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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?

StayOnTarget
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    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 – Ray Butterworth Oct 02 '19 at 13:18
  • The same width? Circles (and spheres) have the same distance from a single point. – RonJohn Oct 03 '19 at 04:37
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    @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. – psmears Oct 03 '19 at 09:34
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    @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. – RonJohn Oct 03 '19 at 12:48
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    @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... – psmears Oct 03 '19 at 15:36
  • @psmears not arguing, but I've never heard the term "width" used in this context. Of course, I'm not a mathematician or geometer. My first instinct would be how to verify that the radius is always the same. – RonJohn Oct 03 '19 at 15:50
  • @psmears one way to do this is to measure the sides of inscribed triangles and then calculate the circumradii of the triangles, which should be the same for all inscribed triangles. I actually tried to make this an answer, but (possibly because I got tangled up in typos) it was downvoted leading me delete it. – Oscar Lanzi Oct 05 '19 at 23:27

3 Answers3

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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.

StayOnTarget
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Nuclear Hoagie
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    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. – Organic Marble Oct 02 '19 at 16:53
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    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. – Jack B Oct 02 '19 at 19:51
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    @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). – Nuclear Hoagie Oct 02 '19 at 20:25
  • I finally re-found it, yay! Will post a supplemental answer. – Organic Marble Oct 03 '19 at 01:12
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    @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. – RonJohn Oct 03 '19 at 04:44
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    @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. – Ilmari Karonen Oct 03 '19 at 07:37
  • @IlmariKaronen I would think of an object having a certain expected circular radius using a comparison test object would be the way to go. For instance, a carefully machined cone could be inserted and then any gaps measured to be within some predetermined tolerance. Perhaps corrected for conditions / temperature / etc. – StayOnTarget Oct 03 '19 at 11:37
  • @IlmariKaronen great point! The tool actually used on the SRBs worked on the diameter, not the radius. – Organic Marble Oct 03 '19 at 12:14
  • @IlmariKaronen right. Simpler, but fails the examples demonstrated by JackB, whereas radius succeeds. – RonJohn Oct 03 '19 at 12:45
  • Why would roundness, or the lack of it, prevent an O-ring from sealing? For an extreme example, the valve covers and other parts of my cars' engines are sealed with O-rings, even though their shapes are anything but round. (Rectangles with some squiggles, basically.) – jamesqf Oct 03 '19 at 18:24
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    @jamesqf It's not directly because of the lack of roundness, but because of the difference in roundness between the 2 segments pinching the o-ring. from the Rogers report "If the very tight tang-to-clevis assembly gap did persist to time of launch, it could have resulted in near maximum compression of the O-rings. Such compression, in conjunction with cold temperatures, joint dynamics, and the variable performance of the insulating putty has been shown to have detrimental influences on the joint's ability to seal. " – Organic Marble Oct 03 '19 at 19:51
  • @Organic Marble: So it's not really a matter of roundness, but of the two segments not fitting properly? – jamesqf Oct 04 '19 at 01:35
  • @jamesqf shrugs if they were perfectly round and the same size, I guess they would fit properly. – Organic Marble Oct 04 '19 at 01:37
  • @jamesqf - it's a matter of being able to transport them on train cars, otherwise they'd have been welded and not have o-rings to fail. #thelowestbidder – Mazura Oct 04 '19 at 01:38
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    @Mazura more like #theadminstratorwasfromUtah – Organic Marble Oct 04 '19 at 01:38
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    @Organic Marble: And if the segments were all perfectly square, they'd fit together properly, too. – jamesqf Oct 04 '19 at 17:21
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    @NuclearWang: Actually, constant-width shapes can have an even number of arcs (got this citation from Wikipedia; see fig 3 and the text below: "It is worth noting here that a curve of constant width may have any number of corners..."). It's the regular-polygon-based Reuleaux shapes that have to have odd numbers of arcs. – Tim Pederick Oct 05 '19 at 05:41
  • Commissioner Joe Sutter pointed me the following exchange in March 1986 which puts the nail in the Cold O-ring coffin:

    MR. LEE: That says if it's not squeezed any more than it should be, then at 25 degrees and up the resiliency issue doesn't come into play, but at the .004 gap it means you squeeze the heck out of it. Do you understand what it is doing there?

    MR. SUTTER: Yes.

    MR. LEE: So it doesn't recover from an over-squeeze.

    Temperature was not the overriding factor and the squeeze, not the resiliency is the key to joint performance.

     – Challenger Truth Nov 07 '19 at 14:54
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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.

SE - stop firing the good guys
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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.

Organic Marble
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    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.... – Mazura Oct 04 '19 at 01:42
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    @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... – Oscar Bravo Oct 04 '19 at 05:55
  • 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. " https://www.nasa.gov/sites/default/files/atoms/files/19930303_james_e._kingsbury_oral_history_interview.pdf 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. – Challenger Truth Nov 07 '19 at 14:46
  • 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. – Challenger Truth Nov 07 '19 at 15:01
  • @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.) – Challenger Truth Nov 07 '19 at 15:34
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    @Organic Marble The answer above is incorrect. This is not a drawing of the "rounding tool" It is actually a drawing of the VAB lifting crane with the 4 attachment points which were used to raise and lower the segments. This process is described in Appendix C as "4 point or 2 point" hang. This was done on the 51L segments RH aft segment which leaked as an initial attempt to reduce ovality.
    The rounding tool consisted of a long threaded rod with wooden block on each end. The tool was tightened manually or later with hydraulics to "squeeze" the segment across one specific diameter.     – Challenger Truth Nov 08 '19 at 14:19