Bob, I can email you a Mathematica notebook which generates Reuleaux tetrahedra. It isn't well documented or intended for general use, but it will generate a two-parameter family of rollers based on a rounded isosceles triangle. George http://georgehart.com/ On 1/8/2013 6:45 PM, Robert Baillie wrote:
Does anyone have the equations for either a Reuleaux or a Meissner tetrahedron that could be plugged into Mathematica?
Bob Baillie ---
Bill Gosper wrote:
Yow, we didn't know about this! Not even the revolved Reuleaux. MoMath neither?
Karzes>I wonder if Meissner tetrahedra (true constant width) would work better or worse than Reuleaux tetrahedra (approximate constant width) for this purpose.
Tom ----------- Better: Presumably, a flat plate resting on the slightly bogus Reuleaux tetrahedron will have a small but finite angle of repose.
Worse: If we round just one edge of a face on which the solid rests, it presumably will roll onto the adjacent face after < 60° tilt. But then it could pivot on that face and stagger back onto the previous one, etc., but,
Better: because one of the Meissners has a face with three sharp edges and an apparently lower center of gravity, with presumably > 60° repose, if placed carefully.
Actually, the abovementioned staggering wouldn't be necessary--once over an edge it would run away, under plausible scenarios of rolling friction and atmospheric viscosity.
Tom, thanks for the wakeup!
So the Meissner is really a rounded HEPtahedron. Here <http://gosper.org/roundtet.png>'s a rounded tetrahedron (monohedron?), unlikely to be constant width, with an extremely simple formula. --rwg
Bill Gosper writes:
MLB>
And what exactly does "smooth" buy us? Let's just ask instead: how would we quadrangulate a rounded tetrahedron? ------ I nearly slid through a stopsign recently. It was on a sharp downgrade, preceded by gravel washed off a hillside. Had the gravel been perfectly spherical, I wouldn't even have slowed down. Except spherical gravel would have rolled away. What about Reuleaux tetrahedra (http://www.youtube.com/watch?v=jK7xPo1YXzY 3:18)? Neil 3D printed a lovely one, ~1.5", and then split for CES w/o photographing it. It's eerie to roll a book over. Why aren't they sold at science museum gift shops? But would one stay put on a hill? Neil claims, with sufficient friction, an amazing 60°! They're much less round than you might think. This has 007esque possibilities, not just for foot chases and booby-traps. Imagine pursuing Bond on a winding road where he releases a few pounds of Reuleaux gravel. You say
http://www.zombiesurvivalwiki.com/thread/3341691/Liquid+Banana+Peel+-+Make+a...
is cheaper? OK, but it leaves a residue. For the perfect crime, use dry ice Reuleaux gravel. --rwg http://gosper.org/reuleauxtet.png
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