[math-fun] Cubic sections
(Simple stuff I couldn't visualize w/o plotting.) Rotate a unit cube about a major diagonal. Its intersection with a plane containing that diagonal varies (with period pi/3) between a rhombus (of side rt(5)/2) and a 1 by rt2 rectangle. --rwg Who might be able to tell me that "my" Fourier expansion of Jacobi am is a) Centuries old, or b) Just what we've been looking for. ?
On Sat, Dec 11, 2010 at 7:35 PM, Bill Gosper <billgosper@gmail.com> wrote:
(Simple stuff I couldn't visualize w/o plotting.) Rotate a unit cube about a major diagonal. Its intersection with a plane containing that diagonal varies (with period pi/3) between a rhombus (of side rt(5)/2) and a 1 by rt2 rectangle. --rwg
Forgot to add: It's always a parallelogram. See http://gosper.org/cubewedges.pdf for the rotating cube dodecasected into congruent tetrahedra by the plane, showing that the (in fact nonexistent) azimuthal dependence of the center of oscillation of a cube swung by a corner would be tiny at most. (The larger faces are halves of the aforementioned rectangles and rhombi. Note seeming pi/3 symmetry is really 2pi/3.) --rwg
Who might be able to tell me that "my" Fourier expansion of Jacobi am is a) Centuries old, or b) Just what we've been looking for. ?
The obvious place was dlmf, but their "Fourier" series wasn't real and was inapplicable for the pendulum (m>1, K(m) imaginary case.)
On Mon, Dec 13, 2010 at 12:07 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sat, Dec 11, 2010 at 7:35 PM, Bill Gosper <billgosper@gmail.com> wrote:
(Simple stuff I couldn't visualize w/o plotting.) Rotate a unit cube about a major diagonal. Its intersection with a plane containing that diagonal varies (with period pi/3) between a rhombus (of side rt(5)/2) and a 1 by rt2 rectangle. --rwg
Forgot to add: It's always a parallelogram. See http://gosper.org/cubewedges.pdf for the rotating cube dodecasected into congruent tetrahedra by the plane, showing that the (in fact nonexistent) azimuthal dependence of the center of oscillation of a cube swung by a corner would be tiny at most. (The larger faces are halves of the aforementioned rectangles and rhombi. Note seeming pi/3 symmetry is really 2pi/3.) --rwg
Optional: Enclose the dodecasected cube in two concentric lucite shells with clock hands painted on. Who might be able to tell me that "my" Fourier expansion of Jacobi am is
a) Centuries old, or b) Just what we've been looking for. ?
The obvious place was dlmf, but their "Fourier" series wasn't real and was inapplicable for the pendulum (m>1, K(m) imaginary case.)
Oh foo, it's just something called Jacobi's Real Transformation, which basically reciprocates the parameter and changes dn to cn. And am to something perhaps unnamed: JacobiAmplitude[u, m] == ArcSin[JacobiSN[Sqrt[m] u, 1/m]/Sqrt[m]] == ( ArcCos[JacobiDN[Sqrt[m] u, 1/m]] JacobiSN[Sqrt[m] u, 1/m])/( Sqrt[m] Sqrt[1 - JacobiDN[Sqrt[m] u, 1/m]^2]) Well these large m series, which really are Fourier series, certainly should be listed next to the small m Fourier series, which simply fail for m>1. The large m series represent the pendulum beautifully, but seem unknown to pendulologists webwide. --rwg
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Bill Gosper