Physically Straightening the Circle

Continuing my riff on whether arc lengths are fully commensurate with radial or linear lengths. We certainly seem to have the ability to physically straighten out a circle. Take a circular object like a hula hoop, cut it, and simply flatten it into a straight line. Measure the distance with a ruler and voila, we have converted arc length into linear length.Or dip the hoop in paint and roll it one complete revolution on a flat surface. Measure the length of the mark left on the surface and voila we have done it again. Arc length converts into radial length. Problem solved right?

Not precisely. Any physical object we can mold into a circle and then straighten out again is going to have a characteristic stretch to it (I believe the technical term is "ductile"). The length when it is straightened is going to depend on how ductile the material composition is. Different materials will stretch out differently. Which one will give us the correct linear conversion of a circular object of a fixed radius?

Similarly the ability of a circular object to move on a flat surface depends on static friction. No friction, no forward motion. Picture a tire spinning on ice. But the friction between a surface and a wheel can vary and hence the forward motion in one rotation will not be precisely the same for any two differently composed wheels or surfaces.

Yes I know they will be close and if most circular objects of fixed radius unbend and yield close to the same liner distance, why quibble? Close enough for government purposes, right? Always the querulous quibbler.

Maybe, but maybe not. Maybe the fact that the constant of proportionality, Pi ,by which we relate a liner measure ( radial length) to arc length (circumference) cannot be represented by a fixed or even repeating decimal reflects something fundamental about the physical universe?

Clarification on Fundamental and Derivative Physical Measurements: What about Clocks?

I made, in the last post, what was probably a rash and incorrect speculation: I said that the straight edge and protractor are the fundamental physical measuring devices and that all others are derivative from these. Well clearly we need to add clocks to the list. Or do we?

Well a straight edge and a protractor are easily constructed and they are literally all the same (or perhaps I should say that they are all geometrically similar). But clocks are a great deal more complex.

Clocks all seek incremental measurements of time in an analogous manner to the way straight edges seek to measure length and protractors seek to measure angles. But unlike the cases of length and angle, the measurement of time increments is a much more derivative and tricky affair. We have direct measurement access to lengths and angles but we do not have direct access to time. Only through the observation of some natural or engineered process are we able to infer increments of time. By its very nature a process involves the movement or change of some observable object over time.

However a merely observable process will not suffice to serve as a clock. For a process to be utilized as a clock an object must go through repetitive or cyclical change. The beatings of our heart, the movement of a shadow in the sun, or the vibrations of a spring are the traditional components of both natural and engineered clocks.

Does the use of a clock require the use of a meter stick or protractor? Obviously not. To use a clock to determine an increment of time entails simply being able to observe an end point to a process. For example if we have an hour glass all we have to observe is that all the sand at the top of the glass has flowed into the bottom part in order to infer that an hour of time has elapsed. To see that five minutes have elapsed on a traditional non digital watch we need only observe the beginning and end of the movement of the minute hand through the arc length between any standard demarcated five minute interval.

So a clock is a fundamental measuring device as opposed to a derivative one.

Clarification on Fundamental and Derivative Physical Measurements: What about Clocks?

I made, in the last post, what was probably a rash and incorrect speculation: I said that the straight edge and protractor are the fundamental physical measuring devices and that all others are derivative from these. Well clearly we need to add clocks to the list. Or do we?

Well a straight edge and a protractor are easily constructed and they are literally all the same (or perhaps I should say that they are all geometrically similar). But clocks are a great deal more complex.

Clocks all seek incremental measurements of time in an analogous manner to the way straight edges seek to measure length and protractors seek to measure angles.
But unlike the cases of length and angle, the measurement of time increments is a much more derivative and tricky affair. We have direct measurement access to lengths and angles but we do not have direct access to time. Only through the observation of some natural or engineered process are we able to infer increments of time. By its very nature a process involves the movement or change of some observable object over time.

However a merely observable process will not suffice to serve as a clock. For a process to be utilized as a clock an object must go through repetitive or cyclical change. The beatings of our heart, the movement of a shadow in the sun, or the vibrations of a spring are the traditional components of both natural and engineered clocks.

Does the use of a clock require the use of a meter stick or protractor? Obviously not. To use a clock to determine an increment of time entails simply being able to observe an point to a process. For example if we have an hour glass all we have to observe is that all the sand at the top of the glass has flowed into the bottom part in order to infer that an hour of time has elapsed. To see that five minutes have elapsed on a traditional non digital watch we need only observe the beginning and end of the movement of the minute hand through the arc length between any standard demarcated five minute interval.

So a clock is a fundamental measuring device as opposed to a derivative one.