We live in an unknown and unknowable universe, we don’t know where the boundaries are, we don’t know whether it will end or not, or even if it ever really started - although there are many theories.
Everything we study seems to render greater detail or greater senses of scale, apparent boundaries are constantly dissolving or expanding, and we are on a journey of discovery of which we cannot see the limits.
Example: a straight thing, a border line drawn on a map. We say it is straight, it clearly delineates a finite area. Yet when we look more closely we see that it has a fractal structure, it meanders and wanders - even if it appears straight at first glance.
We apply a microscope and see ever more detail, the border gets ever longer until we reach the molecular scale and it seems to dissapear as a border line and become something else.
But is that the end ? We go further down to the subatomic realm and more structure is revealed. Is there an end ? And how would we know ?
Was that finite border so finite after all, or was that an illusion ?
[Non-jokingly, even an infinitely long fractal will typically encompass a finite area; if the border never wanders outside some particular rectangle, then, well, its area is bounded by that of the rectangle.
Not to mention other finities… e.g., I have one and only one sister, regardless of her fractal composition. This is the 3rd post in the thread. There are 26 English letters. A chessboard has 64 locations, 32 starting pieces, and two colors, regardless of what the chess set is made of (whether wood or plastic or an abstraction drawn with pixels on a monitor), and in the same way, a tic-tac-toe board has but 9 locations, regardless of how it is realized. Etc.]
But that’s just a model version, yes ? In reality any border is always moving even just a little bit if it is made from matter, and the border becomes a much less distinct entity at the sub atomic level ?
No, that’s the reality version. In reality, if something is on a plane and bound by a bounding box, regardless of the length of its border, or whether the border moves, or whether it’s fuzzy, its area is less than or equal to the bounding box.
If it’s not on a plane, then the bounding box can be extended to 3 (or more, pick your number) dimensions. If you insist on infinite dimensions then you win, by fiat.
That’s both in theory and in reality. In practical reality (for some physical object) the border can’t be infinitely long, if it’s a physical thing, because of the limitations of the planck length etc. But the border can be infinitely long if it’s defined by some rule rather than mere physical object(s).
I don’t think the Planck length, as such, necessarily implies anything about fractal borders. If there’s one thing I’ve learnt from Chronos, it’s that, despite the common perception that the Planck length is the smallest measurable distance, or the grid-scale on which the world is quantized, or various such things, the Planck length is not actually known to have any particular physical significance; it’s just the length that falls out of the relations between other constants. It’s not unreasonable to guess that it will eventually turn out to have some significance in itself, but if there is any, it is as of yet unknown.
No he said he didn’t know if the universe is pixelated. The Plank length might turn out to be a fundamental attribute of the universe, in which case we might say that, yes, the universe is pixelated. Or it might not be.