Our eyes receive a two-dimensional image but our brains build a three-dimensional one. Could two three-dimensional images create a four-dimensional one, ad infinitum?
Simon McLeish
Lechlade, Gloucestershire, UK
Yes, it is possible to create a three-dimensional image of a four-dimensional object: a hypercube is the four-dimensional analogue of a cube, and it has a well-known three-dimensional image that looks like a cube with another cube nested inside it. It is even possible to represent this shape in two dimensions, as can easily be seen with a search online for 鈥渉ypercube鈥�.
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These are perspective images, and they are distorted in length and angle so that a flat painting creates the appearance of space. It seems likely that n-dimensional beings with a sense of sight could use similar distortions to make sense of n+1 dimensional objects.
More abstractly, any finite dimensional equivalent of a cube can be represented with Cartesian coordinates, where each vertex is defined by a set of numbers. A 2D square can be written as (0,0), (1,0), (1,1), (0,1), while a 3D cube would be (0,0,0), (1,0,0), (1,1,0), (0,1,0), (0,0,1), (1,0,1), (1,1,1), (0,1,1). As the number of dimensions increases, the number of coordinates required to define each point goes up.
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