# Level of complexity

Level of complexity, often abbreviated as LOC, is a measure of a polytope model's complexity introduced by Jonathan Bowers. It is defined as value/half-order, in which value represents the sum of values of a polytope's external pieces, where the value of a dyad is counted as 1, 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 more directly.

## 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.