Wythoff symbol (InstanceTopic, 3)

From Higher Dimensions Database

Wythoff symbols are a way of describing uniform polyhedra by decorating the symbol of the Schwarz triangles.

Schwarz triangles are spherical triangles, that by reflection in the side of the triangle, lead to a finite cover of the surface of the sphere. Schwarz proposed and solved this problem.

The designation of the triangle is by the three corner-angles. These are fractions of the semicircle, and are usually designated by the denominator of the fraction that go into the semicircle, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All Schwarz triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or multiple-cover, comprised of several copies of one of these triangles.

The decoration of this symbol gives the location of the vertex, relative to the symmetry, with the vertex either off | on the particular mirrors. So p | q r would have its vertex off the mirror opposite the angle 180/p, and on the mirrors opposite 180/q and 180/r.

The resulting polyhedron is constructed by Wythoff's construction, where edges are formed by perpendiculars to mirrors that the vertex is off. Polygons might form around the three corners of the triangles, either of side 0p, p or 2p. In the case of 0p, this causes the derived string to repeat p times. One simply counts how many times the letters other than p occur before the bar sign.

So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the icosahedron. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and decagons, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie squares, hexagons and decagons. This is the truncated rhomboicoahedron.

The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3. One can derive the snub figure, by alternating the vertices of p q r |. When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.

When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles.