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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><b><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>From:</span></b><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'> extropy-chat [mailto:extropy-chat-bounces@lists.extropy.org] <b>On Behalf Of </b>Adrian Tymes<br><b>Subject:</b> Re: [ExI] tumbling pyramids again<o:p></o:p></span></p><p class=MsoNormal><o:p> </o:p></p><div><div><div><div><div><p class=MsoNormal style='margin-bottom:12.0pt'>>…I'm not sure, but I *think* this reduces to "keep the CG equidistant from all vertexes".<o:p></o:p></p></div></div></div></div><div><p class=MsoNormal style='margin-bottom:12.0pt'>…<o:p></o:p></p></div><div><p class=MsoNormal>>…3h/4 = sqrt( 2/4 + h^2/16 )<o:p></o:p></p></div><div><p class=MsoNormal>square:<o:p></o:p></p></div><div><p class=MsoNormal>9h^2/16 = 2/4 + h^2/16<o:p></o:p></p></div><div><p class=MsoNormal>multiply by 16:<o:p></o:p></p></div><div><p class=MsoNormal>9h^2 = 8 + h^2<o:p></o:p></p></div><div><p class=MsoNormal>subtract h^2 then divide by 8:<o:p></o:p></p></div><div><p class=MsoNormal>h^2 = 1<o:p></o:p></p></div><div><p class=MsoNormal>square root:<o:p></o:p></p></div><div><p class=MsoNormal style='margin-bottom:12.0pt'>h = 1<o:p></o:p></p></div><div><p class=MsoNormal>>…So if I have my math and assumption right, that comes to a height of 1 cm if the square base is 1 cm on a side. Adrian<o:p></o:p></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>Ja, this is an approach I took back in the same timeframe I was first starting to read ExI-chat, about 24 yrs ago. What I discovered back then is that the pyramid on its triangular face is higher off the table than when on its square face (.25 cm vs ~.335 cm) so the potential energy is lower when resting on the square face than on any of the triangular faces. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>Here’s my calc on that:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><img width=380 height=193 style='width:3.9583in;height:2.0104in' id="Picture_x0020_2" src="cid:image001.jpg@01D28830.0BC4CBA0"><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>At the risk of being considered by the Greek philosophers a crass empiricist, I took a superball and carved a square based pyramid with height approximately equal to side length, and tumbled it on my kitchen floor fifty times:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><img width=317 height=296 style='width:3.302in;height:3.0833in' id="Picture_x0020_1" src="cid:image002.jpg@01D28830.0BC4CBA0"><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>…and verified what I already suspected, that this shape is way more likely to land on the square face.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>I made a second one of dimensions h = 1.5 and tossed that one around a bit. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>However… this was kinda inconclusive by itself, never mind statistical significance for now. Reason: superball material is way too hyperactive for this application. (What the heck is that stuff? Google says it is called Zectron. By fun coincidence, the Google page right up top shows the kind of superball I carved up, with the stars embedded: <a href="https://en.wikipedia.org/wiki/Super_Ball">https://en.wikipedia.org/wiki/Super_Ball</a> )<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>OK so I just contacted my buddy with the 3D printer. He agreed to make me a set of square-based pyramids of various dimensions. We already know that h =1 is too short, and we can pretty much intuit that h =2 is too tall. Shall we create a betting pool? Doesn’t hafta be real money; they can be street cred points. If we do it that way, we can share lines of reasoning. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>Here’s mine: if we equalize the potential energy in either configuration, I get h = sqrt(2) or sqrt(10)/2 which in decimal is h = 1.414 or h = 1.581 but my experience so far is that both of those answers seem to be still too low. The 1.581 may not be low by much.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'>spike <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif'><o:p> </o:p></span></p></div></div></div></body></html>