Archive for the ‘Hedronometry’ Category

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 […]

Posted 18 July, 2012 by Blue in Hedronometry, Open Question

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 determinantĀ 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 […]

Posted 17 July, 2012 by Blue in Hedronometry, Open Question

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 […]

Posted 17 July, 2012 by Blue in Hedronometry

Pseudofaces of Tetrahedra

The hedronometric (“area-based”) Pythagorean Theorem for Right-Corner Tetrahedra generalizes to an unsurprising Law of Cosines: for face areas , , , and dihedral angles , , .Ā I recently learned (while browsing Boyer’s A History of Mathematics at a Barnes & Noble) that, as early as 1803, the mathematician Carnot was aware of this result —which […]

Posted 17 July, 2012 by Blue in Hedronometry

About the “Hedronometry” Category

By the end of high school, most students should be familiar with the “three-dimensional” Pythagorean theorem (aka, “the distance formula”) that relates the length of the diagonal of a shoebox to the lengths of its edges. (Students should also know that this generalization extends to relate diagonals and edges in “any-dimensional” shoeboxes; always, the square […]

Posted 16 July, 2012 by Blue in Hedronometry