The five joint-longest cycles (length 48) with three-way rotational symmetry. The first of these is unusual for lying wholly on the surface of its bounding cube. The second one is unusual for having a vertex of degree 4 in all three projections.
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The six joint-longest cycles (length 42) with two-way symmetry. In the first four cases, the symmetry is reflection in an edge-to-edge diagonal plane. In the last two, it's rotation about a line betwen two opposite edge-centres.
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The longest cycles of all that fit in a 3×3×3 cube have length 50. There are 911 of them in total (up to symmetry), and they're all totally asymmetric. These four have the unusual feature of being entirely on the surface of the bounding cube (like the one at the top of the page):
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And these two (also of length 50) have the unusual feature of reaching all 8 corners of the bounding cube. (There are none that do that and live entirely on the surface, alas.)
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Going the other way from that, here are three cycles with 3-way symmetry that don't reach any of the corners of the bounding cube (even though they stretch to every face of it). The first has length 42, and the other two length 36.
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The shortest cycles that live entirely on the surface of a 3×3×3 cube. Not all that interesting: both are stretched versions of the 2×2×2 Rickard cycle, one stretched so as to have 3-way symmetry, and the other asymmetrically.
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And the shortest cycles (length 44) that reach all 8 corners of the cube.
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These four cycles (all of length 40) have the curious feature that two of their three projections are not merely trees but simple paths, with no branching. No cycle has all three projections of that form.
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Oppositely to that: there are 38 cycles which manage to have a degree-4 vertex in all three projections. One of the longest is at the top of this page (the second of the length-48 3-way symmetric batch). At the other end of the spectrum, here are the two joint shortest, with length coincidentally also 38.
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