Archive for the ‘Open Question’ Category

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 […]

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

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 […]

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

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 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 […]

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

Spectral Realizations of Graphs

“Spectral realizations” of a (combinatorial) graph have two important properties: they are harmonious (each graph automorphism induces a rigid symmetry) and eigenic (replacing each vertex with the vector sum of its neighbors yields the same result as scaling the figure). My paper “Spectral Realizations of Graphs” describes a straightforward way of generating the spectral realizations of any graph, using […]

Posted 17 July, 2012 by Blue in Harmonious Figures, Open Question