Mathematica(@mathemetica):The wedge a ∧ b is the oriented parallelogram the two vectors already span. The Hodge star, using the metric and a choice of orientation, sends that plane element to the unique line standing perpendicular to it. Only in three dimensions does the dual of a bivector become another vector; in every other dimension the same operation yields a complementary multivector. The ordinary cross product is therefore a three-dimensional accident that Grassmann’s exterior algebra of 1844 already contained and that Gibbs later isolated.

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Mathematica@mathemetica

2026.09.23 13:25

The wedge a ∧ b is the oriented parallelogram the two vectors already span. The Hodge star, using the metric and a choice of orientation, sends that plane element to the unique line standing perpendicular to it. Only in three dimensions does the dual of a bivector become another vector; in every other dimension the same operation yields a complementary multivector. The ordinary cross product is therefore a three-dimensional accident that Grassmann’s exterior algebra of 1844 already contained and that Gibbs later isolated.