Hodge dual: Difference between revisions
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{{wikipedia|Hodge star operator}} | {{wikipedia|Hodge star operator}} | ||
In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]], and vice versa. | In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]] (a kind of "comma wedgie"), and vice versa. | ||
== Definition == | == Definition == | ||
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In <math> \mathbb{R}^3 </math>, the Hodge star <math> \star: \Lambda^2 \, \mathbb{R}^3 \to \Lambda^1 \, \mathbb{R}^3 </math> acts on basis elements as: | In <math> \mathbb{R}^3 </math>, the Hodge star <math> \star: \Lambda^2 \, \mathbb{R}^3 \to \Lambda^1 \, \mathbb{R}^3 </math> acts on basis elements as: | ||
:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\star(e_1 \wedge e_2) &= e_3 \\ | \star(e_1 \wedge e_2) &= e_3 \\ | ||
\star(e_1 \wedge e_3) &= -e_2 \\ | \star(e_1 \wedge e_3) &= -e_2 \\ | ||
\quad \star(e_2 \wedge e_3) &= e_1 | \quad \star(e_2 \wedge e_3) &= e_1 | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
Applying this to <math> v_1 \wedge v_2 </math>: | Applying this to <math> v_1 \wedge v_2 </math>: | ||