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Cells
Here we consider "cubical" cells.
Given a 2D grid $[0,N] \times [0,M]$, we define cells for all integers n,m as follows:
- a vertex is $\{n \} \times \{m \}$, for all $0 \leq n \leq N, 0 \leq m \leq M$;
- an edge is $\{n \} \times (m, m + 1)$ or $(n, n + 1) \times \{m \}$, for all $0 \leq n \leq N-1, 0 \leq m \leq M-1$;
- (the inside of) a pixel is $(n, n + 1) \times (m, m + 1)$, for all $0 \leq n \leq N-1, 0 \leq m \leq M-1.$
Red vertices, green edges, blue pixel:
A voxel and its boundary cells:
In other words all cells are "open".
This gives us a uniform terminology for all dimensions:
- a vertex is a $0$-cell,
- an edge is a $1$-cell,
- a pixel is a $2$-cell,
- a voxel is a $3$-cell,
- etc.
In 3D we have:
- $0$-cell $\{n \} \times \{m \} \times \{k \}$;
- $1$-cell $\{n \} \times \{m \} \times (k, k + 1)$, or $\{n \} \times (m, m + 1) \times \{k \}$, or $\{n \} \times \{m \} \times (k, k + 1)$;
- $2$-cell $(n, n + 1) \times (m, m + 1) \times \{k \}$, or $\{n \} \times (m, m + 1) \times (k, k + 1)$, or $(n, n + 1) \times \{m \} \times (k, k + 1)$;
- $3$-cell $(n, n + 1) \times (m, m + 1) \times (k, k + 1)$;
- etc.
This notation can and should be simplified. The simplest way might be to use $(n)$ for $(n,n+1)$.
More generally, a cell in the $k$-dimensional space is the product of points and edges: $$C = A_1 \times A_2 \times A_3 \times \ldots \times A_k,$$ where $A_i = \{n \}$ or $(n, n + 1)$.
Cells are used to form cubical complexes. See also Cell and Cell complexes.