Abstract: In this paper we investigate the nature and existence of (at most) n-to-1 space filling curves f that map [0,1] onto [0,1]2. In particular, we find an at most 3-to-1 space filling curve similar to the at most 4-to-1 Hilbert Curve and show there is no at most 2-to-1 space filling curve. In addition, we provide a plethora of examples of n-to-1 space filling curves and show that every space filling curve has a “space filling core,” a minimal perfect set that the function maps onto [0,1]2 and on which the function is space filling on every portion. We then use the Lebesgue space filling curve to illustrate how a complete arithmetization of a space filling curve can be done on any of its space filling cores.
Publication Year: 2021
Publication Date: 2021-08-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 3
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