Plücker coordinates: Difference between revisions
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{{Wikipedia|Plücker embedding}} | {{Wikipedia|Plücker embedding}} | ||
'''Plücker coordinates''' (also known as the '''[[wedgie]]'''), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space. | '''Plücker coordinates''' (also known as the '''[[wedgie]]'''), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space. | ||
[[File:Plucker_embedding.png|thumb|400px|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]] | |||
== Definition == | == Definition == | ||
We have a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> consisting of the k-dimensional subspaces of <math>\mathbb{R}^n</math>. | We have a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> consisting of the k-dimensional subspaces of <math>\mathbb{R}^n</math>. | ||
The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes. | The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes. | ||