Wikipedia:Reference desk/Archives/Mathematics/2012 May 3

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May 3

Octahemioctahedron

Octahemioctahedron

Why is the octahemioctahedron the only orientable hemipolyhedron? Double sharp (talk) 12:23, 3 May 2012 (UTC)

It seems from the article that there are only 9 hemipolyhedrons (I don't know why these are the only ones). It's not hard to check that the octahemioctahedron is the only orientable one. Look at the faces that don't pass through the center. Ones that meet at a vertex have the opposite orientation. In order for the polyhedron to be orientable, there can't be any odd walk through the outside faces sharing vertices. This only happens on the octahemioctahedron (which is due to the fact that the outside faces are arranged like the vertices of a cube, which is bipartite). Rckrone (talk) 05:37, 5 May 2012 (UTC)

Thank you for your clear explanation. It's not hard to prove that there are only 9 hemipolyhedra, by the way. The hemipolyhedra are closely related to the quasiregular polyhedra. Some edges of the quasiregular polyhedra with four faces at a vertex form the faces passing through the centre of the hemipolyhedra: for example, the cuboctahedron can generate regular hexagons. To keep the polyhedron having two faces at an edge, one set of faces must be discarded: discarding the triangles gives the cubohemioctahedron, while discarding the squares gives the octahemioctahedron. This doesn't work for the quasiregular polyhedra having six faces at a vertex (the three ditrigonal polyhedra), because they don't have "equators". Since there are 5 quasiregular polyhedra with four faces at a vertex (the rectified Platonic and Kepler-Poinsot solids), and each can generate 2 hemipolyhedra (except the octahedron as a tetratetrahedron, because both types of faces are the same, so it can only generate one). Hence there can only be (4 × 2) + 1 = 9 hemipolyhedra. (Other polyhedra, such as the regular dodecahedron, can also have facets that pass through the centre, but these facets are not regular polygons and so no new uniform hemipolyhedra are obtained.) Double sharp (talk) 03:23, 6 May 2012 (UTC)

4-fold and 5-fold rotational symmetry

Why is it that Euclidean polyhedra cannot have both 4-fold and 5-fold rotational symmetry? What about polychora (4-polytopes) and higher? Double sharp (talk) 12:31, 3 May 2012 (UTC)

I think is due to the fact that if 4 pentagons met at a vertex, the total angle here would be 4 × 108° = 432. Slightly more rigorously would be to consider Möbius triangles. As 1/2 + 1/4 + 1/5 < 1 the corresponding Möbius triangle with angles π/2, π/4, π/5 will lie in the hyperbolic plane giving the Order-5 square tiling or Order-4 pentagonal tiling. Whats confusing me at the moment is Schwarz triangle with angles π/2, π/4, 2π/5, here 1/2 + 1/4 + 2/5 > 1 so it should be spherical, but it does not seem to exist.--Salix (talk): 07:37, 5 May 2012 (UTC)

Perhaps it covers an irrational fraction of the sphere? Double sharp (talk) 03:46, 6 May 2012 (UTC)

How widespread is the notation

$b){\overline {~a~}}$ in elementary schools around the world? (meaning a divided by b).— Preceding unsigned comment added by ‎ OsmanRF34 (talk • contribs) 21:51, 3 May 2012

According to the Long division article, its the notation used in English speaking countries, China, Japan and India. There are several different notations used one in Brazil, Venezuela and Colombia; one in the rest of Latin America; one in Spain, Italy, France, Portugal, Romania, Turkey, Greece and Russia; one in Germany, Norway, Macedonia, Poland, Croatia, Slovenia, Hungary, Czech Republic, Slovakia and in Bulgaria. --Salix (talk): 22:49, 3 May 2012 (UTC)

However, $b){\overline {~a~}}$ looks to me like a rather bad attempt to replicate, using available symbols, something that ought to look more like this, for example. 86.160.209.216 (talk) 02:38, 9 May 2012 (UTC)