Recently I’ve had occasion to think of this old post of mine, which I’ve rewritten here for you. Never mind why, just enjoy.
In geometry, a point is a zero-dimension object, having no extension in any direction.
A line is a one-dimensional object defined by two points (that is, for any two points, there is one and only one line that can be drawn through both). A segment of a line is bounded by two points – the endpoints.
A square is a two-dimensional object that is bounded by (or defined by) four lines.
Finally, a cube is a three-dimensional object bounded by six squares.
Notice how each object is defined by an object of one lower dimension: the cube by the squares, the square by the lines, the line by the points. Also notice the progression: two points make a line, four lines make a square, six squares make a cube.
I therefore posit a four-dimensional object that is defined by eight three-dimensional cubes.
We’ve all seen a representation of a cube drawn in two dimensions, right? [There, now you have.] Therefore we should be able to create a representation of a four dimensional ‘something’ in three dimensions. Here, however, is where I run up against the cognitive limitations of my species.