AMoore curve (afterE. H. Moore) is acontinuousfractalspace-filling curve which is a variant of theHilbert curve. Precisely, it is theloop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves combined in such a way to make the endpoints coincide.
Because the Moore curve is plane-filling, itsHausdorff dimension is 2.
The following figure shows the initial stages of the Moore curve:
The Moore curve can be expressed by arewrite system (L-system).
Here,F means "draw forward",− means "turn left 90°", and+ means "turn right 90°" (seeturtle graphics).
There is an elegant generalization of theHilbert curve to arbitrary higher dimensions. Traversing the polyhedron vertices of an n-dimensional hypercube inGray code order produces a generator for the n-dimensional Hilbert curve.[1]
To construct the order N Moore curve in K dimensions, you place 2K copies of the order N−1 K-dimensional Hilbert curve at each corner of a K-dimensional hypercube, rotate them and connect them by line segments. The added line segments follow the path of an order 1 Hilbert curve. This construction even works for the order 1 Moore curve if you define the order 0 Hilbert curve to be a geometric point. It then follows that an order 1 Moore curve is the same as an order 1 Hilbert curve.
To construct the order N Moore curve in three dimensions, you place 8 copies of the order N−1 3D Hilbert curve at the corners of a cube, rotate them and connect them by line segments.[2]