Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals

Hornfeck, W.

doi:10.1107/S2053273323008276

Figure 7
Triangular lattice paths representing the first 12 steps of the binary inclination sequences [alternating functions α, see equation (7

)] resulting from the parameter values μ = 1 to μ = 4. The binary inclination sequences specify the relative directions (zero = red arrows = 2π/6, one = green arrows = −2π/6) in which a turtle has to turn for its next step on the lattice. Note that while the very first step is in an arbitrary direction (here, it is the rightward horizontal direction) it fixes the direction relative to which the next step proceeds, and by this all following steps in a recursive manner. Note that the same binary inclination sequences define a self-avoiding walk on their respective $[{\bb Z}]$ module, which is not true in the case of the triangular lattice (the reader might imagine the case μ = 5 to find the first counterexample). Note also that the definition of a spiral self-avoiding walk on a triangular lattice as used in the literature is different from our models, at least in a strict sense, unless one treats successive alternating steps as quasi-straight steps (dashed blue arrows), since direct straight steps are not allowed in our models otherwise. The binary inclination sequences are given for the first 36 steps, with single and double bars highlighting the pattern of μ times repeating, perfectly alternating, binary maximal subsequences of odd length.

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ISSN: 2053-2733

Volume 79| Part 6| November 2023| Pages 570-586

https://doi.org/10.1107/S2053273323008276

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