# Talk:Skew apeirohedron

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## Fundamental changes

### Uniform infinite polyhedra

Wachmann, Burt and Kleinmann, Infinite polyhedra, Technion, 1974, describes not only the few regular examples known to Coxeter, but also nearly a bookful of uniform infinite polyhedra – which may usefully be summarised here. -- Cheers, Steelpillow (Talk) 14:36, 26 February 2009 (UTC)

### Skewness

In two dimensions, a zig-zag apeirogon is only regular if it is understood to be skew, with a screw angle of 180 deg. Like Petrie polygons, "skew" indicates extension in an extra, third dimension. But the same figure can also be understood as a non-regular apeirogon in the plane whose angles alternate between positive and negative angles of equal magnitude. This second interpretation is not skew. All the infinite polyhedra described in this article extend only in three dimensions, and so are not skew. For this reason Wachmann et.al. (ibid) refer to them merely as "infinite polyhedra". This modern usage is more precise and understandable than the older names, and I would suggest that this page be retitled accordingly. -- Cheers, Steelpillow (Talk) 14:36, 26 February 2009 (UTC)

### Other names and articles

The lead mentions various names for these figures.

As sponges, these figures are more accurately polyhedral sponges, since any surface characterized by periodic hyperbolic holes is a sponge, whether smoothly curved or made of flat polygons.

Similarly, there are many hyperbolic tessellations, i.e. tilings of the hyperbolic plane, which are not sponges.

The documented apeirohedra comprise the plane tilings and these 3-D infinite polyhedra. When we consider how small the Apeirohedron article is, I think that it would be useful to merge the two articles. -- Cheers, Steelpillow (Talk) 14:36, 26 February 2009 (UTC)

### Gott

In his paper on pseudopolyhedrons, Gott proposed a relaxed definition of regularity which allowed some novel structures. This definition has not stood the test of time (for example his novel structures are not transitive on their flags). His theorems need to be explained in this context, and to be moved out of the "regular" section. -- Cheers, Steelpillow (Talk) 14:36, 26 February 2009 (UTC)

Aperion in the greek sense, has the meaning of endless as the sea or the desert. It is built of two stems, a- = without, and peri = perimeter. The notion that this means "infinite", rather than say "immense" is something to do with the confusion between number and extent that allowed Cantor to advance a theory of infinity that elsewhere would had been dismissed as Q.E.A..

The figure shown here is neither apeiric (where the faces are solid in the containing space) nor skew (where there is a reversal that inverts the surface), but only infinite. --Wendy.krieger (talk) 07:49, 11 December 2009 (UTC)

## Examples

Can we change the current figure to either an infinite skew polyhedron or a Menger sponge?199.212.11.84 (talk) 17:50, 22 March 2010 (UTC)