20160601, 14:23  #1 
"Mike"
Aug 2002
2^{2}·3^{2}·223 Posts 
June 2016

20160603, 01:28  #2 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·7·11·13 Posts 
I can solve this for maximum filled volume. But I am stumped on the variable volume optimization.

20160603, 17:33  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·7·11·13 Posts 
This is how I figure the solution.
Notes: * The actual angles are unreadable (I hope) * Instructions for viewing XEyed 3D images can be found on the web: http://www.starosta.com/3dshowcase/ihelp.html 
20160604, 14:29  #4 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2·7·11·13 Posts 
Well my submission is not showing up, which probably means there is a better (more accurate) optimization possible. I did a successive approximation style search by checking the CenterofMass' offset by checking 16 successive fill levels (2^4). There were 2 offsets to the right (towards the edge), one above and one below the center of the icosahedrons. The one below was larger so I made a more precise locating by further successive approximation(all manually).
I assume to find an optimization with a greater CenterofMass offset, one could write a script to check COMs for a much higher number of fill percentages. But That would be too much work for me. 
20160604, 16:44  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×7×11×13 Posts 
I examined CentersofMass at 64 different heights of filled volume. I don't think there can be a better optimization than what I already had.

20160622, 02:05  #6 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×7×11×13 Posts 
There are some names up. For the record I had to resubmit the answer with a higher precision/accuracy.
* The fully loaded container will always have its COM at the same point in the center of the symmetrical geometry. * A partially loaded container however is not (necessarily) symmetrical * Unlike a fully loaded container whose COM remains constant at any tilt angle, a partially filled container will have its COM shift dynamically as the container is tilted and hence the geometry transforms. 
20160629, 01:26  #7 
Jun 2016
2^{3} Posts 
Hi, I have just submitted a solution but it is very close to the deadline and it sometimes takes a while to find out if your solution is correct or not. Could someone who has already got a correct solution PM me so I can give them my numerical solution and check if it correct or if I have made a mistake somewhere?

20160704, 13:59  #8 
"Mike"
Aug 2002
2^{2}·3^{2}·223 Posts 

20160705, 02:12  #9 
Romulan Interpreter
Jun 2011
Thailand
2×4,679 Posts 
Ha! Looking through that guy's paper, I would actually call that "an elegant solution"!!
I don't know how this problem could be solved more elegant, beside some animation/movie. You can't skip the formulas part anyhow. We also solved this "by hand", using AutoCad and got the right tilt and fill, but were too lazy to put it in a nice form. We were thinking to a small lisp program to show it in acad and capture the animation, but we were too busy to write it (or too stupid to write it fast, in the limit of the available time). Something like "tilt it in this direction, section the thing a little, project the mass center, section the thing a little bit more, etc, if the center of the mass projects inside of the base, tilt a bit more, etc, then change the tilt direction by one degree and start from the beginning (only 36° due to symmetry of the tilting direction); record the max angle in all cases, send the minimum angle and the section (volume) to the guys. But we only dreamed to do this, all the month... never did it... Last fiddled with by LaurV on 20160705 at 02:18 
20160705, 03:46  #10  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5×1,877 Posts 
Quote:
Link #1 is not a valid PDF file  Quote:
Quote:


20160705, 07:59  #11 
Romulan Interpreter
Jun 2011
Thailand
2×4,679 Posts 
Your "linker" is confused. (or linkator? or pdf viewer? hehe). The first link is a valid pdf link, with a nice Wolfram Mathematica solution.

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