My previous tentative prospect for the maximal regular simplex inscribed within a unit hypercube in 6-space with side sqrt(5/2) = 1.581138830084 ... has been toppled by a tantalising new candidate with side sqrt(11 - 6 sqrt(2)) = 1.585786437627 ... and vertices [ d, d, d, d, d, d ], [ b, 0, c, 0, 1, 1 ], [ 1, b, 0, c, 0, 1 ], [ 1, 1, b, 0, c, 0 ], [ 0, 1, 1, b, 0, c ], [ c, 0, 1, 1, b, 0 ], [ 0, c, 0, 1, 1, b ]; where d = 0.0257642624106 , c = 0.7071067811865475 , b = 0.4142135623730950 satisfy b = sqrt2 - 1, c = sqrt/2, d^2 - (c + 1/3)d + (c 4/3 - 11/12) = 0 . Symmetry is 6-cyclic; the fixed vertex lies diagonally from the origin, separated from it by merely 1 part in 40 . One might reasonably expect this to be outperformed by some 6-cyclic contender having one vertex squarely at the origin. But it seems one would be mistaken: an exhaustive search found just three such candidates, with approximate sides 1.498028 , 1.511416 , 1.551629 . These constructions give only lower bounds on the maximum side obtainable. A straightforward upper bound sqrt(n+1)/2 = sqrt(7/2) = 1.870828693387 comes via the circumradius of the hypercube dominating that of the inscribed simplex. Less trivially, Charles Greathouse's Hadamard observation leads to slightly improved upper bound ( (1/80) / (sqrt(7/128)/720) )^(1/6) = 1.837453732988 . Neither is likely to be anywhere near as close as the current lower bound, which I expect be sharp to at least 2 decimals. Fred Lunnon On 10/1/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I got a little bit lucky for n = 9 , guessing a configuration which plonks one simplex vertex just outside the hypercube --- but scaled down to fit, still gives lower bound 9*(2*Sqrt(5) - 3*Sqrt(2)) = 2.065457... on the maximal side length --- but that is unlikely to be sharp.
The geometric problem I formulated earlier doesn't appear amenable to conventional "quadratic programming" or "convex programming" techniques, which apparently cope only with quadratic inequalities. It can be formulated in a nutshell: given the values of some subset comprising more than half the coordinate components of the vertices of a regular simplex, complete its coordinates.
Fred Lunnon
On 9/30/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Still contemplating Dan's n-space regular simplices inscribed in a unit hypercube, the way one does, for n = 9 now.
Given that for n = 7,8 the (conjectural) max side equals 2 , it was not a promising start that my rotation-jiggling search program took 12 hours to find just a couple of examples exceeding that: but the good-ish news is that for n = 9 a lower bound on the maximal side is 2.00452263 .
Such configurations as warrent further investigation do not seem to have a great deal evident in the way of symmetry. More promising is their preponderance --- in one case two-thirds --- of coordinate components near --- within say 0.02 --- of an integer --- 0 or 1 --- which, one may assume, are on the way to eventually converge to interval endpoints.
The freedom 1 + (n+1)n/2 similarity transforming current into limiting configuration is therefore by this stage well over-determined. If only there were an effective algorithm to compute it ...
I ran up program which forces these near-endpoints to endpoints, in the process, destroying the regularity of the simplex, then attempts to restore regularity by minimising the side-length variance, with respect to individual cooordinate components iteratively. This works a treat on smaller cases already solved, but disappointingly fails to progress on the problem to hand.
A less direct approach --- solving for the almost orthogonal projective matrix of the final transformation --- involves large numbers of simultaneous quadratic equations. This looks fairly horrible --- "quadratic programming", perhaps?
Fred Lunnon