Limits are about the journey …
A Ceva-like Theorem for Tetrahedra
A question at Math.StackExchange asked about “Ceva’s Theorem in three dimensions”.
The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with “triple-ratios” of triangle areas, to satisfying effect. Whereas the ratios in Ceva’s Theorem reduce to \(1\) (or “\([1:1]\)”), my triple-ratios reduce to \([1:1:1]\).
I like the result enough that I’ve tidied the derivation, added a smidgeon of context, and touched-on the (current) lack of a corresponding Menelaus-like theorem. See my note: “A Ceva-like Theorem for Tetrahedra” (PDF).
Proof without Words: Angle Sum and Difference Formulas
One of my favorite images:
What more I know about hyperbolic tetrahedra
I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF), with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.)
The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough that I won’t attempt to render it here.
The Open Question: As one might expect, the formula involves an integral. One of the limits of integration is the subject of a Conjecture. Again, the notion is too complicated to describe here, but the gist is that I *believe* that, by appropriately assigning names to the faces (and pseudo-faces), we guarantee that the lower limit is simply one-quarter of a particular pseudo-face area. (If I’m mistaken, then that limit is a less-obvious root of a trigonometric equation.) Numerical experiments in Mathematica suggest that the conjecture is true, but I don’t have even non-constructive proof. (Nevermind that the conjecture wouldn’t really be helpful without a practical way to determine what an “appropriate assignment” of names would be.) When (if?) a properly symmetric formula is finally discovered, the order of face names won’t matter at all; but for now, it makes for an irksome little wrinkle in the formula.
This … is … “Maximal Jeopardy!”
What’s the largest possible amount of money you can win in a single game of TV’s “Jeopardy!”?
Read my note This … is … “Maximal Jeopardy!” to find out.