"James Waldby" > > I think your area formula went awry -- the values I see from it are > about 0.444, (presuming 11 Pi s / 21600 etc are in radians) rather > than in the neighborhood of 1.2990381, the area of a triangle > inscribed in a circle of radius 1. But that aside, although > 05:49:09.1233840851 and 06:10:50.8766159149 aren't too bad, > both 02:54:34.56169071* and 09:05:25.43830908 are much better, > giving areas about .000008 bigger. (More exactly, > area 1.2990353071, angles 119.831845018 120.082130171 120.086024811 > vs. 1.29902692089, angles 120.336310208 119.835723690 119.827966102) > > * About the same time that Robert Israel mentioned, in his > derivation via analysis and number theory. > > The four times mentioned above are the only ones that have neighborhoods > giving area > 1.299; from an exhaustive binary search, there are 16 time > neighborhoods that give areas between 1.298 and 1.299, eg 08:00:20.17865 > with area 1.29870. > -jiw
Ah well, my formula is nearly correct. I made a typo with one of the signs, it should be
(Sin[11 Pi s / 21600] + Sin[59 Pi s / 1800] - Sin[719 Pi s / 21600] ) / 2
This I get to be 1.2989558 when s = 20949. This is not too far from 3 Sqrt / 4 = 1.29903811
I then made the following mistake. A simple search using integer seconds values gave 20949 sec as a maximum. I then assumed incorrectly that using that value as a starting point would give the best value.
For a clock that only displays seconds, like many clocks, I think that the values 20949 & 22251 are quite good. However for a clock that displays seconds smoothly (for example, like mains driven ones) then the solution afforded by Robert Israel is correct.
I once repaired a quartz clock that incremened in 1/2 seconds. I think the best times for these are 10474.5 & 32725.5 = 02:54:34.5 & 09:05:25.5.