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.