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Home » the hobbit the desolation of smaug - Imagemakers

the hobbit the desolation of smaug - Imagemakers

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 z_k = \text{cis}\left( \frac{2\pi k}{6} \right) = \text{cis}\left( \frac{\pi k}{3} \right), \quad k = 0, 1, 2, 3, 4, 5.

📌 $k=0$: $1$ → $(1, 0)$

📌 $k=1$: $\text{cis}(\pi/3) = \frac{1}{2} + i\frac{\sqrt{3}}{2}$ → $(0.5, \approx 0.866)$

📌 $k=2$: $\text{cis}(2\pi/3) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ → $(-0.5, \approx 0.866)$

📌 $k=3$: $\text{cis}(\pi) = -1 + 0i$ → $(-1, 0)$

📌 $k=4$: $\text{cis}(4\pi/3) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$ → $(-0.5, \approx -0.866)$

📌 $k=5$: $\text{cis}(5\pi/3) = \frac{1}{2} - i\frac{\sqrt{3}}{2}$ → $(0.5, \approx -0.866)$

📌 The node **opposite to $A = (1, 0)$** is the one diametrically opposite, i.e., rotated $180^\circ$, so $k = 3$:

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