Between them Michael Reid and Warren Smith seem to have triangles satisfactorily stitched up. Now if I may, I'll throw in my next two-penn'orth: I have inspected every Heronian tetrahedron (ie. with integer edge-lengths, face areas, volume) having diameter up to 1000, and lo! they can all be posed on the integer lattice. [There are just 27 primitive, non-flat cases; data is available on request. Quickly, before James Buddenhagen goes off to promptly quadruple it.] So, for my next (rather more dubiously supported) conjecture --- Is every Heronian tetrahedron posable on the 3-space integer lattice? And finally and even more wildly --- are we eventually going to discover octonions lurking around the corner somewhere in this discussion? Fred Lunnon On 11/18/11, Warren Smith <warren.wds@gmail.com> wrote:
Re Michael Reid's proof, I had earlier sent Lunnon email explaining a proof idea which I believe coincides with Reid's proof idea (I did not have a proof, but I did have this underlying idea), [snip] Lunnon now wonders: "What next? On to tetrahedra?" Well, maybe... I point out that in 4 dimensions, the "Hurwitz integers" are the quaternions A+B*i+C*j+D*k such that either (A,B,C,D) or (A+1/2, B+1/2, C+1/2, D+1/2) all are integers. http://en.wikipedia.org/wiki/Hurwitz_quaternion These enjoy a Euclidean-like GCD algorithm, albeit be careful since quaternion multiplication is noncommutative, Any 3D rotation+scaling can be expressed using quaternions as follows: Let (x,y,z), the 3-vector being rotated, be regarded as X=0+x*i+y*j+z*k and perform X --> Q*X*Qconjugate. [Also any 4D rotation+scaling can be expressed using quaternions as follows: X --> Q*X*R.]
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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