A Little More about Higher Dimensions

[This is related to my earlier post, Plato's Shadows and Higher Dimensions - please read that first...]

Flatland was a novel about dimensions. The narrator, a square, lives is 2-D world. Suddenly he got a visit from Lord Sphere from 3-D world. Lord Sphere showed him 1-D world in a dream. Square met the king of the 1-D world who do not believe in the existence of higher dimensions. Square tried to explain - to no avail - that there is another direction; that the world is not a single line but a flat plane. When Lord Sphere tried to explain 3-D world (space), Square was as stubborn and unbelieving as the king. Lord Sphere demonstrated things that can only be done in higher dimension: disappearing and reappearing, seeing the inside of Square, changing size (small to large circle as seen in 2-D world.) Then Lord Sphere reminded him to his experience with the king. Lord Sphere finally brought Square to space and finally Square believe Lord Sphere.

As mentioned above, Square is the reflection of us in the 3-D world. We may be unbelieving about existence of the higher dimension as Square was. Indeed Square rebuked Lord Sphere when the latter said that there is no higher dimension than 3. Looking at Square and Sphere we may be conceited and he looked ridiculous but there is a possibility that creatures of 4-D are looking at us the same way.

We can't say that fourth dimension does not exist because we cannot see it. [if we of 3-D world can see it, then it is not of 4-D world, duh...]

In a book titled 'Constants of Nature' (by John Barrow if I am not wrong...) it is said that our world that has 3 dimensions of space and 1 dimension of time is the most stable. Too high up or down, it will be unstable. Again, this is only a prediction of us in the 3-D world trying to perceive higher dimension. Can we imagine that there is another axis in our space beside x, y, z? Imagine you are a square and all your life you are moving along the x-y plane when out of the blue revelation comes and you are told that there is another direction: the z-axis.

Indeed it is hard to imagine the 4-D world. It is possible to predict the number of vertices and sides of objects of higher dimensions, though.

Let us consider a square of 2-D world.

In 1-D world it is a line.

In 3-D world it is a cube.

In 0-D world it is a dot.

Follow? Just remember casting shadow reduce dimension by one.

Dimensions	0	1	2	3	4
Vertices	1	2	4	8
Sides	0	2	4	6

For number of vertices, the geometric pattern tells us that a 4-D 'cube' should have 16 vertices.

Likewise, the arithmetic pattern of number of sides tells us that a 4-D 'cube' (the proper term is tesseract or octachoron) has 8 sides.

I have difficulty imagining this: after all we live in 3-D space; but isn't it interesting to know?