If we were divide a one kilogram ingot of gold we could cut it down to size and divide it into tenths, hundredths, thousandths, millionths, billionths, and so on. But we cannot divide the ingot indefinitely, the smallest unit of gold we can have is one atom, and there are approximately 3.057x10^24 atoms of gold in a one kilogram ingot. From that bar we can have no more than around 3 trillion trillion pieces of gold. Divide it any further, and what you have is no longer discernibly gold. Given that the properties of matter and energy and various chemical reactions require certain scales before they can arise (even if those scales seem imperceptibly small), it seems reasonable to assume that there exists a fundamental lower limit on size. At some point we must reach a limit below which there would be no properties at all, since all properties would necessarily arise from the interactions occurring at such a scale. Viewed this way, a fractal theory of the universe seems untenable: how can universes be packed within each other when scale is so important to physical nature? However, it is still possible that beyond the scale that we can currently observe, past the fundamental building blocks of our universe, lay the outer reaches of embedded fractal universes, and these universes have analogous but dissimilar properties that operate on different magnitudes of scale.
It does not seem counter-intuitive to suppose that our universe has limits, that there is a fundamental unit that defines all physical properties, or that there is a finite amount of matter that can ever exist. But to the human mind, vastness implies infinity, a library that adds a new book to its collection as soon as you have read one of the existing books always has a finite amount of books, but you will never be able to read them all (by definition, supposing you doubled your reading rate, the rate of new books added would simply double as well) and so the effect is the same as if the library's collection of books is infinite. An objection to this simple analogy might be that we could still look up the finite number of books rather than reading them all, but then the library need only add new books as quickly as we are able to look up existing books. It is understood by definition that the universe contains the set of all that is, it extends far beyond our own world and past whatever strange worlds exist far beyond our own. To think that such a set is finite, is in effect containable, seems somewhat perplexing. It would seem we can imagine more than what exists.
If one assumes that the universe is composed of common
component parts that represent a lower limit on size, that is, that there is
finity of depth even though this does not imply finity of numeracy, then one
can further postulate the fundamental shape that these components take, and
from this develop a complex theory that describes a universe or set of
universes. It may not be unreasonable to think of these fundamental components
as being point-like. In mathematics, logic, and formal systems, a point is a
primitive notion, it is not defined by other concepts, and stands on its own as
a basic unit upon which we can build more complex concepts and geometry. The
fundamental components of our universe should also be axiomatic, and unreliant
on simpler units. But supposing that what is logically axiomatic must also be
physically axiomatic is a significant leap. What evidence do we have that our
universe can even support points? The question might seem incredibly stupid,
after all, points are everywhere. But look closely, very closely, and I contend
that you cannot show me a point. If you could demonstrate to me a point that
could not be broken down, could not be made any simpler, then you would be
demonstrating the fundamental components of our universe. If the points were
not really points but clumps of particles or matter with sub-components, then
we have not come any closer to determining what the basic blocks of our
universe are. If one starts with a rock, a three-dimensional solid, and breaks
it into small chunks, one obtains compact discrete rocks occupying
three-dimensions. If one then grinds these rocks further until they become a
fine sand, one might approximate this sand from a far distance as
two-dimensional, but close inspection, or simply touch would reveal that each
grain of sand is itself like a tiny rock, occupying three dimensions. We can
then break down the sand further to the molecular level, but as we all know,
molecules are of course three-dimensional as well. When we move further down
the scale things get more interesting, but far from the components becoming
simpler, they only seem to get weirder, more complex. The smaller we go, the
more confused we are as the less it resembles the world we are familiar with.
Further, as we probe with higher energy levels it seems we are in a game to
determine how deep the rabbit hole goes rather than on a quest to discovering
the simplest component of the universe itself. One might propose that the
fundamental components are not point-like and therefore one-dimensional in
nature, but perhaps two-dimensional, say string-like. But again with strings,
we know of no such things, two-dimensionality does not exist in our universe as
near as we are able to so far detect. Note that we must also determine the
number of dimensions that our fundamental components can take shape in. A
two-dimensional component could be string-like, but it could also be surfacic,
in three dimensions it could wrap in on itself. Strings can vibrate, surfaces
can ripple, we can imagine the fundamental motions based on shape that will
determine the properties of our universe. But we never said we were describing
our universe, although that is the goal, and in our universe it seems things
have three dimensions. So suppose then that the fundamental components are
three-dimensional. Then they could be thick surfaces, or spheroidal, or any
other imaginable shape. Note that the dimensionality of our components serves
as a restriction, if we suppose they are three-dimensional, then there can be
no two-dimensional components (since two- and not three- would then be the
fundamental dimensionality), so once we move on to three-dimensional
components, any surfaces must necessarily be thick. We may find that for
practical purposes it is not so restrictive since a three-dimensional object
can do a good job of impersonating a lower-dimensional object by having one of
its dimensions sufficiently smaller than the others. Volumes can support all
kinds of motion, they can be pinched or pulled, they can support ripples,
oscillations, vortices, and the like, and they can have surfacic properties
such as ripples on a pond, which are themselves distinct from their volumetric
properties, such as underwater currents. The universe that would arise from
such fundamental components seems daungtingly complex. In theories that suppose
the fundamental components are points, or lines such as strings, essentially
lower-dimensional objects, it is supposed that they can move through higher
dimensions, a point can be anywhere in three-dimensional space and time despite
having no volume, it is a singularity.
