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

Proposition: Positive face areas

Cellulation and the isolated particle event › Edges, lengths, and areas · lines 770–805 · proof included or linked

VII-F@1#prop:areas

The induced areas are

face V_{i} U_{i}^{-} U_{i}^{+} Q_{i} s < 0 \frac{3}{2} inactive inactive \frac{7}{2} s > 0 \frac{3}{2} - \frac{15}{22} s^{2} \frac{9}{11} s^{2} \frac{9}{11} s^{2} \frac{7}{2}

In particular, for $0 < s \leq 1$ ,

Area (V_{i}) \geq \frac{9}{11} > 0.

Proof. On a graph plane with gradient $a$ , induced area is Euclidean projected area multiplied by $1 - ∣ a ∣^{2}$ .

For $V_{i}$ , this factor is $1/3$ . The original projected triangle has area $9/2$ . Each new projected inner triangle has area

\frac{1}{2} a_{s} u_{s} = \frac{45}{44} s^{2} .

Subtracting the two inner projected areas and multiplying by $1/3$ gives the stated area of $V_{i}$ .

For either inner face, the area factor is $1 - 9/25 = 4/5$ , giving $9 s^{2} /11$ . The collar quadrilateral has projected area $(16 - 9) /2 = 7/2$ and is horizontal.

Displayed formulas in this passage

unlabeled · line 773
$face V_{i} U_{i}^{-} U_{i}^{+} Q_{i} s < 0 \frac{3}{2} inactive inactive \frac{7}{2} s > 0 \frac{3}{2} - \frac{15}{22} s^{2} \frac{9}{11} s^{2} \frac{9}{11} s^{2} \frac{7}{2}$
unlabeled · line 784
$Area (V_{i}) \geq \frac{9}{11} > 0.$
unlabeled · line 795
$\frac{1}{2} a_{s} u_{s} = \frac{45}{44} s^{2} .$

Proposition: Positive face areas

Displayed formulas in this passage

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