Most of the world around us is empty space. From the sub-microscopic world of atoms to the vast reaches of the cosmos, there is, in fact, almost nothing there.
This is perhaps easiest to imagine as we gaze into the night sky. We see a vast ocean of darkness with pinpoints of light; but even there we cannot truly grasp the emptiness we are viewing. Proxima Centauri, the star nearest our Sun, is 30 trillion miles away; in other words, if the Sun were the size of a basketball located in Los Angelos, Proxima Centauri would be located in New York City, with empty space all around and in between. The Universe is so vast and empty that if all the matter of all the stars were broken up and dispersed evenly throughout space, the average density would be about one hydrogen atom in each cubic yard of space.
In our Solar System, even though we live in a relatively "crowded" part of the Universe, it is over 99.9% empty space. The distance between the Earth and the Sun, for example, is 10,000 times the diameter of the Earth. If the Earth were the size of a tennis ball, the Sun would be over half a mile away. And the space in between is a more nearly perfectly empty vacuum than the best science and engineering can produce on Earth.
What about the things we see and touch around us? They are all made of atoms fitted together into molecules. Every atom consists of a central nucleus with clouds of electrons orbiting relatively far away from it. In fact, the diameter of an atomic nucleus (which has substantially ALL of the mass of the atom) is roughly 0.01% of the diameter of the atom as a whole. In other words, each atom, even in the most dense substances--in lead or in gold--is, like the astronomical cosmos, more than 99.9% empty space.
Why does the substance of material things around us seem so tough, so impenetrable? That is because of the enormous forces that the tiny sub-atomic particles wield. Just as a very powerful magnet could push around another magnet in an adjoining room, so sub-atomic particles through powerful electromagnetic repulsion keep other sub-atomic particle at arm's length. When atomic particles with like charges (hence, repelling charges) are occasionally crushed together--as in the nuclear fires of exploding stars--other, short-range forces, the nuclear forces, overcome the electromagnetic repulsion and larger atomic nuclei are formed. But even these larger nuclei are crushed in on themselves (by the powerful nuclear forces) into incredibly small volumes so that, even with large atoms and large molecules, most of the space occupied by the atom or molecule is empty.
Some scientific theories speculate that there are multiple, overlapping universes. Clearly, if the forces acting in one universe do not affect the forces of another, there is plenty of room for a large number of overlapping universes without any danger of their parts bumping into one another.
Moreover, some scientific theories postulate that there are more than three spacial dimensions. What does this mean? Consider for a moment the shadow of a coffee cup you hold over your desk. That shadow is the appearance of a three-dimensional object (the coffee cup) projected onto a two-dimensional surface (your desk top). Another shadow from another coffee cup held in your other hand might overlap the first shadow. When seen in two dimensions, they seem to collide, but in three dimensions, the two coffee cups may actually be a foot or more apart.
In a similar way, objects in four or five or six spatial dimensions might appear to collide in their three-dimensional "shadows" which we can see, while in their higher dimensions (which we cannot see) they are actually far apart.
Even our three-dimensional universe is mostly empty space. When we add in the concept of higher dimensions, the room for parallel universes that are actually right under our noses becomes infinite.
Bun Gladieux, president of the Presssure Positive Company, has a blog with an interesting series of topics.
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