Orbits

if i = 0: 0-cell(d) = ⟨{ β j o β k with 1 ≤ j &lt; k ≤ N }⟩(d)
else: i-cell(d) = ⟨β 1 , β 2 , ..., β i-1 , β i+1 , ..., β N ⟩(d)

We define orbits as a set of darts that are accessible from a given dart, using a certain set of beta functions. For example:

⟨β₁⟩(d) refers to all darts accessible from d using β₁ recursively any number of times.
⟨β₁, β₃⟩(d) refers to all darts accessible from d using any combination of β₁ and β₃.

i-cells

A specific subset of orbits, referred to as i-cells are defined and often used in algorithms. The general definition is the following:

In our case, we can use specialized definitions for our dimensions:

i	Geometry	2-map	3-map
0	Vertex	⟨β₁ o β₂⟩(d) or ⟨β₂ o β_-1⟩(d)	⟨β₃ o β₂, β₁ o β₃⟩(d) or ⟨β₃ o β₂, β₃ o β_-1⟩(d)
1	Edge	⟨β₂⟩(d)	⟨β₂, β₃⟩(d)
2	Face	⟨β₁⟩(d)	⟨β₁, β₃⟩(d)
3	Volume	-	⟨β₁, β₂⟩(d)