John Conway explored generalized polyhedron operators, defining what is now called Conway polyhedron notation, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a *semi-snub*.

In this notation, snub is defined by the dual and gyro operators, as *s* = *dg*, and it is equivalent to an alternation of a truncation of an ambo operator. Conway's notation itself avoids Coxeter's alternation (half) operation since it only applies for polyhedra with only even-sided faces.

In 4-dimensions, Conway suggests the snub 24-cell should be called a *semi-snub 24-cell* because it doesn't represent an alternated omnitruncated 24-cell like his 3-dimensional polyhedron usage. It is instead actually an alternated truncated 24-cell.

Coxeter's snub terminology is slightly different, meaning an alternated truncation, deriving the snub cube as a *snub cuboctahedron*, and the snub dodecahedron as a *snub icosidodecahedron*. This definition is used in the naming two Johnson solids: snub disphenoid, and snub square antiprism, as well as higher dimensional polytopes such as the 4-dimensional snub 24-cell, or s{3,4,3}.

A regular polyhedron (or tiling) with Schläfli symbol,
{
p
,
q
}
, and Coxeter diagram , has truncation defined as
t
{
p
,
q
}
, and and snub defined as an alternated truncation
h
t
{
p
,
q
}
=
s
{
p
,
q
}
, and Coxeter diagram . This construction requires *q* to be even.

A quasiregular polyhedron
{
p
q
}
or *r*{*p*,*q*}, with Coxeter diagram or has a quasiregular truncation defined as
t
{
p
q
}
or *tr*{*p*,*q*}, and Coxeter diagram or and quasiregular snub defined as an alternated truncated rectification
h
t
{
p
q
}
=
s
{
p
q
}
or *htr*{*p*,*q*} = *sr*{*p*,*q*}, and Coxeter diagram or .

For example, Kepler's **snub cube** is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol
{
4
3
}
, and Coxeter diagram , and so is more explicitly called a **snub cuboctahedron**, expressed by a vertical Schläfli symbol
s
{
4
3
}
and Coxeter diagram . The snub cuboctahedron is the alternation of the *truncated cuboctahedron*,
t
{
4
3
}
and .

Regular polyhedra with even-order vertices to also be snubbed as alternated trunction, like a *snub octahedron*,
s
{
3
,
4
}
, (and *snub tetratetrahedron*, as
s
{
3
3
}
, ) represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry. The *snub octahedron* is the alternation of the truncated octahedron,
t
{
3
,
4
}
and , or tetrahedral symmetry form:
t
{
3
3
}
and .

Coxeter's snub operation also allows n-antiprisms to be defined as
s
{
2
n
}
or
s
{
2
,
2
n
}
, based on n-prisms
t
{
2
n
}
or
t
{
2
,
2
n
}
, while
{
2
,
n
}
is a regular n-hosohedron, a degenerate polyhedron, but a valid tiling on the sphere with digon or lune-shaped faces.

The same process applies for snub tilings:

Nonuniform polyhedra with all even-valance vertices can be snubbed, including some infinite sets, for example:

Snub star-polyhedra are constructed by their Schwarz triangle (p q r), with rational ordered mirror-angles, and all mirrors active and alternated.

In general, a regular polychora with Schläfli symbol,
{
p
,
q
,
r
}
, and Coxeter diagram , has a snub with extended Schläfli symbol
s
{
p
,
q
,
r
}
, and .

A rectified polychora
{
p
q
,
r
}
= **r{p,q,r}**, and has snub symbol
s
{
p
q
,
r
}
= **sr{p,q,r}**, and .

There is only one uniform snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol,
{
3
,
4
,
3
}
, and Coxeter diagram , and the snub 24-cell is represented by
s
{
3
,
4
,
3
}
, Coxeter diagram . It also has an index 6 lower symmetry constructions as
s
{
3
3
3
}
or s{3^{1,1,1}} and , and an index 3 subsymmetry as
s
{
3
3
,
4
}
or sr{3,3,4}, and or .

The related snub 24-cell honeycomb can be seen as a
s
{
3
,
4
,
3
,
3
}
or s{3,4,3,3}, and , and lower symmetry
s
{
3
3
,
4
,
3
}
or sr{3,3,4,3} and or , and lowest symmetry form as
s
{
3
3
3
3
}
or s{3^{1,1,1,1}} and .

A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and or sr{2,3,6}, and or sr{2,3^{[3]}}, and .

Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and or sr{2,4^{1,1}} and :

The only uniform snub hyperbolic uniform honeycomb is the *snub hexagonal tiling honeycomb*, as s{3,6,3} and , which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, . It is also constructed as s{3^{[3,3]}} and .

Another hyperbolic (scaliform) honeycomb is an snub order-4 octahedral honeycomb, s{3,4,4}, and .