Talk:Small cubicuboctahedron

wanted: non-Schläfli symbol

The corresponding tiling of the hyperbolic plane, with vertex figure 3.8.4.8 (triangle, octahedron, square, octahedron) may be denoted by the (extended) Schläfli symbol t_0,1{4, 3, 3} .... -User:Nbarth

I understand what is intended, but that's the extended Schläfli symbol for the rectified tesseract. —Tamfang (talk) 20:31, 13 May 2010 (UTC)

oops, truncated. —Tamfang (talk) 22:03, 9 July 2023 (UTC)

That's why the Wythoff symbol is superior, so the tiling is represented as 3 4 | 4. The Schläfli symbol is limited to families that have regular polytope members (i.e. a right angle fundamental domain). Tom Ruen (talk) 08:04, 14 May 2010 (UTC)

TBH I never really liked Wythoff symbols, finding CD diagrams more intuitive (and general). Double sharp (talk) 15:15, 18 July 2012 (UTC)

How can this be a tiling of the Klein quartic???

The article says"

". . . the small cubicuboctahedron can be interpreted as a coloring of the regular (not just uniform) tiling of the genus 3 surface by 20 equilateral triangles, meeting at 24 vertices, each with degree 7. This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface . . ."

But there is no such tiling of the surface of genus 3. (For, the number of vertices would have to be 3*20 / 7, which is not only unequal to 24, but it is not even an integer.)Daqu (talk) 13:46, 16 August 2012 (UTC)

This is *not* a polyhedrally "immersed" surface -- nor can that be "Stated alternatively" to say it is a uniform tiling (although it is)

The article states:

"As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface. Stated alternatively, it corresponds to a uniform tiling of this surface."

This is not an immersion, polyhedral or otherwise, of a surface. For, at each of its 24 vertices, the picture locally is that of a cone on a figure-8 (just as the cross-cap has one of these), and even one of these means the polyhedron is not immersed in 3=space. The underlying polyhedron does correspond to a uniform tiling of the surface of genus 3. But the second sentence's claim that it is the first sentence "Stated alternatively", is also incorrect.Daqu (talk) 14:04, 16 August 2012 (UTC)