A tetrahedron has freedom 12, since each its 4 vertex points has freedom 3; but 4 lines on a quadric regulus have freedom 4 + 9 = 13. Therefore there must be a single further constraint on the 4 altitudes, besides their lying on a quadric. What could this be? Regarding the altitudes as 4 points on the Grassman quadric (4 quadratic constraints) in projective 5-space, the quadric constraint is equivalent to their also lying on a plane (3 further linear constraints). The missing constraint must surely be (skew-)symmetric in the altitudes, and rational involving quadratic or quartic polynomial; it must also be projectively (semi-)invariant, at any rate with respect to 3-space. One (slightly asymmetric) possibility might have been the "cross-ratio" of lines K,L,M,N defined by (K^L)(M^N) / (K^M)(L^N), where "^" denotes the Grassman wedge inner product: however this turns out to be the squared ratio of two polynomials, each quartic in the faces' components. Anyone got any more suggestions? Fred Lunnon On 10/30/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
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