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In a Math Overflow post about mathematical fallacies it was stated that:

Richard Feynman regarded the mistake that a "circle is the only figure which has the same width in all directions" as one reason for the space shuttle Challenger disaster.

I haven't been able to find any references to this myself. Is it an accurate statement and if so, what is it referring to?

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    $\begingroup$ You might want to add an explicit description as to why a "circle is the only figure which as the same width in all directions" is incorrect. Curve of constant width - Wikipedia $\endgroup$ – Ray Butterworth Oct 2 at 13:18
  • $\begingroup$ The same width? Circles (and spheres) have the same distance from a single point. $\endgroup$ – RonJohn Oct 3 at 4:37
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    $\begingroup$ @RonJohn: Yes, the same width - if you measure the horizontal distance from the leftmost point to the rightmost point, then for a circle it's the same whichever way you orient the circle (twice the radius). By contrast this isn't true for a square (which will have the least width when its sides are vertical, and the most when they're at 45 degrees). But, perhaps surprisingly, the circle isn't the only shape for which the width is the same in any orientation. $\endgroup$ – psmears Oct 3 at 9:34
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    $\begingroup$ @psmears my comment should have been "from the center point". JackB gave some examples of shapes having the same width, but they fail at having the same radius everywhere. $\endgroup$ – RonJohn Oct 3 at 12:48
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    $\begingroup$ @RonJohn: Yes - but isn't that the whole point? The (potential) issue was that the checks they were performing on the shuttle parts checked constant width, but Feynman pointed out that didn't guarantee circularity... $\endgroup$ – psmears Oct 3 at 15:36
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This was indeed an avenue of investigation for Feynman. From his autobiographical book What Do You Care What Other People Think?:

Then I investigated something we were looking into as a possible contributing cause of the accident: when the booster rockets hit the ocean, they became out of round a little bit from the impact. At Kennedy they're taken apart and the sections... are packed with new propellant... During transport, the sections (which are hauled on their sides) get squashed a little bit - the softish propellant is very heavy. The total amount of squashing is only a fraction of an inch, but when you put the rocket sections back together, a small gap is enough to let hot gases through: the O-rings are only a quarter of an inch thick, and compressed only two-hundredths of an inch!

He then describes the procedure used to ensure the roundness of tanks, which was to check that the diameter was consistent at different angles around the tank - but then notes that this does not guarantee roundness, an arbitrary shape can have the same diameter at multiple different points, and there are even non-circular shapes that have a consistent diameter at every point.

Having tank sections slightly out-of-round may have contributed to the O-ring failure, and the method they used to ensure roundness was not theoretically sound, as it relied on an incorrect assumption that a circle is the only shape with a fixed diameter at all points.

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    $\begingroup$ I recently ran across a nice drawing of the circumferential tool used to "round off" the SRB casings during stacking, but I can't seem to find it again, grrrr. $\endgroup$ – Organic Marble Oct 2 at 16:53
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    $\begingroup$ As an aside, a good example of a shape which appears to have the same diameter everywhere but which isn't circular is the British 50p coin. They are that shape so coin machines can measure them. A more extreme example is this one. $\endgroup$ – Jack B Oct 2 at 19:51
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    $\begingroup$ @JackB Good example. You'll notice that most, if not all, non-circular coins in the modern day have an odd number of "sides" for this reason - you can't get a consistent diameter with an even number of "sides" (sides in quotes because the edges aren't straight segments). $\endgroup$ – Nuclear Wang Oct 2 at 20:25
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    $\begingroup$ @JackB "which appears to have the same diameter everywhere but which isn't circular". I caught that immediately in the Feynman quote. You need to test the radius. $\endgroup$ – RonJohn Oct 3 at 4:44
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    $\begingroup$ @RonJohn: …which requires you to first find (what you believe to be) the center point and then somehow accurately keep track of it throughout the measurements. Which can be easier said than done, if you're measuring something like a pipe section or, indeed, a rocket booster segment or any other similar hollow 3D cylinder. A diameter measurement is much simpler (just pick any point on the edge and find the most distant point from it on the other side) but, as noted, not sufficient to prove circularity. $\endgroup$ – Ilmari Karonen Oct 3 at 7:37
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In addition to Nuclear Wang's answer, Feynman also mentions this during a PBS Newshour interview with Jim Lehrer.

(the relevant part starting at 7:30)

While he doesn't directly mention the mathematical fallacy, he describes how the width-preserving properties that's usually observed in the automobile industry usage of o-rings, does not necessarily hold true, and how this affected the shuttle.

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Supplemental answer -

Here is a diagram of the Circumferential Alignment Tool that was used during stacking when the SRB segments were "severely" out-of-round.

enter image description here

This diagram is from Volume 2 Appendix L of the Rogers Commission Report, the report of the STS 51-L Data & Design Analysis Task Force Accident Analysis Team.

There is a lengthy writeup in Volume 1 Appendix C describing the out-of-round problems and the use of the tool in an attempt to correct them.

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    $\begingroup$ Supplemental comment, Re. "reason for the Challenger disaster?" o-rings. Why were there o-rings? So that they could be transported dissembled to fit on train cars. Why? because the company that built them is in Utah. Why? because the administrator was from Utah. Why.... $\endgroup$ – Mazura Oct 4 at 1:42
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    $\begingroup$ @Mazura Good point. The modular design was a consequence of NASA having to farm out manufacturing around the country. They had to spread the wealth to ensure congressional support for their funding. Law of Unintended Consequences at work... $\endgroup$ – Oscar Bravo Oct 4 at 5:55
  • $\begingroup$ Jim Kingsbury, Head of Engineering at MSFC in 1986, was quoted: "Would you think it mattered that I told you the leak occurred between two segments which when they went to put them together got a mismatch of an half of an inch. This one was egg-shaped this way and this one was egged shaped that way. " nasa.gov/sites/default/files/atoms/files/… Using the same "common sense" logic that lead to the cold caused the rubber to leak, which is more likely explanation for a single point leak, O-ring cold at that spot or too tight. $\endgroup$ – Challenger Truth Nov 7 at 14:46
  • $\begingroup$ As an ironic side note, the Circumferential Alignment Tool (rounding tool) was originally designed by Roger Boisjoly, the guy who later was the leading proponent of the Cold O-ring theory. $\endgroup$ – Challenger Truth Nov 7 at 15:01
  • $\begingroup$ @Mazura The modular design was not the problem, The problem was in the processing of the segments. If you put them together properly they won't leak. Put them together improperly and you create a possibility of a problem. It flew successfully 134 times. Challenger accident was just like Apollo 13, the Oxygen tank was not a bad design, simply damaged in assembly and processing. If you must point fingers, point them at the company that did the SRB stacking, Lockheed Space Operations. And interestingly, the chief legal counsel for that company was Rogers and Wells (as in William P.) $\endgroup$ – Challenger Truth Nov 7 at 15:34

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