Quote:
Talya: Pi, when expressed in any numerical system with an integer-base, has infinite decimal places that do not repeat.
Talya: so
Corolinth: That's the prevailing theory. We're still trying to check it out.
Talya: this means that the measurement of either circumference or diameter will also, if taken precisely, have infinite decimal places that do not repeat.
Corolinth: Yes.
Talya: Matter has a lower limit on distance calculations.
Corolinth: Yes. According to our current understanding.
Talya: does this not conflict with pi?
Corolinth: No.
Talya: see, at a certain point, the added decimal places become impossible when measuring distance.
Math is not concrete. It is abstract. I should clarify by pointing you to the Pythagorean Theorem. The hypotenuse of a right triangle is often a real thing that we wish to measure. When I say this, I mean real in a physical sense, and not a Khrosstonian sense, nor a mathematical sense. (The concept of "real" and "imaginary" in mathematics is somewhat misleading, because both have a real physical meaning. The interplay between the real and imaginary units dominates the very physical law that allows us to communicate over the Internet.) This physical thing exists, and could be measured, but either we do not have a long enough ruler, or it's in an inconvenient place to hold a ruler. What we can do is measure two orthogonal components and put them together in such a fashion that we arrive at a value equivalent to the one we hoped to find but could not measure directly.
I bring this up because it's important to understand the distinction between the physical world and the mathematical world.
Pi has more physical meaning than simply the ratio of circumference to diameter for a circle. That was simply the first method discovered, because it was most readily available to early mathematicians. For instance, we can get pi to pop out of measurements of a vibrating spring. Physically, there is no circle, just a spring bobbing up and down in a straight line. There is no circumference or diameter to measure - only forces required to stretch or compress a spring, the length of said deformation, and the time to complete a full vibration. Had we first discovered pi in this manner, we would have come up with Kaffis's favorite alternate circle constant: 2pi. Mathematically, this is because the cycle of a spring's vibration is modeled with a circle. This is where the abstraction comes in. Many different mathematical objects are valid models even when that geometric shape is not present in the physical world.
Now, what you were more directly interested in hearing was specifically related to the circle, since circle is the shape the layman always (correctly) associates with pi. As you've proposed, the current understanding of physical space is that it's quantized into discrete packets. If we had a circle small enough for that to matter, we would have a diameter of one Plank length. This is the lower constraint. A circle with a circumference of one Plank length couldn't exist, because the diameter would have to be smaller, and that can't happen. The circumference is pi Plank lengths. This isn't a problem, because the entire circle fits inside of a single Plank area (which, due to the way three dimensional space works, is spherical rather than rectangular).
It may seem problematic that the circumference is not a whole number of Plank lengths, but remember that the circumference is wrapped around inside a space that is one Plank length wide. If I were to cut the circle open and unroll it, then the pi Plank length would get stretched out into four Plank lengths. (Space being quantized means it comes in discrete packets, much like energy.) Does that mean the circumference is actually four, and not pi? No, it does not. Once you cut the circle open and unroll it, it's no longer a circle. It is a straight line. The relationship between circumference and diameter no longer applies. It's similar to how subatomic particles gain and lose mass when their atomic bonds change. That difference in length between four and pi that gets lost when the line rolls back up into a circle is like the mass that converts to bonding energy when two protons and a neutron form Helium-3.
But, we also have to get back to the idea that math is an abstraction, and is not the physical world itself. We did not come up with pi through direct measurement. Back then, it was really hard to measure bendy stuff. Instead, we used straight lines, and we analyzed patterns that emerged with polygons both inside and outside a circle as we increased the number of sides on the polygon. The beauty of mathematics is that it holds even when the physical world gets too large or too small to accurately observe, or holds a significant quantity of pieces that simple arithmetic and counting would be too cumbersome. To quote Richard Feynman, "Mathematics is a tricky way of doing something that would otherwise be laborious."
Because we begin early childhood education with counting objects, people get the false impression that math is counting, numbers, and arithmetic calculations. This perception gives rise to ignorant opinions like idea that math is imaginary because you can't count an irrational number, and you can't get to one with a fraction involving integers. Unfortunately, you can't count a rational number, either, because your chances of cutting an apple into halves, thirds, fourths, or any recognizable fraction is infinitesimally small. Your pieces will not be of equal size. Period. Ancient Greeks had this problem with √2, and there was a period of time where people were executed for heresy for holding the notion that such a number could exist. For a number of years, they had a huge problem with right isosceles triangles until they finally accepted that there had to be numbers which couldn't be counted. This all happened before the birth of Jesus. The Greeks knew the Earth was round, which is one of the reasons why they were so interested in circles to begin with. Unfortunately, Western society is still reeling from the collapse of the Roman Empire.
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