[math-fun] generalized Pythagorean theorem
By the way, can anyone give me a reference for the generalization of the Pythagorean theorem that says that squared k-dimensional volume of a flat k-dimensional object sitting in n-space is the sum of the squared k-dimensional volumes of the k-dimensional shadows that the object casts in the n-choose-k different directions associated with an orthogonal coordinate frame? For instance, spin a coin in the air; the shadows that it casts on the floor and two perpendicular walls will rapidly change, but their squared areas will add up to a constant, namely the squared area of (one side of!) the coin. I seem to recall that this is due to Cayley or Sylvester or someone like that. But I'm more interested in where an interested math major might read about this result, rather than who first noticed it. (Surely there's an American Mathematical Monthly or Mathematics Magazine article about this!) Jim Propp
Jim & others, Only the faintest of recollections, and perhaps way off, but I think you're right about Cayley. Look in a book by Blumenthal Distance Geometry(???) perhaps Menger matrix? R. On Wed, 31 Oct 2007, James Propp wrote:
By the way, can anyone give me a reference for the generalization of the Pythagorean theorem that says that squared k-dimensional volume of a flat k-dimensional object sitting in n-space is the sum of the squared k-dimensional volumes of the k-dimensional shadows that the object casts in the n-choose-k different directions associated with an orthogonal coordinate frame?
For instance, spin a coin in the air; the shadows that it casts on the floor and two perpendicular walls will rapidly change, but their squared areas will add up to a constant, namely the squared area of (one side of!) the coin.
I seem to recall that this is due to Cayley or Sylvester or someone like that. But I'm more interested in where an interested math major might read about this result, rather than who first noticed it.
(Surely there's an American Mathematical Monthly or Mathematics Magazine article about this!)
Jim Propp
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I don't remember noticing this result in Blumenthal, and didn't actually know it --- but I think it probably follows immediately from the Grassman exterior product (or maybe it's the interior, I can never remember) representing the linear closure of a set of points: the associated magnitude gives the squared content of the simplex formed by their convex hull, and the individual components are the contents of the projections. [In a different guise, the same formula turns up in Geometric Algebra ...] WFL On 10/31/07, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Jim & others, Only the faintest of recollections, and perhaps way off, but I think you're right about Cayley. Look in a book by Blumenthal Distance Geometry(???) perhaps Menger matrix? R.
On Wed, 31 Oct 2007, James Propp wrote:
By the way, can anyone give me a reference for the generalization of the Pythagorean theorem that says that squared k-dimensional volume of a flat k-dimensional object sitting in n-space is the sum of the squared k-dimensional volumes of the k-dimensional shadows that the object casts in the n-choose-k different directions associated with an orthogonal coordinate frame?
For instance, spin a coin in the air; the shadows that it casts on the floor and two perpendicular walls will rapidly change, but their squared areas will add up to a constant, namely the squared area of (one side of!) the coin.
I seem to recall that this is due to Cayley or Sylvester or someone like that. But I'm more interested in where an interested math major might read about this result, rather than who first noticed it.
(Surely there's an American Mathematical Monthly or Mathematics Magazine article about this!)
Jim Propp
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I think I learned about this from Dan Klain and Gian-Carlo Rota's lovely book _Introduction to Geometric Probability_. But my math library is in boxes these days, so I can't be sure or offer a more precise pointer. --Michael Kleber On 10/31/07, James Propp <jpropp@cs.uml.edu> wrote:
By the way, can anyone give me a reference for the generalization of the Pythagorean theorem that says that squared k-dimensional volume of a flat k-dimensional object sitting in n-space is the sum of the squared k-dimensional volumes of the k-dimensional shadows that the object casts in the n-choose-k different directions associated with an orthogonal coordinate frame?
For instance, spin a coin in the air; the shadows that it casts on the floor and two perpendicular walls will rapidly change, but their squared areas will add up to a constant, namely the squared area of (one side of!) the coin.
I seem to recall that this is due to Cayley or Sylvester or someone like that. But I'm more interested in where an interested math major might read about this result, rather than who first noticed it.
(Surely there's an American Mathematical Monthly or Mathematics Magazine article about this!)
Jim Propp
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participants (4)
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Fred lunnon -
James Propp -
Michael Kleber -
Richard Guy