Groan department --- for "simplex" read "regular simplex". WFL On 10/23/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
During a hunt for cyclic configurations improving the current lower bound 2.065457 on the side of a maximal simplex inscribed in the unit hypercube in (n = 9)-space, a specimen emerges equipped with intriguing symmetry. Its n+1 Cartesian vertices comprise n cyclic shifts of seed rows
[a, a, a, a, a, a, a, a, a], [c, 0, 0, 1, b, 1, b, 0, 1]
where
a = 0.008275807801733629 or 0.9490527540582451 , b = 0.5549581320873712 , c = 0.1980622641951620
satisfy
(horrible sextic), b^3 - 2*b^2 - b + 1 = 0 , c = b^2 - 2*b + 1 .
Cyclic permutations of seed rows obviously leave the simplex invariant, and reversals yield another congruent simplex. But besides such trivial symmetries, the permuted second row seeds
[c, 0, 0, 1, b, 1, b, 0, 1]; [c, 1, 0, b, 0, 0, 1, 1, b]; [c, 0, 1, 1, 0, 1, b, b, 0];
yield a triad of essentially distinct, congruent inscribed simplices; each has the same set of 4 formally distinct, numerically equal squared sides
4 b^2 - 2 b c - 2 b + 2 c^2 - 2 c + 4 , 2 b^2 - 2 b + 2 c^2 + 4 , 9 a^2 - 4 a b - 2 a c - 6 a + 2 b^2 + c^2 + 3 , 4 b^2 - 6 b + 2 c^2 - 2 c + 6 .
Sadly, the side evaluates to a disappointing 1.893277 , which will hardly set any pulses racing (even mine).
Still maybe something interesting is going on here --- while superficially it might appear that one should expect a shedload of locally maximal configurations, perhaps there is some deeper reason why in practice far fewer can actually turn up.
Fred Lunnon