# 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[edit | edit source]

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 2*n* sides of alternating lengths and LOC=2, and semi-uniform star polygons can have LOC up to 6.

## 3D[edit | edit source]

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 6*n* to account for the individual faces' pieces - and half-order 2*n*, which gives LOC=6*n*/2*n*=3. Star prisms' pieces split much like their base star polygons, and so have LOC=6. Convex antiprisms have 2*n*+*2n*=4*n* pieces, doubled for the same reason as the convex prisms', and half-order 2*n*, for LOC of 8*n*/2*n*=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=6*m*, and those with crossed-trapezoidal vertex figures appear to have LOC=6*m*+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+[edit | edit source]

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.

## External links[edit | edit source]

- Bowers, Jonathan. "Glossary" (LOC)