It's geometry. You find the point in the center of the lower dimension, and extend outward in a new direction. For 0 dimensions is a point. You extend outward and form a line which is the first dimension. From the center of the line you extend outward in a new direction, and form an equilateral triangle which is the second dimension. From the center of this triangle, you extend outward in a new direction, and form an equilateral pyramid, which is the 3rd dimension. Likewise, from the center of this pyramid, you extend outward in a new direction and form an equilateral superpyramid which is the 4th dimension. In the fourth dimension there is more space, and just as the 2nd dimension imprisons and makes up the sides of the 3rd dimensional figure, the 3rd dimension imprisons and makes up the sides of the 4 dimensional figure. So each [side of the] of the 5 sides of 4th dimensional equilateral superpyramid is an equilateral pyramid. You can imagine that each side of the 3rd dimensional equilateral pyramid had another 3-D equilateral pyramid stuck onto it. The center pyramid forms the base of the 4th dimensional superpyramid, and the point of each of the other 4 pyramids folds into the fourth dimension in a new direction so that they meet in a point immediately over, and which extends down to center of the base pyramid, just as the three triangles which are the sides of the 3rd dimensional pyramid meet in a point directly above the center of the original 2nd dimensional triangle.
There is more space in the 4th dimension. This is geometry. 0, 1, 2, and 3 dimensions are geometry, so I do not see why the 4th dimension should not be geometry as well. I see no reason to say the fourth dimension is time. There should be an infinite number of geometric dimensions. I learned this in part by reading the book Flatland by Edwin A. Abbott. There is also a book called Spaceland by Rudy Rucker. But I myself found the points to figure out exactly where the 4th dimension is and should be.
A related fact is that the ancient egyptians buried their pharaohs in the center of the pyramid. The only difference is that those pyramids has squares for bases and so were not actually equilateral pyramids. It is reported that Napolean Bonaparte once spent the night in the center of the pyramid and "when he emerged, it was reported that he looked visibly shaken. When an aide asked him if he had witnessed anything mysterious, he replied that he had no comment, and that he never wanted the incident mentioned again. Years later, when he was on his deathbed, a close friend asked him what really happened in the King's chamber. He was about to tell him and stopped. Then he shook his head and said, "No, what's the use. You'd never believe me."