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 […]

## Archive for the ‘**Hedronometry**’ Category

## What more I know about hyperbolic tetrahedra

## 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 […]

## 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 […]

## What is the Pythagorean Theorem for Right-Corner Simplices in Hyperbolic 4-Space?

The Pythagorean theorem in hyperbolic 2-space is fundamental, and I discovered the hedronometric 3-space analogue years ago (see here), but what about dimension 4 and beyond? As discussed in the Heron-like Strategies for Non-Euclidean Tetrahedral Volume post, my investigations into the 4-space case stalls-out because hyperbolic volume is defined in terms of the thorny Derevnin-Mednykh […]

## Tetrahedra Sharing Volume, Faces Areas, and Circumradius

From the abstract of my note “Tetrahedra Sharing Volume, Face Areas, and Circumradius: A Hedronometric Approach”: Volume, face areas, and circumradius sometimes determine multiple —even infinitely-many— non-isomorphic tetrahedra. Hedronometry provides a context for unifying and streamlining previous discussions of this fact. This was my first attempt to solve someone else’s problem with hedronometry. I’m rather […]