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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><div><div style='border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0in 0in 0in'><p class=MsoNormal><b>From:</b> extropy-chat [mailto:extropy-chat-bounces@lists.extropy.org] <b>On Behalf Of </b>spike<br><b>Subject:</b> Re: [ExI] isaac's number<o:p></o:p></p></div></div><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><b><span style='font-size:14.0pt'>From:</span></b><span style='font-size:14.0pt'> extropy-chat [<a href="mailto:extropy-chat-bounces@lists.extropy.org">mailto:extropy-chat-bounces@lists.extropy.org</a>] <b>On Behalf Of </b>John Clark<br><b>Sent:</b> Wednesday, August 02, 2017 6:00 AM<br><b>To:</b> ExI chat list <<a href="mailto:extropy-chat@lists.extropy.org">extropy-chat@lists.extropy.org</a>><br><b>Subject:</b> Re: [ExI] isaac's number<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><div><div><p class=MsoNormal><span style='font-size:14.0pt'>On Tue, Aug 1, 2017 at 2:10 PM, spike <<a href="mailto:spike66@att.net" target="_blank">spike66@att.net</a>> wrote:<o:p></o:p></span></p></div><div><div><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p></div><blockquote style='border:none;border-left:solid #CCCCCC 1.0pt;padding:0in 0in 0in 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0in;margin-bottom:5.0pt'><p class=MsoNormal><span style='font-size:14.0pt'>>>>...Woolhouse's 11/144 = .076388889, equal to my Monte Carlo solution to about a part per million.<o:p></o:p></span></p><div><p class=MsoNormal><span style='font-size:14.0pt'> <o:p></o:p></span></p></div><p class=MsoNormal><span style='font-size:14.0pt'>>>>…Correction, my sim gives .0763889013, but I found a much faster way to do the calc by generating only three points and calculating an area.<o:p></o:p></span></p></blockquote><div><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p></div><div><div><p class=MsoNormal><span style='font-size:14.0pt'><o:p></o:p></span></p></div><p class=MsoNormal><span style='font-size:14.0pt'>>>…That my friend is impressive! John Clark <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p></div><div><div><p class=MsoNormal><span style='font-size:14.0pt'>>…John as much as I would like to soak up that sentiment, I can’t claim credit. The really impressive feat was the closed form solution found by Woolhouse. spike<o:p></o:p></span></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>OK, news from the geek-with-too-much-time-on-his-hands department: I took my Monte Carlo sim for the triangles and generalized it to 3 dimensions.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><b><span style='font-size:14.0pt'>Original question</span></b><span style='font-size:14.0pt'>: in a unit square with three random points, what is the average area of the triangle?<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><b><span style='font-size:14.0pt'>New question</span></b><span style='font-size:14.0pt'>: in a unit cube with four random points, what is the volume of the tetrahedron?<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>I studied that Woolhouse page BillK found, struggled with the integrals for a while and finally had to admit that I suck. I won’t be channeling The Donald any time soon. But I can write Monte Carlo sims of anything that moves and plenty of things that do not, so I did. I just finished it and haven’t even run it yet. Before I pull the trigger, I want to allow us to make guesses on what the answer will be.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>Last time, my mathematical intuition fell flat, but I want to try again. For reasons I will not explain, I now think this answer will come out somewhere around 0.00849.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>If you want to enter a guess, do so, or if you want to be less committed, just say higher or lower than my 0.00849. The grand prize will be my undying respect for all eternity, or until I forget, whichever comes first. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>If anyone here generalizes the Woolhouse integrals, I will totally bow down before you and declare my grubby self unworthy to gaze upon your brilliance. I will be seriously impressed as all hell and say good things about you.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>Alternative: BillK might find us another miracle page somewhere that answers the new question.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>Guesses? Higher or lower than 0.00849?<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'>spike<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p></div></div><div><div><p class=MsoNormal><span style='font-size:14.0pt'><o:p> </o:p></span></p></div></div><div><div><p class=MsoNormal><span style='font-size:14.0pt'> <o:p></o:p></span></p></div></div><p class=MsoNormal><o:p> </o:p></p></div></div></div></body></html>