On 10/30/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
... A long time ago I came across a charmingly antiquated tome: Nathan Altshiller-Court "Modern pure solid geometry" (1935) which I seem to remember stated the theorem that the altitudes always lie on a quadric; though his notation --- undefined --- for this property was something confusingly obscure. There were several other theorems of a similar nature.
Can anybody confirm this? Has any further work in this direction taken place more recently?
WFL
Using Cl(3,0,1) geometric algebra (DCQ's), Court's theorem that the 4 altitudes lie on a quadric is reduced brutally to triviality. Let the equations of the tetrahedral face planes be represented by generic vectors E = [Eo,Ex,Ey,Ez], F, G, H; the vertices have covectors P = <F G H>_3, Q, R, S, where the product is Clifford; the altitudes have Pluecker bivectors K = <E P>_2, L, M, N, their components quadric polynomials in the faces' components. It is now a matter of computation to verify that the 4x6 matrix of the latter has rank 3, showing that the altitudes are linearly dependent, and therefore lie on a quadric. QED Fred Lunnon