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In the Wolfram definition it says

The helicoid is the only non-rotary surface which can glide along itself.1

1Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231-232, 1999.

I looked up the definition of glide, which is "A product of a reflection in a line and translation along the same line." I'm trying to picture what that would mean on the helicoid. I understand that the helicoid is the only 'ruled' minimal surface other than the plane... is it that ruled line on which the 'gliding' takes place? Is there an intuitive explanation as to what that would mean... the pictures I see of helicoid show a grid on the surface, so I'm not clear which of those orthogonal lines the gliding would be on- radially outwards or snaking downwards barber-pole style. Does gliding mean that locally it can sort of self-reflect a symmetry then break it further down the line? Finally, what would these look like on a hyperbolic helicoid, and what is the application of any of this?

uhoh
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Svenn
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    I think gliding in the helicoid's case means turning it like a screw. Indeed, like you said, "snaking downwards barber-pole style". – Benjamin Wang Oct 26 '23 at 07:04
  • My impression is the same as Benjamin's, but if so "the" ruled helicoid is not unique in this regard, e.g., https://math.stackexchange.com/questions/2070121/self-sliding-surfaces. I don't have access to Steinhaus's book to get more context, however. By helicoid he may have meant any surface swept by a curve under (a one-parameter group of) helical motions of three-space. – Andrew D. Hwang Oct 26 '23 at 14:30
  • Thanks for the link, its description of glide as a ""slide" onto themselves so that you see no change." is helpful. Andrew, Wolfram seems to confirm that the ruled helicoid isn't unique in this regard, but given your link, the helicoid does seem unique in that it's the only non-rotary surface (e.g. cone, sphere, torus). Are you saying that the helicoid mentioned by Steinhaus could be different from the one given in Wolfram's definition? Is a surface swept by a 'curve' of helical motions different from the formal helicoid's sweeping of a 'line' helically? – Svenn Oct 26 '23 at 21:56

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