## 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 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 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 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 in Hedronometry, Open Question

## The Descartes Rule of Sweeps

Something about Descartes’ Rule of Signs had 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 in Misc. Math, Open Question