of rotations, and it means the answer can be read as an axis and an angle rather than as a matrix that has to be interpreted.

For a square of side $a$ and a ball of radius $r$, the prediction is

$$\text{angle} = \frac{a^2}{r^2},$$

and the measurement, for $r = 50$ mm:

The law holds and the departure grows. Fitting the departure against the predicted angle gives an exponent of

$$1.998,$$

so the error is second order in the angle — the next term of the same expansion, exactly what a leading-order law’s first correction does. An integration error would not behave that way; it would be constant in the angle and would fall when the sampling was refined, and this does neither. The composition of exponentials along a straight leg is exact at any number of sub-steps, since the increments about one axis commute, so there is no sampling error in these numbers at all.

Why the axis tilts #

The other half of the measurement says the same thing from a different side and is worth reporting because it is the honest limit of the law.

The rotation the ball comes back with has an axis as well as an angle. For the smallest square the axis is within 0.04% of vertical; for the largest it is 1.4% away. The law angle = area over $r^2$ is a statement about a rotation about the vertical, and the rotation is only about the vertical in the limit.

That is where the second-order departure comes from, and it is not a nuisance: it is the statement that the ball’s configuration space is five-dimensional and curved, and that a closed loop downstairs lifts to a path upstairs whose endpoint is only approximately describable by one number. The one number is the leading term and the tilt is the rest.

A third reading of the same numbers is available and is the one to trust least: the sign. The rotation comes back about $-z$ for a loop traversed anticlockwise, which is a convention question rather than a fact about balls — it depends on which way up the rolling condition was written — and the essay quotes magnitudes for that reason. What is not a convention is that reversing the loop reverses the rotation, which is checked and is the subject of a section below.

Where the area comes in #

The appearance of an area rather than a length is the signature of a bracket, and it is worth connecting to the four-leg manoeuvre explicitly.

The four-leg manoeuvre’s gain is $\varepsilon^2$ — the product of two leg lengths, which is an area in the space of controls. Here the two controls are rolling in $x$ and rolling in $y$, and a rectangular loop of $a$ by $b$ is a manoeuvre whose two legs are $a$ and $b$. The gain is $ab/r^2$: the product, divided by a length squared to make it an angle.

So the ball’s holonomy is the same phenomenon with the constant made explicit. What is special about it is that the constant is exactly $1/r^2$ and the loop can be any shape rather than a rectangle — the area is the enclosed area, whatever the boundary — which is a statement no four-leg manoeuvre can make.

The reason it can is that the ball’s rolling is a local isometry: the trace the contact point makes on the ball’s surface has the same length and the same enclosed area as the path it makes on the table. So rolling a ball round a loop of area $A$ transports the ball’s orientation round a spherical loop enclosing area $A$, and the rotation that results is the enclosed area divided by $r^2$ — which is Gauss–Bonnet on a sphere of radius $r$, arrived at from a mechanism.

What a radian of twist costs in rolling #

The law says the ball turns by the loop’s area over the square of its radius, and that is a rate. The quantity a designer of a ball drive actually spends is rolling distance, and it follows from the law in two lines with the same shape as every other cost in this field.

A square loop of side $a$ wins $a^2/r^2$ radians and costs $4a$ of rolling. One radian of twist therefore needs $r^2/a^2$ squares and $4r^2/a$ of rolling in total. For the 50 mm ball above, driven in 2 mm squares, that is 625 squares and five metres of rolling for a single radian. In 12 mm squares it is seventeen squares and 833 mm.

The $1/a$ is the same inverse law the parking manoeuvre and the four-leg wiggle both produce, and its appearing here for a third time is the point rather than a coincidence: a direction two brackets deep costs the reciprocal of the room available, whatever the mechanism. What differs between the three is only the constant, and here the constant is $4r^2$.

That constant says something the angle law does not, and it inverts the usual instinct about size. A bigger ball is more forgetful and more expensive. The holonomy per unit area goes as $1/r^2$, so a large ball barely twists at all for a given loop — and the rolling needed for a fixed twist goes as $r^2$, so getting it to twist deliberately is quadratically worse. Doubling the ball’s radius quarters the twist per square and quadruples the distance to a radian.

The scale-freedom also explains why the effect is so rarely met despite being universal. Loops that are large compared with the rolling body are common in a laboratory and rare in a machine, where a ball is usually driven a fraction of its own diameter at a time and its accumulated twist over a working cycle is correspondingly tiny. The phenomenon is not hidden by being small; it is hidden by machines being built at the ratio where it is small, and a mechanism designed at the other ratio meets it immediately.

The design consequences follow directly and they sort real mechanisms cleanly. A spherical robot or a ball drive that needs to control its heading wants a small ball and a large working area, and the ratio of the two is the whole of its manoeuvring budget. A ball-transfer unit in a conveyor wants the opposite: the ball should carry a load in any direction and should not accumulate an orientation, so a large ball is the right answer and the twist it never acquires is a feature. A trackball sits between them and resolves the tension by cheating, since it is allowed to spin about the vertical and therefore has no bracket to climb at all.

One thing the expression does not contain is worth marking, because it is what makes the law a law rather than a measurement. Nothing in $a^2/r^2$ or $4r^2/a$ has a unit in it once the two lengths are given, so the whole result is scale-free: scale the ball and the loop together and the twist per loop is identical and the rolling per radian scales linearly, exactly as a length must. There is no size at which the effect switches on. A ball bearing rolled round a millimetre square and a planet rolled round a continent obey the same expression, and the only question either of them poses is how large the loop is compared with the ball.

The other holonomies on this site #

Two, and they are the same structure rather than analogies.

A redundant arm taken round a closed loop in its task space comes back with its joints in a different place. The tool returns to $10^{-14}$; the joints have moved a third of a degree along the self-motion curve and stay moved. The evidence that this is a holonomy rather than accumulated error is refinement: 0.305°, 0.299°, 0.296° at 90, 180 and 360 steps — converging rather than shrinking.

The cat that lands on its feet does it by changing its shape round a closed loop with no external torque anywhere. The shape comes back; the orientation does not; and the amount it changes by is a bracket of the shape changes.

The three cases have different names in different literatures — geometric phase, anholonomy, the falling cat — and one structure: a closed path downstairs, an open path upstairs, and a discrepancy that is the area enclosed times a curvature. It is worth being able to see the ball’s version as the mechanism it is, because the mechanism is the one case where the discrepancy has an exact closed form and a radius that can be measured with a rule.

What this means for a machine #

A ball driven by rollers is a real mechanism — it is the inside of an old mouse, run backwards, and it is the drive of a spherical robot and of some omnidirectional platforms — and the growth vector says what such a machine can and cannot do easily.

Position is cheap. The two contact-point coordinates are directly available: roll the ball and it goes where it is rolled. First order in everything.

Orientation about the vertical is expensive. It is two brackets deep, so a manoeuvre of amplitude $\varepsilon$ wins $\varepsilon^2$ of it, and a machine that has to deliver a specified twist has to trace out an area to get it. To turn a 50 mm ball by one radian takes a loop of 2,500 mm² — a 50 mm square — which is a large motion of the ball to achieve one rotation the ball could have been given directly if twisting were allowed.

Which is why twisting is allowed wherever it can be. A ball-transfer unit, a trackball and a ball-and-socket castor all permit spin about the vertical, and permitting it takes the machine from a growth vector of $2, 3, 5$ to one of $3, 5$. That is a design decision with a measurable consequence and it is usually made without anybody noticing there was one.

Rolling one way and rolling back #

There is a version of the experiment that anybody can do with a snooker ball and a marked spot, and it is worth describing because the result is counter-intuitive in the direction that makes the point.

Roll the ball 100 mm east, then 100 mm north, then 100 mm west, then 100 mm south. It is back where it started. Now do the same four legs in the other order — north, east, south, west. It is back where it started again, and its mark is in a different place from where the first order left it, by twice the holonomy.

That is the antisymmetry of the bracket, made physical. The two orders differ by exactly the loop traversed twice, once each way, and the rotation is proportional to the signed area — so reversing the sense of the loop reverses the rotation. A ball rolled clockwise round a square comes back turned one way and anticlockwise the other, by the same angle.

It also settles a question that the word memory invites. The ball does not remember the path; it remembers the area, which is much less. Two completely different loops enclosing the same area leave the ball in the same orientation, and a loop that crosses itself contributes its two lobes with opposite signs — so a figure-of-eight of equal lobes leaves the ball exactly as it found it, having travelled a long way.

What is not here #

The forces. A ball is held down by its weight and driven by friction at its contact, and everything about whether a real ball actually rolls rather than sliding is a force question. The rows here say what a non-sliding motion is; they say nothing about when sliding starts.

The contact patch. A real ball on a real surface touches over a small area rather than at a point, which means a real ball resists twisting — there is spin friction, proportional to the patch size — and a real ball’s rolling is not quite this mechanism. The idealisation is the standard one and the correction is a contact-mechanics question.

Two balls rolling on each other. The same construction applies to a sphere rolling on a sphere, and the growth vector depends on the ratio of the radii in a way that is genuinely surprising — one particular ratio gives a distribution with an exceptional symmetry group. It is named here and not built, because what it needs is a different object from anything else in this field.

The measurement that would have been wrong #

One correction is worth recording because it changed a number in a caption.

The first version of the holonomy check used loops of 10 to 120 mm on a 50 mm ball, which gives predicted angles from 0.04 to 5.76 radians. At the top of that range the ball has turned most of the way over, the axis has tilted to 15° from vertical, and the predicted angle has passed $\pi$ — so the comparison is between a measured rotation, which is defined modulo a full turn, and a predicted number that is not.

The result was a table with relative errors of 3% at the bottom and 60% at the top, which read as a law that works for small loops and breaks down. That is not what is happening. The law is a limit and every one of those rows was outside it; what the table was measuring was how far outside.

Moving the loops down to 2–12 mm puts the whole range inside the regime the law describes, and the errors then behave like a series remainder — 0.013% to 0.48%, rising as the square. The lesson is the ordinary one about quoting a limit: a leading-order law needs its measurements taken where the leading order leads, and a table that spans the transition measures the transition rather than the law.