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 […]

## Author Archive

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

## 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 […]

## Heron-like Strategies for Hyperbolic Tetrahedral Volume

With two Heron-like formulas for tetrahedral volume in Euclidean space, it makes sense to investigate what happens in non-Euclidean —specifically, hyperbolic— space. After all, a number of hedronometric formulas (for instance, the Laws of Cosines) have counterparts in both geometries. Unfortunately, the search for a hyperbolic hedronometric formula for volume is hindered by the fact […]

## Heron-like Results for Tetrahedral Volume

Heron’s formula provides the area, \(A\), of a triangle from the lengths, \(a, b, c\), of its edges: $$A = s (s-a)(s-b)(s-c) \qquad \text{where}\qquad s := \frac{1}{2}\left(a+b+c\right)$$ The Cayley-Menger determinant generalizes this formula and can provide the “content” of an any-dimensional simplex from the lengths of its edges, but I want something hedronometric. We cannot expect […]

## The Laws of Cosines for Non-Euclidean Tetrahedra

Some time ago, I derived a (the?) hedronometric (“area-based”) Pythagorean Theorem for tetrahedra in Non-Euclidean 3-space. $$\cos\frac{W}{2} = \cos\frac{X}{2} \cos\frac{Y}{2} \cos\frac{Z}{2} \pm \sin\frac{X}{2} \sin\frac{Y}{2} \sin\frac{Z}{2}$$ where, throughout, “” is “” in hyperbolic space, and “” in spherical space. Naturally, this leads to a Law of Cosines: $$\begin{align}\cos\frac{W}{2} = \cos\frac{X}{2} \cos\frac{Y}{2} \cos\frac{Z}{2} &\pm \sin\frac{X}{2} \sin\frac{Y}{2} \sin\frac{Z}{2}S […]