On 2016-09-22 17:59, Allan Wechsler wrote:

Son of a gun.

On Thu, Sep 22, 2016 at 8:05 PM, Bill Gosper <billgosper@gmail.com> wrote:

...

"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges." True or false? --rwg

Easier T|F: "The height of the square pyramid is the circumradius of its base, so that all five vertices are equidistant from that circumcenter." Eh? Obviously. Put the square's points on the equator, separated by 90º of longitude. Then the apex is at the north pole, separated by 90º of latitude, dividing the hemisphere into four equilateral spherical triangles, each with three right angles. Is this why a square pyramid of oranges has the same lattice as a tetrahedral pyramid of oranges? Final T|F: "The solid ∠ at the π/3,π/3,π/3,π/3 apex is exactly twice the solid ∠ of each π/3,π/3,π/2 base point." Spoiler: the pyramid is half an octahedron. (Whose solid ∠s = 4 ArcTan[1/√8] = 2 ArcCos[7/9] = 4 ArcCos[√8/3] = 4 ArcSin[1/3] = 8 ArcTan[√(Cot[5π/24] Tan[π/24]) Tan[π/8]] =ArcTan[56√2/17] == ArcSin[56√2/81] == ArcCos[17/81].) --rwg