in The Factory Floor, 19 secs ago, by
Secret One can draw some interesting philosophical connections of these with mathematics. For example, Wheel and axle work as they do because of the mathematical properties of a circle
Circles are an example of a compact set (under the usual topology in $\Bbb{R}^2$, or when viewed as a curve, some kind of 1-manifold)
Compact sets are interesting because they can "bound" infinities, meaning that you can place something infinite in them and the overall result will retain many properties of being finite.
Actually, for a topologist, we distinguish between those two objects by using disk to refer to a "solid, filled in circle" and circle for the boundary, or the closed 1-manifold.
Abstract circles with different values of $\pi$:
Recall that $\pi = \frac{C}{d}$ where $C$ is the circumference and $d$ is the diameter
Now, suppose the paper we drew a circle on has variable number and size of pixels (paper grains?), meaning that as we change the value of $\pi$ around, the figure will not look different to our eyes
Suppose we held $d$ fixed, then the circumference must changed.
You can see that as we vary $\pi$, while the circumference does not look changed, given an ant that walk on it by 1 unit/s, the ant can only cover a much shorter distance compared to when it walked on the circle with smaller value of $\pi$
Roughly speaking, this can be understood as the measure being used to calculate the arc length between the two circles are different. This makes sense because the circle has uncountably many points and thus you can always biject it with multiples of itself and still does not change the cardinality. (thus look the same in the diagram)
Therefore, the difference in travel distance by the ant that is observed is because we are using a different measure
If we insists we are using the same measure, then the difference can be explained by the circle with larger $\pi$ is actually in a non-euclidean geometry
Now, let us imagine the scenario when $\pi \to \infty$, what will happen:
and now suddenly, the ant gone stuck and can go nowhere
Therefore, one immediate consequence of being in a universe with a different value of $\pi$ is that you need to use more/less material to build some circle with a desired circumference
and this can affect the length of rope you need to use for your pulley system, or the number of charges you can place in a storage ring as well many other things...
(NB The word "immediate" is used because we all knew that when the fundamental constants were varied just a little bit (without varying other constants to compensate for it), there will even be intelligent life developed to e.g. discuss about circles)
In other words, the physical significance of $\pi$ (I think) is it controls the geometry of circles and periodic phenomenon in the universe
I have been working on some math proofs and general theory stuff in a paper linked below and I have hit a mental block on something.
Piecewise Constant Functions in Differential and Functional Equations
On page 22 there is a definition given for the set of alternate derivatives associated with ...
