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Conway's Notation

John H Conway suggested a notation of transformations to convert the regular figures into the remaining uniform figures, and also their duals. This appears to be largely a mechanical process.


The various operations are treated as prefix-functions, as applied to whatever is left of what is to the right. There is a stiring of it in the forms like sr{3,5} for the snub dodecahedron. Conway would read this as s(r(I)) or srI.

Because the duals are also being done in the same set, there are dual operators. Indeed, the d operator is among the list.

The seeds are the regular polyhedra, prisms Pn, antiprisms, An, and pyramids Yn. But you can write the prisms and antiprisms from a single n-gonl dihedron Dn, as prisms are eDn, and antiprisms sDn.

The prefixes in this system are "active": that is, they have general application outside this set. It is "open" in that it can be extended, and it does not allow for "complete enumeration" except by following the tree.


The following operators are used in this notation.

dual: This creates the dual of the figure it is applied to.

truncation: This removes the verticies, creating a new face where the vertex was. A partial truncate removes only those verticies with n sides.

"kis": Raise pyramids on each face, the sloping sides based on the face-edge, and the vertex over the centre. This is the dual of t: kX=dtdX

ambo: This produces a polyhedron whose verticies are in the midpoint of the original's edges. If a figure and its dual are placed so that their edges cross, the ambiant is the common intersection.

join: The join operator is dual to the ambo operator, and is thus the convex hull of duals with crossing edges.

expand: This is Mrs Stott's expansion operator. Each edge is moved out radially, but keeps its same size. New faces appear over the old margins, etc, the vertex giving way to the dual of the vertex figure. In three dimensions, eX=aaX

snub: This snub operator introduces a pair of triangles instead of the rectangle that "expand" creates. Since all the faces can be given a "clockwise" direction, the edges of eX point in opposite ends, and so the diagonals connect arrow-heads or arrow-tails. The snub is thence chiral.

gyro: This is the dual of the snub. All of the faces of the target are marked with a clockwise direction, and the centre of each face connects to 1/3 point of each edge. The 2/3 point on the edge is the one-third point on the opposite side. This produces a series of pentagonal faces.

bevel: This appears to be a name for truncated ambio.

ortho: This addes to each old edge, a new vertex in the middle, and to each face, a new central vertex. New edges connect the centre to the midpoint, and the new faces are kite-shaped figures that go from the old centre to a midpoint, to an original vertex, to the next edge midpoint, back to the face centre. oX=jjX. This is the dual of "expand".

meta: This is the duak of the bevel operator. On each face, draw new verticies on the edge midpoints, and the face-centre. The new edges connect the central vertex to the old verticies and the edge midpoint-vertex.

reflect: Take the mirror image of a figure. This has no effect if the figure already has symmetry.

propeller: Makes each n-gon into a propeller of an n-gon surrounded by quadralaterals. This operator communtes with a, j, i and e, so that paX=apX


The notation is quite extensable: George Hart added the last two. It produces lots of interesting polytopes, not necessarily uniform. The prefixes in it are "live", in as much as they can be applied to any given shape. The seed-set is restricted to the regular figures, and to

Copyright 2002 Wendy Krieger

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