OFFSET
0,1
COMMENTS
Smith & Smith conjecture that this is the Steiner ratio rho_3, the least upper bound on the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 3. Diaconis & Graham offer $1000 for proof (or disproof) of this conjecture.
This is an algebraic number of degree 4; the minimal polynomial is 25x^4 - 17x^2 + 1.
REFERENCES
Persi Diaconis and R. L. Graham, Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, Princeton University Press, 2011. See pp. 212-214.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 504.
LINKS
Warren D. Smith and J. MacGregor Smith, On the Steiner ratio in 3-space, Journal of Combinatorial Theory, Series A 69:2 (1995), pp. 301-332.
FORMULA
(3*sqrt(3)+sqrt(7))/10.
EXAMPLE
0.7841903733771222471083954778156877526538674944513592064535755...
MATHEMATICA
RealDigits[(3*Sqrt[3]+Sqrt[7])/10, 10, 120] // First (* Jean-François Alcover, May 27 2014 *)
PROG
(PARI) (3*sqrt(3)+sqrt(7))/10
(PARI) polrootsreal(25*x^4 - 17*x^2 + 1)[4] \\ Charles R Greathouse IV, Jan 05 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Charles R Greathouse IV, Dec 11 2012
EXTENSIONS
Formula and name simplified by Jean-François Alcover, May 27 2014
STATUS
approved