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feat: implement Cubical complex with canonical orientations #16

@FiberedSkies

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@FiberedSkies

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Similar to #12 and #13, we should implement a builder pattern for cubical complexes with canonical orientation. For cubical complexes, we work with elementary cubes. An n-cube can be written as a product of intervals $I_1 \times I_2 \times ... \times I_k$ where each $I_i$ is either:

A degenerate interval $[v_i, v_i]$ (dimension 0)
A non-degenerate interval $[v_i, v_i + 1]$ (dimension 1)

The incidence $[b:a]$ between an (n+1)-cube $b$ and an n-cube $a$ is defined when $a$ is a face of $b$. The canonical incidence $[b:a]$ is defined as:

  • $[b:a] = (-1)^k$ where $k$ is the number of non-degenerate intervals in $b$ that precede coordinate $i$
  • If $a$ is the "lower" face (has $[v_i, v_i]$), then $k$ counts intervals before position $i$
  • If $a$ is the "upper" face (has $[v_i + 1, v_i + 1]$), then $k$ counts intervals up to and including position $i$

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    enhancementNew feature or requestmaththis issue requires implementing pure mathspriority: lowuximproves user experience

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