#### experimentalist

##### Plastic

- Joined
- Jan 5, 2024

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:

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:

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:

- To understand the STEP file, I must first touch upon the system I propose we use to measure the part.

- We would use a Renishaw SP25M scanning Probe, which will drag itself along and around the outer rim of your circular part (depending on the roughness of the surface) Click the BLUE link to see a sample video

- While the probe drags itself along the circumference of your part, it collects points along the arc, in a pre-determined resolution. This point density can be set before measurement (ex each point can be 1mm apart, or as tight as 0.1mm, although that would add a lot of time to the scan)
- The complete scan STEP file can be imported into a CAD software, and when looked at as a whole it may resemble an unbroken line, but when zoomed in, it is apparent that the circle is made up of points closely together
- This same STEP file is what is used to calculate the max circumscribed and min inscribed size
and determine the circumference in this fashion, andWe can measure a straight line between each pointthis would be the most accurate we can get with our measurement

- Please note that, although the measurements themselves will be within about 4 microns of reality,
per se, but ratherwe would not be measuring the circumference(point resolution to be determined, but I believe the limit for our system is 0.1mm apart)the sum of straight line lengths between each point

__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.we would not be measuring the circumference per se

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,

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

__not__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!