“And it must follow as the night the day” that the second day of the work week (today) follows the first (yesterday).  This is what happens with ordinal numbers; they delineate (if that’s the word I want) the order of things.

Of course, as with numbers on the standard clock, or cycles of days of the week, or months of the year, the first day (or whatever) also follows the second day.  Indeed, in a small enough cycle, they can follow each other by exactly the same amount.  This is what happens in a binary cycle, like soldiers marching (or any biped walking).  The left foot starts off, and the right foot follows immediately after.  But then the left foot follows the right foot immediately after that.  And so on.

Of course, the weekdays have seven members, so that cycle may be more akin to the movement of the heptapods from Arrival.  And there are more months still, and more days in months than that, but the cycles still cycle.

Some cycles are surprising, of course.  For instance, if you go for one 360-degree turn around a Mobius strip, you’re not back where you started; you’re on the opposite side of the strip, “underneath” where you began.  To return to your starting point, you must take another 360 degree (or 2 pi radians, if you prefer) run around the strip.

This is a good analogy with (and a way of making some intuitive sense of) the spins of fermions, i.e., particles with half-integer spins, such as electrons and quarks.  When we say they have half-integer spins, we mean if you “rotate” one 360 degrees, it will end up in the opposite spin configuration to where it started; it requires another 360 degrees to return to its beginning.

That’s in contrast to the integer-spin bosons, which behave more “intuitively” in that they require 360 degrees to come back to where they started, like numbers on a clock.  Photons are probably the most obvious examples of spin-1 particles.

There is also a hypothetical spin-2 particle, the graviton (still a boson, because it has an integer spin).  This is easier to model in one’s head, because it “looks” the same after a 180 degree rotation.  It’s pretty easy to imagine 3-dimensional shapes to model that.  Think of football shapes, or even just rectangles.

Of course, it would be even easier to imagine “spin 3” or “spin 4” things:  think of equilateral triangles and of squares.  One can even quite easily imagine a spin-infinity object.  That’s just a circle; no matter how you spin it (maintaining its center point) it looks the same.

Of course, if you’re Archimedes (you’re not; I checked), you get closer and closer to finding out the value of pi (the ratio of the circumference of a circle to its diameter) by bounding a circle’s circumference between regular polygons that fit exactly within and exactly around the circle and taking their perimeters.  It’s hard to do that for a triangle, or at least it’s hard for me to do in my head, but a set of squares*, one nested in a unit circle**, the other in which is nested a unit circle, can be done.

The perimeter of a square in which is nested a unit circle is easy, because each side of the square is length 1, just like the diameter, so the perimeter is 4.  The interior square has diagonal length 1.  So any two sides make the legs of a right triangle, and a^2 + b^2 = c^2.  So side^2 plus side^2 equals 1^2.  Or, 2 times the side-length squared equals 1, or a single side squared is 1 over 2, and the length of a side is 1/√2 (we only use the positive root, since we’re talking about physically realizable lengths).  Thus, the perimeter of that square is 4 x (1/√2), or 4/√2, which is 2√2.  That ends up being around 2.8 ish.

So, it seems pi is between 2.8 and 4.  Which it is.  That doesn’t narrow it down much, but if you go from there to ever-increasing numbers of (equal) sides in nested/nesting figures, as Archimedes did, you can get arbitrarily close to the actual value of pi.

That’s all you can do, since pi is an irrational number.  There is an infinite amount of information encoded in the digits of pi, and indeed in any irrational number (making each irrational number its own version of the Library of Babel in a sense).  But pi is a fundamental geometrical number, so that fact hits home with it perhaps more than with any other specific number.

If the universe really is continuously divisible (this possibility is far from certain), then each spot in the universe contains a potential uncountable infinity of info.  If that is the case, nothing can truly, fully simulate the universe in detail.  But we can still make useful models.

I could get into the restrictions imposed by the Planck length and the Planck time and so on, but that’s a whole can of bees I don’t want to open just now.  Instead, I’ll call this to a close for today.  I hope you have a good one.

*A very crude starting point, but it’s good for demonstration.

**In this case I’m using that to mean a circle with diameter 1.  Sometimes one sees it used to refer to a radius-1 circle.  It doesn’t matter that much, but it’s very important to be clear which one you mean.  Just as it’s useful, even crucial, to define terms before beginning a discussion to avoid arguments over nonissues***.

***For instance, if you get two people to agree on what they mean by “sound” before you ask them, “If a tree falls in a forest and no one hears it does it make a sound?” then there is unlikely to be any disagreement, because the answer will be trivial (either way).  Thus, this is not actually a philosophical conundrum, despite its reputation; it is merely a demonstration of the folly of poor communication.