What's the difference between zig-zags and a helixes?

Question

I've been reading through the Polytope-Wiki entry on helices. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$ is a blend of itself with a point and so is a one-dimensional helix.

All of these polygons have the same graph-structure. Every vertex joins two edges and every edge meets two vertices. There are no cycles, so their Hasse diagrams are isomorphic. This means that all the helices are isomorphic as abstract polytopes.

As far as I can make out, their symmetry groups are isomorphic. Every symmetry of $\{\infty\}$ corresponds to a symmetry of the $n$-gonal helix and vice versa.

So how do these shapes differ?

All the helices are isomorphic as abstract polygons and they've got isomorphic symmetry groups.

The only way I can see to distinguish them is via "undoing" the blending. The definitions of blending that I've come across all reference the ambient space, where we can project blended polytope onto certain subspaces to obtain the polytopes that we originally blended. This doesn't really satisfy me because I feel there should be a way to distinguish helices in a "more abstract" manner.

Can it really be that the same abstract regular polytope can have such qualitatively different realisations?

Am I missing some simple way to distinguish the helices from each other and from the zig-zags and apeirogons?

A zig-zag, image from Wikipedia.
Part of an apeirogon, image from Wikipedia.
Side view of hexagonal helix, a screenshot from https://cpjsmith.uk/regularpolyhedra.|

Final note: The faces of the petrial mu-octahedron are triangular helices. The faces of the trihelical square tiling are square helices. Is it reasonable to say that "skew apeirogons" are the faces of both of these polyhedra?

Answer 1

A helix is a non-planar shape. Here is a much better picture of a helix:

A zig-zag is a planar shape.

Therefore a helix is not a zig-zag.

Answer 2

I think the short answer to the question

Can it really be that the same abstract polytope can have such qualitatively different realisations?

Is "yes".

Even a square (as an abstract graph) has some qualitatively quite different embeddings in the plane. Many more in space.