[ExI] tumbling pyramids again

spike spike66 at att.net
Thu Feb 16 16:38:28 UTC 2017



From: extropy-chat [mailto:extropy-chat-bounces at lists.extropy.org] On Behalf Of Adrian Tymes
Subject: Re: [ExI] tumbling pyramids again


>…I'm not sure, but I *think* this reduces to "keep the CG equidistant from all vertexes".


>…3h/4 = sqrt( 2/4 + h^2/16 )


9h^2/16 = 2/4 + h^2/16

multiply by 16:

9h^2 = 8 + h^2

subtract h^2 then divide by 8:

h^2 = 1

square root:

h = 1

>…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





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.  


Here’s my calc on that:




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:



…and verified what I already suspected, that this shape is way more likely to land on the square face.


I made a second one of dimensions h = 1.5 and tossed that one around a bit. 


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:     https://en.wikipedia.org/wiki/Super_Ball  )


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. 


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.






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