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A348637 Largest clock triangle area.

%I A348637 #41 Jan 30 2023 19:01:02
%S A348637 1,2,9,9,0,3,5,3,0,7,1,0,7,3,3,2,6,3,2,7,3,9,7,2,0,1,1,3,8,8,2,4,9,6,
%T A348637 6,9,5,5,5,6,7,5,6,0,6,6,5,9,5,3,3,3,9,3,6,9,4,2,9,1,3,7,3,1,6,8,5,7,
%U A348637 8,8,4,1,6,8,5,6,0,5,6,7,7,7,5,7,1,2,4,8,2,5,5,1,9,4,6,4,3,1,3,6,3,7,8,6,8
%N A348637 Largest clock triangle area.
%C A348637 Consider an analog clock face to be a unit circle, with unit-length clock hands; the endpoints of the hands lie on the unit circle and form the vertices of a Clock Triangle inscribed within the circle.
%C A348637 The area within this Clock Triangle has maximum value 1.2990353071..., which occurs around 02:54:35 and at its mirror image around 09:05:25.
%C A348637 At time T seconds after 00:00:00, the clock hands are at angles
%C A348637       S (seconds hand) = T/60 * 360,      (degrees)
%C A348637       M (minutes hand) = T/60/60 * 360,
%C A348637       H (hours hand) = T/60/60/12 * 360.
%C A348637 The clock cycle repeats every 12 hours = 43200 seconds.
%C A348637 The second 6 hours of the cycle is a mirror image of the first 6 hours.
%C A348637 The area within the Clock Triangle at any time is equal to
%C A348637       F(T) = abs(sin(H-M) + sin(M-S) + sin(S-H))/2.
%C A348637 (The derivation of this equation is not overly-complicated.)
%C A348637 The hour and minute hands are exactly 120 degrees apart at times
%C A348637       T = 14400/11*(3k+1)   and   T = 14400/11*(3k+2)  for integer k.
%C A348637 There are 22 such times during every 12-hour cycle.
%C A348637 Empirically examining the relative extrema of F(T) near these 22 times, it is found that the largest F(T) occurs near T = 10475 (02:54:35), and near its mirror image T = 32725 (09:05:25).
%C A348637 Using Newton's iterative method to solve for Tmax in F'(Tmax) = 0,
%C A348637       Tmax = 10474.561690797181984...
%C A348637       F(Tmax) = 1.299035307107332...
%C A348637 Note: an equilateral triangle has area sqrt(3)*3/4 = 1.2990381056...
%D A348637 H. E. Dudeney, Amusements in Mathematics, Dover, 1958, pages 11 and 154; Problem #63, "The Stop-Watch", notes that at times 02:54:35 and 09:05:25 the clock hands are "nearly equidistant" and that "exact equidistance for the three hands is not possible", but does not point out that these two times are the most nearly equidistant times in the 12-hour clock cycle. Of the 430 puzzles in the book, 4 are represented on the front cover illustration; the stop-watch appears in the very center.
%H A348637 Henry Ernest Dudeney, <a href="https://archive.org/details/amusementsmathem00dude_881/page/n15/mode/2up">Amusements in Mathematics</a>, London, New York, Nelson, 1917.
%H A348637 Henry Ernest Dudeney, <a href="/A348637/a348637.pdf">Cover illustration of Amusements in Mathematics with clock</a>, New York, Dover, 1958.
%t A348637 f[x_] := (Abs[Sin[2*Pi*x*(1/43200 - 1/3600)] + Sin[2*Pi*x*(1/3600 - 1/60)] + Sin[2*Pi*x*(1/60 - 1/43200)]])/2; RealDigits[FindMaximum[f[x], {x, 10475}, WorkingPrecision -> 110][[1]], 10, 100][[1]] (* _Amiram Eldar_, Oct 27 2021 *)
%Y A348637 Cf. A104954, A347040, A350141.
%K A348637 nonn,cons
%O A348637 1,2
%A A348637 _Robert B Fowler_, Oct 26 2021