Wythoff symbol (English)
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Definition Niveau 4
A way of specifying any uniform polyhedron using three rational numbers and a bar - . If two of the numbers are equal to 2, the third is arbitrary. Otherwise we have the restriction that the numerators can only be 2, 3, 4, or 5 - and 4 and 5 are not allowed to occur together. There are a few cases:
Regular polyhedra have a pattern r, r, r, ..., r of polygons round each vertex (where 3, 3, 3, 3 would mean triangle-triangle-triangle-triangle). By inserting 'non-faces' with two sides, we can write this as 2, r, 2, r, ..., 2, r. We write this as p - 2r, where p is the number of times the pattern is repeated (for example, 3, 3, 3, 3 would become 2, 3, 2, 3, 2, 3, 2, 3 and be written 4 - 23).
Quasi-regular polyhedra have a pattern q, r, q, r, ..., q, r of polygons round each vertex, (where 3, 4, 3, 4 would mean triangle-square-triangle-square). We write this as p - qr , where p is the number of times the pattern repeats itself (so 3, 4, 3, 4 would be written 2 - 34).
Semi-regular polyhedra have pattern p, 2r, q, 2r of polygons round each vertex, where p may be 2, to represent a 'non-face'. This is written with Wythoff symbol pq - r.
Even-faced polyhedra have pattern 2p, 2q, 2r of polygons round each vertex. This is written with Wythoff symbol pqr - .
Snub polyhedra have pattern 3, p, 3, q, 3, r of polygons round each vertex. This is written with Wythoff symbol - pqr.
The Great Dirhombicosidodecahedron has eight faces round each vertex; the Wythoff method can only deal with up to 6 faces round a vertex, so there is no true Wythoff symbol for this uniform polyhedron. It is denoted with the pseudo-Wythoff symbol ( - 3/2 5/3 3 5/2).
For the purpose of making Wythoff symbols, a fraction a/b means a star made from a points by joining every bth point. For instance, a five-pointed star can be made from five points by joining up every second point; this is written 5/2.
Regular polyhedra have a pattern r, r, r, ..., r of polygons round each vertex (where 3, 3, 3, 3 would mean triangle-triangle-triangle-triangle). By inserting 'non-faces' with two sides, we can write this as 2, r, 2, r, ..., 2, r. We write this as p - 2r, where p is the number of times the pattern is repeated (for example, 3, 3, 3, 3 would become 2, 3, 2, 3, 2, 3, 2, 3 and be written 4 - 23).
Quasi-regular polyhedra have a pattern q, r, q, r, ..., q, r of polygons round each vertex, (where 3, 4, 3, 4 would mean triangle-square-triangle-square). We write this as p - qr , where p is the number of times the pattern repeats itself (so 3, 4, 3, 4 would be written 2 - 34).
Semi-regular polyhedra have pattern p, 2r, q, 2r of polygons round each vertex, where p may be 2, to represent a 'non-face'. This is written with Wythoff symbol pq - r.
Even-faced polyhedra have pattern 2p, 2q, 2r of polygons round each vertex. This is written with Wythoff symbol pqr - .
Snub polyhedra have pattern 3, p, 3, q, 3, r of polygons round each vertex. This is written with Wythoff symbol - pqr.
The Great Dirhombicosidodecahedron has eight faces round each vertex; the Wythoff method can only deal with up to 6 faces round a vertex, so there is no true Wythoff symbol for this uniform polyhedron. It is denoted with the pseudo-Wythoff symbol ( - 3/2 5/3 3 5/2).
For the purpose of making Wythoff symbols, a fraction a/b means a star made from a points by joining every bth point. For instance, a five-pointed star can be made from five points by joining up every second point; this is written 5/2.
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