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 Field:  Geometry

Zig-zags and helixes[edit]

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)

Spherical zigzags[edit]

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)

Apeirogons in 4-space and higher[edit]

I was thinking the skew forms have a higher degree of freedom in 4-space (or higher) . In 3-space they can "oscillate" in one or two orthogonal axes, while in 4-space, they can oscillate in three orthogonal axes. I'm not sure how this is defined, but probably something like 2 independent "screw axes"? (Incidentally, my interest in the question isn't merely trivia, but represents the skew Petrie polygons of regular polytopes and skew regular Petrie apeirogons of regular honeycombs. For example, the n-hypercube and Hypercubic honeycomb families.) Tom Ruen (talk) 20:26, 26 August 2008 (UTC)

P.S. I see we don't have any definition of a Regular skew polygon at Skew polygon which ought to exist in parallel to the skew apeierogons here. Tom Ruen (talk) 20:39, 26 August 2008 (UTC) ... Okay, I aded a sentence and redirect for a start. Tom Ruen (talk) 21:00, 26 August 2008 (UTC)

There are no ends![edit]

"But whereas an ordinary polygon has no ends because it is a closed circuit, an apeirogon can also have no ends because you can never make the infinite number of steps needed to get to the end in either direction".

A bit chatty and children's bookish! And self-contradictory to say the ends don't exist because you can't reach them. Much better I think just to say that, like a (finite) polygon, an apeirogon is unbounded, i.e. has no ends. SteveWoolf (talk) 02:58, 7 August 2010 (UTC)


The first paragraph states "It is the limit of a sequence of polygons with more and more sides." It is the limit under which topology? It seems important that this object is the result of some sort of limit on the set of polygons, there should probably be more explanation of this. Maybe not in the intro paragraph, but could someone who knows more about this topic put in some more information about this. (talk) 17:58, 18 November 2011 (UTC)

Move skew content to skew polygon?[edit]

I'm considering moving the skew content to skew polygon, for both finite and infinite cases. Coxeter uses the word apeirogon exclusively for the linear form, and the rest he calls "generalized polygons". Tom Ruen (talk) 18:00, 22 February 2015 (UTC)

Hmmmm... Peter McMullen in Abstract regular polytopes (p.217) calls {p}#{} as a skew polygon, and {∞}#{} for a zig-zag apeirogon, blending a linear one {∞} with a segment {} or a helix?! It still might make more sense there as a section skew polygon#skew aperigon. Tom Ruen (talk) 19:11, 22 February 2015 (UTC)

I decided to be bold and moved the content, and I think it fits much better with the regular skew polygons. Tom Ruen (talk) 22:44, 22 February 2015 (UTC)

"The regular apeirogon can also be seen within the edges of . . ." ???[edit]

This statement appears in the section titled Regular apeirogon:

"The regular apeirigon can also be seen within the edges of 4 of the regular, uniform tilings, and 5 of the uniform dual in the Euclidean plane."

It is followed by some pictures of regular, uniform, and dual tilings of the plane.

But I do not understand what the quoted statement is trying to say. Can someone who understands it please explain that statement clearly in the article? Thanks.Daqu (talk) 23:57, 27 February 2015 (UTC)

countably infinite[edit]

So can you have a polygon with uncountably many sides? Double sharp (talk) 15:23, 15 March 2015 (UTC)

A single discrete generator of a regular apeirogon means countably infinite, but I suppose you can say a circle has an uncountably infinite number of sides? Tom Ruen (talk) 15:45, 15 March 2015 (UTC)
It might then be like an infinitely dense polytope, where each arc (edge) covers an irrational fraction of the circle's circumference and it never closes up. (You'd then have a dense collection of vertices: if you can have one at every single point on the circle, you'd have made an apeirogon with infinitely many sides.) Double sharp (talk) 16:05, 15 March 2015 (UTC)
I thought about that, but its still a discrete generator, so countable! A single irrational fraction still can't generate all irrational numbers. Tom Ruen (talk) 17:01, 15 March 2015 (UTC)
Oops! You're right. It would have to be non-Wythoffian, then, placing a vertex at every point on the circle. But then all the edges are zero length if it's not a star (which is silly), and if it is a star it can't be regular (because that would use a discrete generator). Double sharp (talk) 14:26, 16 March 2015 (UTC)

hyperbolic pseudogon more reference needed[edit]

I split of the bit about the hyperbolic pseudogon, seems something seperate, also there is only one reference to it ( Norman Johnson, Geometries and symmetries, (2015), Chapter 11. Finite symmetry groups, Section 11.2 The polygonal groups. p.141) and I could not even find this book. is more known about it?

Also I noticed that previously (around 15 februari 2015) there was much more on this page , should we reinstate more back? WillemienH (talk) 08:06, 10 August 2015 (UTC)

The book is in preprint. The other material was split into skew apeirogon. Tom Ruen (talk) 03:41, 12 August 2015 (UTC)

Thanks , think we have to wait till it is published, not sure about it all: What is an pseudogon? , the illustration doesn't show a lot (only when you look at the original file you see the pseudogon boundary, a thin red line) but even then is it just the red segmented arc, is it the area above this arc,is it the area below it, or are both half planes pseudogons? WillemienH (talk) 21:09, 13 August 2015 (UTC)