Level of complexity

Level of complexity, often abbreviated as LOC, is a measure of a polytope's complexity introduced by Jonathan Bowers. It is defined as value/half-order, in which value represents the number of dyads in a polytope's external pieces, and half-order is that polytope's order of rotational symmetry; if calculated correctly, LOC should always be an integer value.

The use of symmetry in the calculation allows polytopes of various symmetries or dimensionalities to be compared objectively.

2D
Convex regular polygons, which have n sides and n rotational symmetries, always have LOC=1. Regular star polygons' sides are split into two pieces each, so have LOC=2. Convex semi-uniform polygons have 2n sides of alternating lengths and LOC=2, and semi-uniform star polygons can have LOC up to 6.

3D
The Platonic solids all have LOC=1, and the Kepler-Poinsot solids all have LOC=3, excluding the great icosahedron which has LOC=9. As for the rest of the non-prismatic uniform polyhedra, LOC is far more variable than in 2D, the small inverted retrosnub icosicosidodecahedron at the top end of this with LOC=213.

Convex n-gonal prisms have 2n+n edges - doubled to 6n to account for the individual faces' pieces - and half-order 2n, which gives LOC=6n/2n=3. Star prisms' pieces split much like their base star polygons, and so have LOC=6. Convex antiprisms have 2n+2n=4n pieces, doubled for the same reason as the convex prisms', and half-order 2n, for LOC of 8n/2n=4. Star antiprisms with base n/2 have LOC=11. Finally, star antiprisms of the type n/m with convex trapezoidal vertex figures should have LOC=6m, and those with crossed-trapezoidal vertex figures appear to have LOC=6m+6; whether this is true for all n/m-antiprisms has yet to be proven.

The filling method used can dramatically affect a polyhedron's - or any higher-dimensional polytope's - complexity. An example of this is the great dirhombicosidodecahedron, which has LOC=164 under binary filling, but a mere 76 under solid filling.

4D+
Convex regular polytopes always have LOC=1. Otherwise, LOC is extremely difficult to calculate for non-convex polytopes, thanks to the lack of software able to extract their external pieces.