Combining $n$-D images to get an ($n-1$)-D one

Question

Our human brains combine two $2$-D images we get from each of our two eyes to get one $3$-D image. Suppose there is a creature in another $4$-D world that can see in $4$-D. How many $3$-D images should the brain of such a creature receive? Generally, how many ($n-1$)-D images should a creature living in an $n$-D world receive to get at least one $n$-D image, for $n\geq 2$?

Answer 1

Start by asking yourself this: how can we reconstruct a 3D image from 2D data? Technically a main problem in computer vision is matching corresponding points in these two images, but the human brain is very good at that, so let's assume that you know which point in one image corresponds to which in the other. Then each point in each image corresponds to a line in 3D. From one image you know in what direction you're seeing the point but not how far away. Two distinct lines in space need not intersect, but if they do, they will do so in a unique point, so you know the 3D position from the two lines described by the 2D points.

Now go to arbitrary dimension. A $(n-1)$-dimensional image fixes a line in $n$-dimensional space. Two distinct such lines need not intersect, but if they do, then the point of intersection is unique and will be the reconstructed location. So two images is enough. This holds for reconstructing a 2D world from two 1D images as well. Reconstructing a 1D world from two 0D images won't work, though, since in that case you'd get the same line from both eyes, and two identical lines have no unique point of intersection. In any dimension, you can't reconstruct a point which lies on the line connecting your eyes, but in 1D you can't turn your head to fix that problem.