# Holosnub

Holosnubbing or holoalternation is an operation that produces a faceting (possibly compound) of a given polytope. It is closely related to vertex alternation, but does not require a bipartite vertex adjacency graph. It has proven useful for discovering new uniform polytopes.

Holosnubbing is a combinatorial operation defined on the underlying abstract polytope, and does not change the realization mapping vertices to points.

## Definition and examples

The holosnub operation has a terse inductive definition:

1. The holosnub of a polytope retains the vertices of the original polytope.
2. The vertex figures of the polytope (which may be compound) are added as new facets. Here, vertex figures are defined as the result of connecting vertices adjacent to a selected vertex.
3. For polytopes of rank greater than 2, the original facets are replaced with holosnub versions.

The resulting figure is a compound iff the vertex adjacency graph (1-skeleton) is bipartite, and in that case the two components of the compound are the alternations of the polytope.

Polytopes containing quadrilaterals will produce digons under the holosnub operation. Digons and any polytopes containing them do not have a defined holosnub.

### Polygons

Given a base polygon P with n > 2 sides, its holosnub H is constructed as follows. Select a vertex and walk through P, skipping every other vertex until the starting point is reached and connecting these vertices together into a new polygon. There are two cases:

1. If P has an odd number of vertices, all vertices of P have been visited, and the new polygon is H (e.g. changing a pentagon to a pentagram).
2. If F has an even number of vertices, then the resulting figure is one of two alternations of P depending on the starting vertex. In this case, H is a compound comprising both alternations (e.g. changing a hexagon into a hexagram). In the special case where F is a quadrilateral, a compound of two degenerate digons are produced.

Given a vertex v of face F, H has exactly one corresponding edge e(F, v), and it connects the two vertices adjacent to v in the original polygon. This edge is the vertex figure of F at v. The e(F, v) notation will assist with the following definition of polyhedral holosnubs.

### Polyhedra

A holosnub polyhedron H can be constructed from a base polyhedron P as follows. First, all elements are removed except for the vertices, and every face is replaced with its holosnub.

H may then be closed up by the addition of each vertex figure of P. To see why this works, a given vertex v in P has three or more incident faces, and each face F has a corresponding edge e(F, v). These edges connect all the vertices adjacent to v into a closed polygon or polygon compound, which is precisely the vertex figure of P at v. (This polygon is not guaranteed to be planar in general, but for uniform polyhedra it is due to the circumscribed sphere and single edge length.)

If P is isogonal, then so is H. If P is regular, then H is uniform if not degenerate; for example, the holosnub dodecahedron is the small ditrigonary icosidodecahedron. General uniforms have non-uniform holosnubs, but in some cases their edge lengths can nevertheless be adjusted to produce uniform polyhedra, such as the truncated icosahedron producing the small snub icosicosidodecahedron.