The Fourth Dimension and Those Above It
By Stanley Traynor (Y7)
Whenever I think of dimensions, the book ‘Flatland’ springs to mind. From the perspective of a storywriter it is not a very good book, but to a mathematician it is supreme. The story is about a square who lives in a 2D world and one night has a vision of a 1D world which is populated by lines and dots. He is amazed that the inhabitants think that it is all there is to the world, and terrifies them by passing through the ‘line’ that is their 1D world and out again. Later the square is visited by a sphere who amazes him by passing through the plane of the 2D world. Then the sphere knocks the square out of the 2D world, and he can now see the 3D world. Then the square has a brainwave and asks the sphere if there are other dimensions beyond 3. The sphere silences him and tells him that the rulers of the 3D world would kill to stop the inhabitants finding out about a 4D world. The point is that there are an infinite number of dimensions, but we cannot imagine any above us. The idea of another plane than width, length, and depth is unimaginable. We can only see 4D shapes as
a 3D shadow of themselves, just as the 1D people could only see the square as a line. The 4D shapes are represented as 3D shapes as they pass through our domain. If a hypercube (4D cube) were to pass through 3D space face first, we would just see a normal 3D cube appearing and disappearing. Edge first would produce a much wider variety of shapes. It would start as a triangular orientation.
With this new knowledge, can you imagine a hypercube? No. You can’t because no human can. Our brains are simply not made for 4D space. However, we can find out about 4D space through 3D shadows of 4D shapes. The shadows, however, are so complicated that only professional mathematicians can understand them. Think of a sheet of paper. You can display the X and Y axis easily but when you try to put a Z axis into the diagram it all goes wrong,
as the paper you are drawing on is (effectively) 2D. You can easily put a Z axis onto a 3D diagram, but the W axis you cannot, as long as the diagram you are trying to put it on has less than 4 dimensions. Venturing into 4D space would be nearly impossible, as we would have to move along the W axis to get there, but to us the W axis does not exist. We cannot move along the W axis; we are fixed at 0. Another force would have
to move us along the W axis and would also have to move us back. To a 2D being, moving out of 2D space into 3D space would be impossible, and once
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