One of the things that I think about a great deal is... "how much are we biased by base-10?" I mean, if you're looking for meaning in the decimal expansion of π (which I wouldn't really suggest you waste much time on), then you're counting on two (improbable) things:
- there is meaning there
- base-10 will illuminate, rather than obscure, that meaning
If you only ever play with base-10, then you'll think that 1/3 is much messier than 1/2. But, if you instead play in base 3, you'll find it's quite the opposite.
As another example, I've encoded two small text messages below. For both of the messages I started with the simple encoding: 0 = space, 1 = A, 2 = B, ... 26 = Z. I used no punctuation. Then, all I did was represent the numbers in different bases. With a little deduction, you may be able to figure it out. But, the second one is a great deal harder.
It goes much deeper than just the numbering system, too. Think about our units of measure. I cannot come up with even one example of a continuous metric that has a well-founded, universal basis. I can come up with some discrete metrics: three antelope is three antelope no matter how bright it is outside, six-hundred protons is six-hundred protons no matter what the barometric pressure, zero corn cobs is zero corn cobs no matter what the temperature is.
Our units of measure are so incestuous they'd make Jerry Springer blush.
How long is a meter? Well, light travels 299,792,458 of them per second. Why 299,792,458? What's a second? It's the *expected* length of time it takes the cesium 133 atom (when exposed to suitable excitation) to complete 9,192,631,770 oscillations. Expected? Why Cesium 133? What's suitable excitation? Why 9,192,631,770?
At what temperature does water boil? It depends on the barometric pressure. How do I measure the barometric pressure? In millimeters of mercury. What's a millimeter? Why mercury?
How much heat does it take to make water boil? It takes 540 calories per gram. What's a calorie? It's the amount of heat required to raise one gram of water one degree Celsius. How much is a gram of water? A gram of water is one cubic centimeter of water at four degrees Celsius. What's a centimeter? See above. How much is a degree Celsius? Well, zero degrees Celsius is when water freezes and one-hundred degrees Celsius is when water boils (all of that is assuming that you're at standard pressure). What's standard pressure? It's 760 mm of mercury (see above).
I was thinking for a moment that we could define a meter in terms of the radius of a non-accelerating hydrogen atom in a vacuum with no electromagnetic or gravitational forces acting on it. But, then I realized that Quantum Mechanics says that at-best such a hydrogen atom has an expected radius. Wheee...
All of this got me to wondering, is a Planck-length universal? And, if it is, can it be used to measure ( space or time ) or can it only be used to measure ( space-time )?