This is just what I was taught
What Is The Fourth Dimension?
I'm sure a lot of what was said is already right, but one thing that I found is that time is the fourth dimension according to Einstein and physicists, but I think what the original question was about is the fourth-SPATIAL-dimension.
A while back I was in Algebra II (I think) and I was bored with class so I asked my teacher about the fourth dimension. It seemed like he understood what it was pretty well, but I don't really know for sure. Most of what he said was about the spatial fourth-dimension cube, or tesseract.
He started off by saying that there isn't a physical shape to the tesseract, but if you could see it at the most basic level, it would look like a cube (thus the hypercube name). The thing that makes this cube hyper is that every side of the cube touches the exact opposite side. So for example, if the room you are in right now were to be a hypercube, you would fall through the floor, only to come right through the ceiling.
This is also how big-bang theorists view the fourth spatial dimension. Normally people think the big-bang was a very dense point that exploded outward, like a balloon that is inflated. However, the more consistent theory is that the fourth dimension is expanding, which gives more 3D space for the universe. This is like if you had a pool that had 100 fish (planets/galaxies) and very little water (3D space). Let's say there was 100 gallons of water, so for every gallon of water there would be one fish, and all these fish would be very close together. Then there was the "Big-Fill-up" which poured 1,000,000 more gallons of water into the pool. At this time, the fish (planets) would have much more water (space) for themselves.
But that's just what I heard... Lol
-reply by Confucioud
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