Hello,

To help convey what is an otherwise deeply embedded problem, person A & B are having a discussion

about a hypothetical 1 m diameter circular object rolling straight on a flat plane surface. Person C is present & listening.

Person A believes the object will roll no further than 3.1416...m because its circumference is π m according to c = πd for d = 1.

Person B claims according to the Pythagorean theorem, the same object will actually roll no less than than 3.1446...m.

A reminds B of Archimedes' upper- and lower-bounds 223/71 < π < 22/7 with 22/7 < 3.1446... therefore the latter can't be correct.

B reminds A of a property of plane geometry called

*isoperimetric inequality* & its proof finally coming only as recently as the 19th C:

View attachment 422847
source
with it always having applied not only to Archimedes' non-circular approach to circle measurement,

but to any & all similarly exhaustive approaches which neglect to establish isoperimetric

*equality*
between line & curve from the onset.

A does not believe anything B says & instead asks A for the math predicting such a number.

B produces:

View attachment 422841
and suggests any & all carefully designed & controlled real-world experiments

will reliably & reproducibly reflect 3.1446... instead of 3.14159... as per Pythagoras.

Person A still doesn't believe this & orders a simple flat 1m diameter circular disk from a precision engineer

& takes it to a metrology lab to have its diameter & circumference measured. However, when asking

*how*
the circumference would be measured & how accurately, the metrologist responded concerning the use of a STEP file:

__emphasis added__
Person A asks B whether or not this measurement approach is affected by isoperimetric inequality.

Person B states it

*is* & metrologists don't actually

*measure* the circumference... as the one above admits:

because the probed points are afterwards being treated as the vertices of a non-circular polygon whose perimeter is taken instead.

B goes on: in fairness to metrologists, mathematicians dropped the ball on failing to retroactively apply isoperimetric inequality to Archimedes' pi.

Upon proving it, they forgot they've always used polygons to arrive at 3.14159... itself, which is meant to be a perfect circle (it is not).

It is impossible to surround any real unit diameter circlular object with a length of only 3.14159... it will never fully surround.

The isoperimetrically equivalent polygon of sides n is n = 4,

__not__ n → ∞ because each c/4 of c reflects one of four discrete convex sides.

For a circle, its side is convex (ie. curved) but nonetheless discrete: it is contained by two right-angled radii of length 1/2 each.

As it rolls, the convex side translates itself as a linear line onto the flat plane directly. This implies isoperimetric equaltity at n = 4.

Person A notices C has been silent & asks what they think.

C states they don't believe either until they see the results of a properly conducted experiment

involving taking a

*direct* *measurement* of the circumference of a sufficiently circular object

in a way which eliminates any & all possible introductions of isoperimetric inequality.

This means no point probes or connect-the-dots, no use n-gons which diverge away from n = 4.

Accuracy min. req. is nearest circ. mm per diametric m:

If 3.1416... is closer to reality, we expect a roll distance of ~3141.6 mm per full rot.

If 3.1446... is closer to reality, we expect a roll distance of ~3144.6 mm per full rot.

The raw numerical difference we need to capture is 3 mm on the circumference.

If you were (or are) a metrologist or engineer... how would you design & conduct a measurement experiment

which circumvents the problem of isoperimetric inequality while/as being accurate enough to the nearest mm?

You can use any circular object, precision engineer any parts & use any digital technology you want.

I am in the position of C and will be performing the experiment myself and am happy to share when completed.

Thanks for reading & looking forward to different ideas & approaches!