[math-fun] Fwd: Re: generalized law of cosines
Gosper encountered a paper (via Askey?) with a generalized law of cosines for polygons and polyhedra. The paper subsumes "my" observation that the pythagorean theorem works for right-angled tetrahedra (cut from a cube corner). The paper doesn't mention higher dimensions, but presumably it's valid. Another nicety is that cos() is even, so you don't need to worry about assigning signs to the angles. Does the law of sines works for tetrahedra? The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star". Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:
for the nth side of an n-gon. Easy to remember. Per Askey. Rich: Math fun?
http://www.rowan.edu/open/colleges/las_new/departments/math/facultystaff/osl... --Bill
Nice paper. Section 2 claims in passing that his polygon analysis also applies to skew (non-planar) polygons, but I wonder. Is there a standard (or any) definition of "area" for a skew polygon? And his definition of angle between nonadjacent sides doesn't obviously extend to nonintersecting sides. - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Friday, April 06, 2012 5:13 PM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: [math-fun] Fwd: Re: generalized law of cosines
Gosper encountered a paper (via Askey?) with a generalized law of cosines for polygons and polyhedra.
The paper subsumes "my" observation that the pythagorean theorem works for right-angled tetrahedra (cut from a cube corner). The paper doesn't mention higher dimensions, but presumably it's valid. Another nicety is that cos() is even, so you don't need to worry about assigning signs to the angles. Does the law of sines works for tetrahedra?
The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star".
Rich
----- Quoting Bill Gosper <billgosper@gmail.com>:
for the nth side of an n-gon. Easy to remember. Per Askey. Rich: Math fun?
http://www.rowan.edu/open/colleges/las_new/departments/math/facultystaf f/osler/129%20Law%20of%20Cosines%20Generalized%20Published%20Paper.pdf
--Bill
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Presumably the angle between two (3-space) lines should simply be defined as the angle between their respective vectors. Using Stokes' theorem is witty; but cheating a little, given that the standard proof proceeds by applying a limiting process to a polyhedron! WFL On 4/7/12, Huddleston, Scott <scott.huddleston@intel.com> wrote:
Nice paper.
Section 2 claims in passing that his polygon analysis also applies to skew (non-planar) polygons, but I wonder. Is there a standard (or any) definition of "area" for a skew polygon? And his definition of angle between nonadjacent sides doesn't obviously extend to nonintersecting sides.
- Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Friday, April 06, 2012 5:13 PM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: [math-fun] Fwd: Re: generalized law of cosines
Gosper encountered a paper (via Askey?) with a generalized law of cosines for polygons and polyhedra.
The paper subsumes "my" observation that the pythagorean theorem works for right-angled tetrahedra (cut from a cube corner). The paper doesn't mention higher dimensions, but presumably it's valid. Another nicety is that cos() is even, so you don't need to worry about assigning signs to the angles. Does the law of sines works for tetrahedra?
The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star".
Rich
----- Quoting Bill Gosper <billgosper@gmail.com>:
for the nth side of an n-gon. Easy to remember. Per Askey. Rich: Math fun?
http://www.rowan.edu/open/colleges/las_new/departments/math/facultystaf f/osler/129%20Law%20of%20Cosines%20Generalized%20Published%20Paper.pdf
--Bill
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 4/7/12, rcs@xmission.com <rcs@xmission.com> wrote:
... The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star".
Rich
The relation between the sides and diagonals of a plane quadrilateral is that the volume of the tetrahedron with those edge-lengths vanishes. The volume is given by the Cayley-Menger determinant for 3-space, a cubic in the squares of the edge-lengths. See eg. http://en.wikipedia.org/wiki/Distance_geometry [I know most of you have heard all this before several times --- Rich obviously wasn't paying attention.] WFL
participants (3)
-
Fred lunnon -
Huddleston, Scott -
rcs＠xmission.com