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From: Archimedes Plutonium on 14 Mar 2010 00:50


Archimedes Plutonium wrote:
(snipped)
>
> Now I am going to offer some proof of this Conjecture, but before I do
> that I want to remark that in 3D, I wonder if the tetrahedron is the
> maximum unit tiler? I wonder if that is true since, unlike the unit
> sphere that is always going to have gaps and holes between tangent
> spheres, we can pack tetrahedrons as parallelograms in vast reaches of
> 3D as solidly as unit cubes without any gaps in between and then when
> we reach the outer surface of the 3D object with its irregular shapes,
> the pointed ends of the tetrahedron are more likely to fit into those
> irregular shapes.
>
> So I do not know if I can generalize this Conjecture from 2D to 3D
> with equilateral-triangles to tetrahedrons.
> And also, the idea in 3D is that a wedge is the most
> versatile tiling 3D object, for a tetrahedron resembles either a wedge
> or a fulcrum.
>

Apparently I do not have enough tetrahedral dice to actually "seeing
is believing"
and where my mind-images are untrustworthy. I can see in 2D that the
equilateral
triangle forms parallelograms which can be packed solid with no gaps
in between in
2D. But I cannot see whether tetrahedron packed in 3D can end up with
no gaps and
be a solid packing. I can see where triangle on triangle of the
tetrahedron fit together, but
it appears as though, becuase the angles are 60 degrees that there is
a cleave-gap sooner
then later. So apparently tetrahedrons do not make a good tiler in 3D.

Apparently cubes or rectangular solids are better tilers in 3D.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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