Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Vertices and faces

Cellulation and the isolated particle event › Vertices and faces · lines 565–636

Write $A_{i} = 3 v_{i}$ and $D_{i} = 4 v_{i}$ for spatial vertices on the cone. Their lifted time coordinate is $2 + s$ .

For $s > 0$ , set

a_{s} = \frac{3}{2} s, u_{s} = \frac{15}{11} s,

and introduce

B_{i} = a_{s} v_{i}, C_{i} = u_{s} (v_{i} + v_{i + 1}) .

All coordinates in this section are spatial unless a time coordinate is explicitly supplied.

The complete vertex data in one sector are

vertex P A_{i} A_{i + 1} D_{i} D_{i + 1} B_{i} B_{i + 1} C_{i} sector coordinates (0, 0) (3, 0) (0, 3) (4, 0) (0, 4) (\frac{3}{2} s, 0) (0, \frac{3}{2} s) (\frac{15}{11} s, \frac{15}{11} s) T s (s \leq 0), 2 s (s \geq 0) s + 2 s + 2 s + 2 s + 2 2 s 2 s \frac{31}{11} s

The last three rows are regular vertices only for $s > 0$ .

For $s < 0$ , the core face in sector $i$ is the triangle

V_{i} = [P, A_{i}, A_{i + 1}], T = s + \frac{2}{3} (ξ + η) .

For $s > 0$ , replace it by the following three faces:

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 + \frac{3}{5} η, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + η) .

The face $V_{i}$ is a simple, generally nonconvex pentagon.

For every $s$ , the collar-side face is

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

Beyond $ρ = 4$ , retain the fixed unbounded face

Z_{i} = {(ξ, η) \in S_{i} : ξ + η \geq 4}, T = s + 2.

The three exterior rays starting at the $D_{i}$ are also retained.

Displayed formulas in this passage

unlabeled · line 610
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], T T T = 2 s + \frac{3}{5} η, = 2 s + \frac{3}{5} ξ, = s + \frac{2}{3} (ξ + η) .$
unlabeled · line 571
$a_{s} = \frac{3}{2} s, u_{s} = \frac{15}{11} s,$
unlabeled · line 577
$B_{i} = a_{s} v_{i}, C_{i} = u_{s} (v_{i} + v_{i + 1}) .$
unlabeled · line 586
$vertex P A_{i} A_{i + 1} D_{i} D_{i + 1} B_{i} B_{i + 1} C_{i} sector coordinates (0, 0) (3, 0) (0, 3) (4, 0) (0, 4) (\frac{3}{2} s, 0) (0, \frac{3}{2} s) (\frac{15}{11} s, \frac{15}{11} s) T s (s \leq 0), 2 s (s \geq 0) s + 2 s + 2 s + 2 s + 2 2 s 2 s \frac{31}{11} s$
unlabeled · line 603
$V_{i} = [P, A_{i}, A_{i + 1}], T = s + \frac{2}{3} (ξ + η) .$
unlabeled · line 626
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], T = s + 2.$
unlabeled · line 631
$Z_{i} = {(ξ, η) \in S_{i} : ξ + η \geq 4}, T = s + 2.$