# Wythoff symbol (InstanceTopic, 3)

### From Hi.gher. Space

**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.