I try to formulate the question about how Unlimited Detail works like this:
Let D be a database of 3D points, containing information about M points. Let also S be the image on the screen, say with N pixels. Problem:
- reorganize the database D to obtain another database D’ with at most O(M) bits, such that
- starting from D’ and a finite (say 100 bytes) word there exists an algorithm which finds the image on the screen in O(N log(M)) time.
Is this reasonable?
For example, take N=1. The finite word means the position and orientation of the screen in the 3D world of the database. If the M points would admit a representation as a number (euclidean invariant hash function?) of order M^a (i.e. polynomial in the number of points), then it would be reasonable to expect D’ to have dimension of order O(log(M)), so in this case simply by traversing D’ we get the time O(log(M)) = O(N log(M)). Even if we cannot make D’ to be O(log(M)) large, maybe the algorithm still takes O(log(M)) steps simply because M is approximately the volume, so the diameter in 3D space is roughly between M^(1/3) and M, or due to scaling of the perspective the algorithm may still hop through D’ in geometric, not arithmetic steps.
The second remark is that there is no restriction for the time which is necessary for transforming D into D’.