|
In geometry, the Wythoff symbol was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles. A Schwarz triangle is a triangle that, with its own reflections in its edges, covers the sphere or the plane a finite number of times. The usual representation for the triangle is three numbers – integers or fractions – such that π/x is the angle at one vertex. For example, the triangle (2 3 4) represents the symmetry of a cube, while (5/2 5/2 5/2) is the face of an icosahedron. Wythoff's construction in three dimensions consists of choosing a point in the triangle whose distance from each of the sides, if nonzero, is equal, and dropping perpendiculars to each of the edges. Each edge of the triangle is named for the opposite angle; thus an edge opposite a right angle is designated '2'. The symbol then corresponds to a representation of off | on. Each of the numbers ''p'' in the symbol becomes a polygon ''pn'', where n is the number of other edges that appear before the bar. So in 3 | 4 2 the vertex – a point, being here a degenerate polygon with 3×0 sides – lies on the π/3 corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a square (having 4×1 sides) and a digon (having 2×1 sides) of zero area. The special case of the snub figures is done by using the symbol | p q r, which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as p 3 q 3 r 3. This gives an antiprism when q=r=2. Each symbol represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different ''Wythoff symbols'' from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry. It can be applied with a slight extension to all uniform polyhedra, but the construction methods do not lead to all uniform tilings in euclidean or hyperbolic space. == Summary table == There are seven generator points with each set of p,q,r (and a few special forms): | |- align=center | r | p q |''(q.p)r'' | | 2 | p q |''(q.p)² |r||t1 | |- align=center |rowspan=3|truncated and expanded | q r | p |''q.2p.r.2p'' | | q 2 | p |''q.2p.2p |t|| t0,1 | |- align=center | p r | q | ''p.2q.r.2q'' | | p 2 | q | ''p. 2q.2q'' |t|| t0,1 | |- align=center | p q | r |''2r.q.2r.p'' | | p q | 2 |''4.q.4.p'' | rr|| t0,2 | |- align=center |rowspan=2| even-faced | p q r | | ''2r.2q.2p'' | | p q 2 | | ''4.2q.2p'' | tr||t0,1,2 | |- align=center | p q (r s) | | ''2p.2q.-2p.-2q'' | - | p 2 (r s) | | ''2p.4.-2p.4/3'' |colspan=2| | - |- align=center |rowspan=2| snub | | p q r | ''3.r.3.q.3.p'' | | | p q 2 | ''3.3.q.3.p'' |colspan=2| sr | |- align=center | | p q r s | ''(4.p.4.q.4.r.4.s)/2'' | - | - | - |colspan=2| | - |} There are three special cases: * p q (r s) | – This is a mixture of p q r | and p q s |. * | p q r – Snub forms (alternated) are given by this otherwise unused symbol. * | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wythoff symbol」の詳細全文を読む スポンサード リンク
|