Infinite Series of Generalized Gosper Space Filling Curves

We report on computer search for generalized Gosper curve for 37 < N < 61, where N is the degree of the generalized Gosper curve. From the results of the computer search and some geometrical insight, we conjecture that the degree N satisfies N = 6n+1. We investigate the existence of infinite series of generalized Gosper curves. We show how to generate these series and introduce two new methods, the ‘decomposition method’ and the ‘modified layer method’. 1 Generalized Gosper space filling curves The Gosper curve is a space filling curve discovered by William Gosper, an American computer scientist, in 1973, and was introduced by Martin Gardner in 1976 [1, 2]. The curve is constructed by recursively replacing a bold arrow, called initiator, by seven arrows, called generators, Fig. 1(a), Fig. 1(b) and Fig. 1(c) illustrate the curves obtained by replacing the initiator by generators once and twice respectively.