Pi and Infinity Conception Involved in Circularity
LINK TO ALL OTHER POSTS on Pi if there are any
LINK TO “The Sacred Circle” Archetype (apeirogon and ‘trinity’ of 0, 1, ∞) AND Euler’s Identity (most beautiful, brain tested)
LINK TO squaring the circle POST
LINK TO CALCULUS AND INFINITY – spiritual history of POST
LINK TO ANIMATED MATH POST
LINK to Potential Geometry post
read full and take quotes from here
Analysis of Pi leads you to the borderline of the imaginable and the unimaginable. The constructible, and the nonconstructible. The finite and the visually approached infinity.
Does it say anything about the approach to accuracy in the math or physics world?
It was shown that…
A solution is not possible to construct in a finite number of steps, using physical tools.
‘Squaring the Circle’ and ‘Proving the irrationality of Pi’ (transcendental, infinitely extending but never ‘repeating’) both result in a strikingly similar visualization of an ‘approach to infinity’
‘Construction and Verification of Perfect Circles in Technology and on Paper’ and ‘Observance of circles in nature which appear perfect’
In the sense of physics it’s impossible to construct a perfect circle that is made of matter. But that’s only because physics imposes a strict sense of reality where you only believe something if you can measure it. To really “prove” that something is a circle, you would have to be able to measure its curvature in all places simultaneously up to infinite accuracy. This is not possible on paper… — but the mind convinced itself it was possible //because of the visual intuition// ???
Buckminster Fuller (From Hugh Kenner’s biography)
“Generations of circle-squarers attested to the persistent intuition that it ought to have a rational value, but nobody ever found one. Eventually it was proved that none was findable. The decimal sequence for pi is 3.141592653589793 . . . and will go on forever. This appears to mean that infinity will invade any circular system, which feels wrong since circles are closed.”
From Scientific American Blog – https://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/
Pi is irrational. That is, the decimal expansion never ends and never repeats, so any number of decimal places we write out is an approximation. (Of course, we can write the number exactly using just one symbol: π which we both do know and do not know)
Each decimal digit we know makes any computation involving pi more precise. But how many of them do we actually need for sufficient accuracy? Of course it depends on the application. When we round pi to the integer 3, we are about 4.51 percent off from the correct value. If we use it to estimate the circumference of an object with a diameter of 100 feet, we will be off by a little over 14 feet.* When we add the tenths place, and use the approximation 3.1 for pi, our error is only about 1.3 percent. The approximation 3.14 is about ½ percent off from the true value, and the fairly well known 3.14159 is within 0.000084 percent.
If you were building a fence around a giant circular swimming pool with a radius of 100 meters and used that approximation to estimate the amount of fencing you would need, you would be half a millimeter short. Half a millimeter is tiny compared with the total fence length, 628.3185 meters. Being within half a millimeter is surely sufficient, and the tools you are using to make the fence probably introduce more uncertainty into your structure than your approximation of pi.
What about something with higher precision standards over much larger distances? I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that controls and stabilizes spacecraft during missions.
Pi appears most often in formulas involving circles or periodic motion, and it infiltrates some fundamental physical constants. These constants appear all over physics: masses of elementary particles, the number of molecules in a volume of a gas, the forces holding matter together, and so on. (Pi itself is not considered a fundamental physical constant.) The fine-structure constant, or “coupling constant,” which measures the strength of the electromagnetic force that governs how electrons and muons interact with photons, involves pi, and the permeability of free space, which describes how a magnetic field forms in a vacuum, is 4π×10-7. It is important to know highly accurate values of the fundamental constants to make good predictions of phenomena involving physics, and the experimental determination of the constants can even help improve our understanding of the physical laws that govern the universe.
Believe it or not, there is a committee that makes recommendations about the values of these fundamental constants. The Committee on Data for Science and Technology, or CODATA, an interdisciplinary group from the International Council for Science, periodically publishes a set of accepted values of the fundamental physical constants. The most recent version, 2010CODATA, was published in June 2011.
Peter Mohr, a physicist who works for the Fundamental Constants Data Center at the National Institute for Standards and Technology, which is involved in calculating and disseminating the accepted CODATA values, says that the institute uses 32 significant digits of pi in their computations. (For programming geeks, this is called “quadruple precision.”)
So NASA scientists keep the space station operational with only 15 or 16 significant digits of pi, and the fundamental constants of the universe only require 32.