A few years back, I sat in on an amateur talent show for a friend where one among the performers had as his "talent" that he had memorized Pi to something like 200 digits-- an interesting and even impressive feat in principle but actually pretty dull to watch. After sitting through that and doing some thinking on the topic, I came to the conclusion that it is useless to even know Pi to more than maybe 50 digits. And here's why:

To begin with, in May of 2004, the Wilkinson Microwave Anisotropy Probe measured the cosmic microwave background radiation of the Universe, yielding the largest-yet proposed size of the known Universe. Those figures suggested the Universe to be at least 78 billion light years in radius (therefore 156 billion light years in diameter (although it has since been proposed that the the Universe may be an oval). Remember, using the Julian calendar, a light-year is 9,460,730,472,580,800 meters.

If we start from the size of the entire known Universe and move inwards, we can progress like so.... first, assume the Universe in fact fits inside a circle 156 billion light years in diameter, 78 billion light years in radius, and call that size "x".... now, a circle one tenth of that (x times .1) is 7.8 billion light years in radius, still unimaginably large. We will shoot to the next power, (x times .01, or x times .1 to the second power) and get to a circle that is .78 billion light years in radius (or 780 million light years in radius).... if you've ever seen the old "powers of ten" video (awesome, and available here), it actually topped out with a square of one billion light years, which captured a bunch of galaxy clusters and made them look like so much dust..... but just think, they could have gone a mere two more powers and gotten to the size of the known Universe!! There's another one here which doesn't go as big or as small, but has better pictures I think, and yet another here which at least tries to catch a tenth of the Universe in a slide -- at the furthest slide out, it says "we must stop our outward journey here. At the next power of ten we might have to display the curvature of the universe."

The same slides count down those powers-of-ten down to the meter and beyond (each of those sites has its own particular best points in this regard), but the last site has a great slide on quarks which says:

By appropriately combining these three universal constants, it is possible to arrive at a fundamental unit of length called the Planck length. The value of the Planck length is approximately 10-35 meters. Physicists have determined an upper limit on the size of the quark based upon data from high energy collisions: Quarks must be smaller than 10-18 meters. This determination is based on the highest energy collisions currently attainable. There are some who believe that the next level of fundamental particle, if any exists, may be found at the incredibly small scale of the Planck length -- that the universe may be inherently grainy at 10-35 meters.

This reminds me of a passage in the Scott Adams book "God's Debris" (where Adams basically puts forth a theory of Pandeism as an explanation for why we're here at all) where the character of the all-knowing Avatar talks about stripping down the particles to smaller and smaller sizes until we get to the individual units, particles of God. Now, suppose we wanted to make a ring of quarks in a perfect circle that surrounded the entire Universe, and we wanted to know the exact length of the quark string needed to make this ring.... we could never need to know more precisely than Pi to about the 50th digit, or our old x times 1 to the 50th power, because any additional level of precision would make a difference of less than a fraction of the size of a quark!! That's a smaller distance than we have any means to measure!!

Modern mathemeticians have done wonders with inventing big numbers, coming up with mathematical symbolism to indicate ten to the billionth or trillionth or trillion-trillionth power. But these are much larger numbers than could ever be used to describe the Universe and everything in it, even in terms of the smallest of particles, so even as the Universe may be inconceivably big, it has boundaries very low within the realm of conceived mathematical numbers!!

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Addendum: a fellow noder has written to me: "By my calculations, using your numbers, the universe would be 1.48E+63 * Pi quarks in circumference. This indicates that we would need to know Pi to at least 63 places to get the exact length. I know that the difference is less than a micron, but 10E+13 is still a large margin of error." So I may be wrong by some number of digits after all, but consider this: here's Pi to 64 digits:

3.1415926535897932384626433832795028841971693993751058209749445923 which means that the length of my Universe-encompassing string will be off by less than .000000000000000000000000000000000000000000000000005820975 -- which is an error I can live with!!

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