A Peano Coincidence

Paul Humke, Khang Vo Huynh

Abstract:

In Section 2 of this paper, we define a Peano-type curve $g_n: [0,1] \rightarrow [0,1]^n$ using a generalization of Hilbert's original
geometric analysis. In Section 3 we show there are generalized Klein 4 groups that determine much of the inherent geometry of $g_n$ and derive a functional equation generalizing that found by de Freitas, de Lima, and dos Santos for their generalized Peano curve, $f_n \colon [0,1] \to [0,1]^n$. In Section 5 we prove that $f_n = g_n$ for every dimension $n \in \mathbb{N}$.