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Decimal expansion of (3/8) * sqrt(3).
1

%I #60 Dec 24 2024 11:35:11

%S 6,4,9,5,1,9,0,5,2,8,3,8,3,2,8,9,8,5,0,7,2,7,9,2,3,7,8,0,6,4,7,0,2,1,

%T 3,7,6,0,3,5,5,1,9,7,0,1,7,8,8,9,2,7,3,5,5,2,0,9,2,7,6,1,7,2,9,4,4,7,

%U 4,8,8,1,3,4,0,8,0,0,0,1,3,9,0,5,4,2,9,8,2,0,0,3,3,9,6,8,2,1,5,8,7,8,3,5,9,8,0,3,0,3,0,7,7,7,5,1,3,6,3,6

%N Decimal expansion of (3/8) * sqrt(3).

%C This is the area of the regular hexagon of diameter 1.

%C From _Bernard Schott_, Apr 09 2022 and Oct 01 2022: (Start)

%C For any triangle ABC, where (A,B,C) are the angles:

%C sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],

%C cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],

%C and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:

%C (ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).

%C The equalities are obtained only when triangle ABC is equilateral. (End)

%D O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.

%D D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

%H Scott Brown, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv36n5.pdf">Problem 3453</a>, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

%F Equals A104954/2 or A104956/4.

%e 0.649519052838328985...

%t RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

%o (PARI) sqrt(27)/8 \\ _Charles R Greathouse IV_, Apr 09 2022

%Y Cf. A002194 (sqrt(3)), A104954.

%Y Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

%K cons,nonn

%O 0,1

%A _Kritsada Moomuang_, Mar 15 2020