The hedronometric (“area-based”) Pythagorean Theorem for Right-Corner Tetrahedra generalizes to an unsurprising Law of Cosines: for face areas , , , and dihedral angles , , . 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 […]
Author Archive
Pseudofaces of Tetrahedra
Extending a Theorem of Barlotti
Barlotti’s Theorem states that an “affinely regular” polygon is the vertex sum of two regular polygons of the same type. This result can be interpreted as a straightforward decomposition of 2×2 matrices, which in turn can be extended to a decomposition of dxd matrices, which immediately gives rise to the (Extended) Barlotti Theorem for Multiple […]
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 […]
(Almost) Everything you need to remember about Trig …
The “Complete Triangle” figure —which represents the six trig functions as lengths of segments— is something of an obsession of mine, and has been for some time. It’s the basis of my triple-ricochet company logo and name of my iOS-specific brand, tricochet, and it’s the subject of my very first iOS app (one of the […]