The Number That Was Only a Curvature
Two physicists want to convince you that two shapes are secretly the same shape. They have a good argument and they have a number, and the number is the part I want to look at, because the number is doing no work at all.
The shapes are double helices — ropes, DNA, anything with two strands wound on a common axis. There is a one-parameter family of them: wind the strands tighter and the pitch angle changes, and if you insist the strands stay in contact without overlapping, the family becomes a curve. Every ideal rope lives somewhere on that curve. The question is which point, and the honest answer is that it depends what you think the rope is for.
If you think a rope is held together by molecular forces that like evenly distributed matter, you want the point that packs the strands most densely. Call that the close-packed helix. It sits at a pitch angle of 32.5 degrees, and I get 32.4487, which is as close as I am going to get to a number someone rounded.
If you think a rope is a thing that resists being untwisted, you want a different point, defined by a different extremum, and you land somewhere else.
Olsen and Bohr noticed that the somewhere else is not very else. In a short paper from 2018 they identify a configuration they call the dual point, reachable from the maximal-torsion helix by unwinding at constant twist, and they put it at 32.9 degrees. Four tenths of a degree from close-packing. They conclude, reasonably, that for any symmetric double helix there is a near-perfect duality between the two states.
Then they offer the evidence. The close-packed structure has a packing fraction of 0.7694. The dual structure has 0.7693. Agreement to one part in ten thousand.
What a stationary point owes you
Here is the trouble, and it is not subtle once you say it out loud. The close-packed helix is defined as the one that maximises packing fraction. A maximum is flat. That is what the word means.
So take any configuration a small distance from it. The packing fraction there is lower, but only by second order in the distance — the first-order term is zero, because that is the definition of the thing you are standing on. Write the relative drop as one half times a curvature times the squared offset. The curvature is a property of the curve, not of whatever argument brought you to the offset.
I measured that curvature. For the double helix it is 2.7046 per radian squared, stable to five digits when I change the finite-difference step by a factor of eight. The dual point sits 0.4513 degrees from my maximum. Second order and nothing else predicts a drop of 6.457 in units of ten to the minus five.
The drop I actually compute is 6.462.
Four significant figures, from the curvature and the offset, with no duality anywhere in the arithmetic. Whatever that 0.0001 is evidence of, it is not evidence that the two structures are related. It is evidence that 0.45 degrees is a short distance and that the function has a top.
The mirror
I could still be fooling myself here. Maybe the agreement is real and the flatness calculation is a coincidence that happens to land on the same value. There is a cheap test, and I think it is the whole method in one move.
Go the other way. Take the angle 0.4513 degrees below the maximum instead of above it: 31.997 degrees. Nothing distinguishes it. No unwinding at constant twist arrives there, and it is the dual of nothing whatever. It is the mirror of the dual point through the optimum, and that is its only property.
Its packing fraction is lower by 6.454 in the same units.
So the quantity cannot tell the dual configuration from an arbitrary configuration the same distance off the peak. A measurement that returns the same answer for the thing you care about and for its meaningless reflection is not measuring the thing you care about.
This does not kill the claim
I want to be careful, because there is a cheap version of this essay that sneers, and it would be wrong.
The duality might be entirely real. The actual content of the claim is that the dual point lands within half a degree of close-packing, and that is a genuine fact about the geometry — not a trivial one, and not one I could have guessed. Olsen and Bohr found it. I only rebuilt it.
What the flatness changes is the units the claim should be stated in. Half a degree is small compared to a degree. It is not small compared to one part in ten thousand, and quoting the packing fractions invites a reader to feel a precision that is not there. Give the curvature alongside and the claim gets stronger, because now the reader can see that the proximity in angle is the result and the proximity in packing fraction is bookkeeping that follows from it.
There is a related number I think matters more, and it points the other way from the paper that produced it. Olsen and Bohr’s triple helix — their candidate structural motif for collagen — has a close-packed configuration at 43.3 degrees and a zero-twist configuration at 42.8, and they call those practically identical too. They are 0.487 degrees apart, and I get 43.322 and 42.835, so I believe them.
But the two principles are not equally fussy. Taking each objective at its own optimum, the twist condition has a curvature of 4.7807 and the packing fraction 1.6686. Zero twist is the stiffer condition by a factor of 2.87. Put differently: the band of pitch angles within which each objective is degraded by less than one part in a thousand is plus or minus 1.17 degrees for twist and plus or minus 2.01 degrees for packing.
Both bands contain the separation. Neither principle selects a pitch angle to better than about a degree, so neither one can claim the structure. And of the two, packing fraction is the weaker discriminator — which is awkward for an argument that a biological structure sits where it does because of packing, since packing is the criterion least able to tell that position from its neighbours.
Why I think this generalises
Flatness near an optimum is not an obscure fact. It is the first thing anyone learns about maxima and it is in every numerical methods course under the heading of why optimisation is hard and error bars are wide.
And yet it stays invisible in exactly this situation, because the sentence that hides it is so natural. Two structures, and their objective values agree to four decimal places. That sounds like independent confirmation. It reads as two witnesses agreeing. But when the objective is the one whose optimum defines one of the witnesses, the agreement was arranged in advance by calculus, and the second witness never spoke.
The test I keep coming back to is the mirror, because it costs one function evaluation and it cannot be argued with. Find the meaningless twin of your result, the same distance out with no story attached, and compute the same quantity for it. If the number does not change, the number was never about your story.
I ran it on myself twice this week. The first time I had a finding about two optima being genuinely distinct and had written it up before discovering Olsen and Bohr said it plainly in 2012, in a paper that was already on my own disk. The second time I went looking for an email address and found a paper of theirs from 2018 that I had not read, which is the one this essay is about, and which I was four minutes from writing to them without having seen.
Neither of those is a story about being careless. They are stories about the cheapest available check being the one nobody runs, including me, and including on the day I write an essay about running it.
