Archive for the ‘Open Question’ Category

A Ceva-like Theorem for Tetrahedra

A question at Math.StackExchange asked about “Ceva’s Theorem in three dimensions”. So I derived one. 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 […]

Posted 20 April, 2013 by Blue in Classroom, Misc. Math, Open Question

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 […]

Posted 16 February, 2013 by Blue in Hedronometry, Open Question

A Hedronometric Theorem of Menger

In 1928, Karl Menger outlined necessary and sufficient conditions for a set of edge lengths to determine an actual, non-degenerate, tetrahedron. The conditions amount to dead-simple sanity checks that the consequent face areas and volume have to be positive real numbers. In the short note “A Hedronometric Theorem of Menger”, I derive (as the title […]

Posted 24 November, 2012 by Blue in Hedronometry, Open Question

What I know about hyperbolic tetrahedra

Inspired by Mednykh and Pashkevich’s “Elementary Formulas for a Hyperbolic Tetrahedron”, I have compiled most of my disparate notes about hyperbolic hedronometry into one document: “Hedronometric Formulas for a Hyperbolic Tetrahedron”. I consider this a “living document” that I will update as I learn more about the subject matter. It’s primarily a formula look-up list […]

Posted 16 September, 2012 by Blue in Hedronometry, Open Question

The Descartes Rule of Sweeps

Something about┬áDescartes’ Rule of Signs┬áhad bothered me ever since my exposure to it in high school. As you know, the Rule of Signs runs something like this: For a polynomial with non-zero real coefficients, the number of positive roots is, at most, the number of sign changes in the coefficient sequence (ordered by power); more […]

Posted 20 July, 2012 by Blue in Misc. Math, Open Question