Leonardo da Vinci permutahedron picture?
May 1, 2005 10:22 PM Subscribe
Help me find a geometrical drawing by Leonardo da Vinci, of a "permutahedron" made of hexagons and squares. Does it appear in a book (or codex?) I can reference? Are there copyright issues?
In a museum exhibit in Vienna of Leonardo's machines, I saw a picture by him of a 3-d permutahedron. (You can make one by e.g. starting with a tetrahedron, file off all the corners creating 4 new triangles and turning the old triangles into hexagons, then file down all the old edges, making each one a rectangle. When you're done, there will be 8 hexagons and 6 rectangles. In Leonardo's picture they're regular.)
As I'm writing a book where these polyhedra feature prominently, it'd be cool if I could reference Leonardo. Obviously I'd be happiest if someone said "yeah, it's in the Codex Whateverificus, p37, here's an ISBN" but I'd appreciate any suggestion more pointed than "go to a library and look at everything he ever drew".
Separate question: once I find it, what are the copyright issues?
In a museum exhibit in Vienna of Leonardo's machines, I saw a picture by him of a 3-d permutahedron. (You can make one by e.g. starting with a tetrahedron, file off all the corners creating 4 new triangles and turning the old triangles into hexagons, then file down all the old edges, making each one a rectangle. When you're done, there will be 8 hexagons and 6 rectangles. In Leonardo's picture they're regular.)
As I'm writing a book where these polyhedra feature prominently, it'd be cool if I could reference Leonardo. Obviously I'd be happiest if someone said "yeah, it's in the Codex Whateverificus, p37, here's an ISBN" but I'd appreciate any suggestion more pointed than "go to a library and look at everything he ever drew".
Separate question: once I find it, what are the copyright issues?
Sorry. From the second link...
Luca Pacioli, De Divine Proportione, 1509, (Ambrosiana fascimile reproduction, 1956; Silvana fascimile reproduction, 1982).
Illustrated with beautiful solid-edge figures of polyhedra by Leonardo da Vinci, Pacioli wrote in Italian about the beauty of symmetry, proportion, the golden number, and polyhedra. (Available in German, Spanish, or French translation, but not English.)
posted by schyler523 at 11:01 PM on May 1, 2005
Luca Pacioli, De Divine Proportione, 1509, (Ambrosiana fascimile reproduction, 1956; Silvana fascimile reproduction, 1982).
Illustrated with beautiful solid-edge figures of polyhedra by Leonardo da Vinci, Pacioli wrote in Italian about the beauty of symmetry, proportion, the golden number, and polyhedra. (Available in German, Spanish, or French translation, but not English.)
posted by schyler523 at 11:01 PM on May 1, 2005
Possibly available in English next month. Or December. (Or never.) $20.00!
posted by IndigoJones at 3:29 PM on May 2, 2005
posted by IndigoJones at 3:29 PM on May 2, 2005
Response by poster: That does look like the book to reference, thanks! Unfortunately that first link is to a soccerene (with hexagons and pentagons) rather than a permutahedron (with hexagons and rectangles).
For those who are curious about the name, to make a permutahedron take a vector in Rn with all different coordinates (v1 ... vn), and permute the coordinates, getting n! different vectors. Then take the convex hull of these n! points.
Another recipe: take the space of n x n symmetric matrices with a fixed list (v1...vn) of eigenvalues. What can the diagonal look like in such a matrix? Answer: it has to be in the permutahedron above, and every point arises in the permutahedron arises this way. (The corners come from the diagonal such symmetric matrices.)
posted by Aknaton at 10:52 AM on May 3, 2005
For those who are curious about the name, to make a permutahedron take a vector in R
Another recipe: take the space of n x n symmetric matrices with a fixed list (v1...vn) of eigenvalues. What can the diagonal look like in such a matrix? Answer: it has to be in the permutahedron above, and every point arises in the permutahedron arises this way. (The corners come from the diagonal such symmetric matrices.)
posted by Aknaton at 10:52 AM on May 3, 2005
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posted by schyler523 at 10:48 PM on May 1, 2005