Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

The declared cellulation

The retuned polyhedral slicing family › The declared cellulation · lines 897–939

VIII-B@3#eq:fixed-vertices, VIII-B@3#eq:born-vertices, VIII-B@3#eq:faces, VIII-B@3#eq:collar-faces

The compact patch is $ρ \leq 4$ . Define

A_{i} = r e_{i}, D_{i} = 4 e_{i}, T (A_{i}) = T (D_{i}) = 2 + s .

For $s > 0$ , define

B_{i} C_{i} = β s e_{i}, = χs (e_{i} + e_{i + 1}), T (B_{i}) T (C_{i}) = 2 s, = 2 s + b χs .

Vertex symbols mean their lifted graph points when used in cells.

For $s \leq 0$ , the active core face in sector $i$ is

V_{i} = [P, A_{i}, A_{i + 1}], T = s + a (ξ + ζ) .

For $s > 0$ , the sector faces are

U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + b ζ, = 2 s + b ξ, = s + a (ξ + ζ) .

For all $s$ , retain

Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], T = 2 + s,

and the three resting unbounded sector faces beyond $ρ = 4$ , with the exterior ray subdivisions.

Displayed formulas in this passage

VIII-B@3#eq:fixed-vertices · line 900
$A_{i} = r e_{i}, D_{i} = 4 e_{i}, T (A_{i}) = T (D_{i}) = 2 + s .$
VIII-B@3#eq:born-vertices · line 906
$B_{i} C_{i} = β s e_{i}, = χs (e_{i} + e_{i + 1}), T (B_{i}) T (C_{i}) = 2 s, = 2 s + b χs .$
VIII-B@3#eq:faces · line 921
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + b ζ, = 2 s + b ξ, = s + a (ξ + ζ) .$
VIII-B@3#eq:collar-faces · line 933
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], T = 2 + s,$
unlabeled · line 917
$V_{i} = [P, A_{i}, A_{i + 1}], T = s + a (ξ + ζ) .$