Talk:Apeirogon

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[edit] Zig-zags and helixes

I appreciate the consideration of the expanded forms, but curious about definitions. All regular polytopes I know have reflection symmetry on the perpendicular edge bisectors. The first zig-zag type has a 2-fold rotation point mid-edge (a point-reflection at least). The second helix form must have a rotational "twist" applied mid-edge, apparently of "any angle" since no need to close up. I guess the first is an "order-2" helix? Both have a degree of freedom which is also unexpected in "regular" forms. I'd probably at best call them "quasiregular", alternating two types of facets (edges). Tom Ruen 22:09, 3 August 2007 (UTC)

Before you think about doing that, read the Grunbaum reference and decide what precisely is meant by "regular". They are certainly not quasiregular, since all sides lie in the same symmetry group. -- Steelpillow 07:54, 4 August 2007 (UTC)

I wasn't going to change anything, but I don't have the source to read. Perhaps you can offer a quote on the definition of regular? Tom Ruen 21:58, 4 August 2007 (UTC)

From the Regular polygon entry: "A regular polygon is a polygon which is equiangular (all angles are congruent) and equilateral (all sides have the same length)." The two congruency conditions taken together imply the stricter conditions that all angles are within the same symmetry orbit, and all sides are within a single symmetry orbit. This logic does not apply to higher polytopes, for which this transitivity must be stated in the definition; the most elegant expression of this is the definition that a regular polytope (including any regular polygon) is transitive on its flags. Some non-trivial symmetry group of the figure is implied. We may treat a translation as a rotation about a point at infinity. Regular apeirogons have translational symmetry, so they can be thought of as belonging to an infinite rotation group. Reflection symmetry as such is not a requirement, though the simple zig-zag apeirohedron has mirror symmetry about any line through a vertex and orthogonal to the main axis of progression. The regular compound of five tetrahedra is another handed example. Again there is nothing to prevent degrees of freedom from existing. For finite polytopes the requirement for transitivity rules most of these out (though stariness is one which survives); for apeirogons, because the Euclidean plane plays games with infinity, certain other degrees of freedom also survive. Much of what I just wrote is assumed by Grunbaum, rather than made explicit. HTH -- Steelpillow 10:29, 5 August 2007 (UTC)

[edit] Spherical zigzags

I was just noticing the antipodal digons on the sphere have a degree of freedom in their internal angle, so at least that's another example of a regular polygon without a singular form. Well, AND I imagine you can equally call for regular n-gons on the sphere to be zig-zags as well, which perhaps defines a new class of regular skew dihedrons (Containing the vertices of a spherical antiprism)!? Tom Ruen 19:52, 27 August 2007 (UTC)

It might be argued that on the sphere, the zig- or zag- as it were becomes important. For example if you flatten out one of those faces you obtain a star-like polygon which is not regular: flatten any other regular spherical polygon and you get a regular planar polygon (accepting the digon as a "polygon" for this purpose). I think you would also have to understand exactly how you define the corner angles. -- Steelpillow 19:09, 28 August 2007 (UTC)