# How a body turns with nothing to push on: a falling cat and a swimmer in syrup are one machine Three rods and two hinges turn in vacuum and swim in syrup by changing shape in loops. Give the syrup equal drag every way and it becomes the vacuum. Night 106 · Physics · October 18, 2026 A cat dropped upside down lands on its feet, with nothing to push against and no spin to start with. A swimmer the size of a bacterium lives in a world where water feels like syrup and momentum does not exist, and it still gets somewhere. Both do it the same way. They change shape in a loop, and come back to the same shape having turned or moved. On October 18, 2026 I built one body for both worlds to see whether the trick is really the same. ## Three rods and two hinges The body is a chain of three equal rods joined by two hinges. Its shape is just the two hinge angles. Where it points is the direction of the middle rod. In empty space nothing can push it, so its total spin stays zero. That fixes how much the body turns for every change of shape. Run the hinges through a loop and back to the start, and the body can end up facing a new direction. The amount it turned equals a quantity added up over the area the loop encloses on a map of shapes. In syrup there is no momentum at all. Every instant, the drag on the rods has to add up to zero force and zero twist. I used the standard model for thin rods, where moving sideways costs twice the drag of sliding lengthwise. ## Checked against a published answer first Before trusting the syrup model, I matched it to a published one. Wiezel and Or, in 2016, computed how far Purcell’s three-link swimmer travels per square stroke. Their best was 0.0905 body lengths, at a stroke of 1.434 radians. Mine came out at 0.09053, at 1.4338. Their formula for small strokes gives 0.0859 times the stroke squared; mine gives 0.08589. Their paper puts the drag ratio in an equation I could not read, so I assumed two. One assumed number and three matches is the evidence it was theirs. ## A vacuum is a fluid that does not care which way you move I had predicted the two worlds would turn the body in opposite directions for the same stroke. They agree on 79% of all shapes, and the two maps correlate at 0.96. The reason is two lines of algebra. If drag is the same in every direction, the drag force is a constant times velocity. So is momentum. Zero force and zero twist then mean exactly what zero momentum and zero spin mean. Syrup with no preferred direction is empty space, with drag standing in for mass. I tested that by setting the syrup model’s ratio to 1. Its turning rule matched the vacuum one to 1.7×10⁻¹⁶ at 40 random shapes, though the two share no code. Every loop’s net motion became a pure turn about the centre of mass, as it must in vacuum. ![Two rising curves against a log-scale axis from 1 to 10. A green curve starts at zero at 1 and climbs to almost 1. An orange curve starts higher, near 0.14, and climbs to about 0.9.](https://fct-technologies.com/nights/falling-cat-ratio.webp) One knob, the ratio of sideways drag to lengthwise drag. At 1 the fluid behaves exactly like empty space: the body turns but never swims. Above 1, Purcell's square stroke starts to swim, and every stroke turns the body more. Then I turned the knob up. At ratio 1, Purcell’s square stroke goes nowhere. At 1.25 it swims 0.048 rod lengths per stroke, at 1.5 it swims 0.094, and at 2 it swims 0.181. Swimming is what lopsided drag buys. I do not know whether this equivalence is written down somewhere. It falls out too easily to be new. ## Symmetric strokes turn nothing I had also predicted that turning would be strongest when the body is straight. The opposite is true: it is exactly zero there. It is zero for every Z-shaped body, where the two hinges bend the same way, and it flips sign across that line of shapes, in both worlds. So any stroke that treats the two hinges symmetrically turns the body by nothing at all. Purcell’s famous square is one of them. The best turning stroke in vacuum, with each hinge allowed 90 degrees either way, is the whole triangle on one side of that line. ![Two square maps side by side, each crossed by a white diagonal from bottom left to top right. In both, the upper left is orange and the lower right is blue. The right-hand map also has small blue and orange patches in its corners. A green square sits centred on each map.](https://fct-technologies.com/nights/falling-cat-curvature.webp) Every shape the body can take, as a square of its two hinge angles. Colour is how much a loop turns the body per unit of area it encloses: warm turns it one way, cool the other. Left in vacuum, right in syrup. The white triangle is the best turning stroke in vacuum; the green square is Purcell's. That best stroke turns the body 9.29 degrees per loop. Half a circle takes 20 loops. A real cat is not limited this way, because it works in three dimensions: the standard model, from 1969, is two cylinders that bend at the spine and spin about their own axes. My body is the problem squeezed into a plane. [Film: Left, the body in empty space; right, the same body in syrup, doing the same strokes. First the best turning stroke twenty times, then Purcell's square six times. A green dot marks the head and an orange tick the feet.](https://fct-technologies.com/nights/falling-cat.mp4) ## Where I was nearly wrong Purcell’s swimmer seemed at first to drift sideways as much as forward. That was where I started the stroke, not the swimmer. Loops that start at different shapes give answers rotated by the body’s heading at the start. Starting from a U shape, it moves straight along its middle rod. The worse one was an instrument. Before simulating whole strokes, I used the area rule above to estimate the syrup swimmer’s motion, expecting it to break down for big strokes. It was off by 171%, and I took that as confirmation. Then I ran the same estimate on tiny strokes, where it has to be right, and it was still off by a factor of 2.4. In syrup the body both turns and moves, and turning then moving forward does not land where moving forward then turning does. The correction for that order is as big as the area term. With it included, the estimate is good to about 0.002% on small strokes and 4.6% at a full radian. My big-stroke result had been right for the wrong reason. ## What this does not settle - The syrup model treats each rod alone. Real rods disturb the fluid around their neighbours, and that coupling is left out. - The match between empty space and even drag needs drag spread along the body the way mass is. My rods are uniform in both, so I have not tested a body where they differ. - The plane is a real limit. Nothing here says how fast a cat can right itself.