Talk:Projective polyhedron

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 Field: Geometry

Unimportant neologism?[edit]

This term and its minor variations get the total of 3+1 hits on MathSciNet. While possibly a natural generalization of convex polyhedra, I don't see any substantial evidence that they have been studied, at least under this name. Arcfrk (talk) 21:01, 17 April 2010 (UTC)

Being unfamiliar with the term myself, I challenged it on Nbarth's talk page. I have since confirmed that reference may indeed be found in McMullen and Schulte, as stated. An accepted term for this class of polytope is long overdue. To students of elementary geometry and projective geometry this is an important class. Previous studies tended to refer to examples as "hemi" figures (such as the hemicube), but as the subject has developed over the last few decades that terminology has become stretched to the point of inappropriateness. I am willing to bet thet you would find plenty of "hemi" references on MathSciNet. Whether McMullen and Schulte have adopted a neologism or not is immaterial - they lend it full mathematical authority and it is worthy of encyclopedic treatment. I am delighted that such a sensible name has been provided at last. -- Cheers, Steelpillow (Talk) 14:03, 18 April 2010 (UTC)

Hi Arcfrk and Guy – thanks both for weighing in!

Doing a bit more reading, Magnus refers to this class as “tesselations [sic] of the elliptic plane” (1974), and Coxeter makes one mention of an “elliptic tessellation” (1970) (only Google books hit), which seems usual tiling/polyhedron ambiguity (is it a spherical polyhedron or spherical tiling?). I’ve added references to these; if there’s a better or more common name for the class I’m welcome to it, though “projective polyhedron” seems both natural enough and as standard as terms get in the field. (When you have one standard book every 50 years terminology drifts a bit…)

(FWIW, Guy, your use of “elliptic tiling” on Talk:Abstract polytope is currently (on a Wikipedia mirror) the #7 Google hit for that term.)

—Nils von Barth (nbarth) (talk) 00:45, 23 April 2010 (UTC)
"Elliptic" refers to any surface or geometry having positive curvature (by contrast to "flat" and "hyperbolic"). Among 2-dimensional surfaces this includes the 2-sphere, the inversive plane (with point at infinity), projective plane, Boy surface, Roman surface and associated tilings (polytopes). These fall into two topological groups, according to their Euler characteristic:
  • Those isomorphic to a 2-sphere
  • Those isomporphic to the projective plane.
It is quite common for a researcher to discover one, and not realise that the other is also elliptic.
Coxeter would have known better. I suspect that his reference to the latter as "elliptic" may not have been intended to exclude the former (whether or not he got the grammar right - apparently he was known for such little slips, and was always delighted when they were pointed out to him). I have not read Magnus. Mathworld, in its typical way, gets in a mess, defining the elliptic plane as projective and ellpitic geometry as spherical.
In the light of this, I think some recent edits to this article need correcting accordingly.
-- Cheers, Steelpillow (Talk) 20:28, 23 April 2010 (UTC)