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From: Archimedes Plutonium on 13 Mar 2010 17:44
I sort of made the waters muddy when I first blurted out this
conjecture yesterday by using "any shape in 2D". Muddy waters because,
obviously if that shape is a unit square itself that the unit
equilateral triangle is not going to tile it. So what I meant to say
was that given a choice of a unit tiler, because a large number of
shaped 2D objects will be given to you to tile, the question is what
"unit tiler in 2D" is the best choice to make? Do we pick a unit
square? or a unit area circle?
or a unit right-triangle? This conjecture believes that the best
choice to make for a unit tiler is the equilateral triangle. For the
reason that two triangles can be situated to form a parallelogram and
thus imitating the tiling ability of a square. And if the object is
roundish, the equilateral triangle is best capable of tiling a rounded
object since its long pointed end acts as spokes on a wheel, so that a
circle is better tiled by equilateral triangles than is square tiling.

So let me try to write out the Conjecture as clear and
logical as possible:

Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
2D objects to tile by unit tilers, the unit tiler that achieves the
maximum tiling is the unit equilateral triangle.

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.

And let me touch on the topic of how and why I think this conjecture
is the equivalent of Least Action Principle in Physics. I believe they
are equivalent because the residue left over in the packing by
equilateral triangles is considered to be the residue of
maximum energy applied to the overall system. So that
if I can pack the object with maximum unit areas by the equilateral
triangles, then the residue portion is the reversal of what the "least
path in physics" is. So the equivalency is the idea that the reverse
of one is the other.

Possible proofs: of this conjecture of Equilateral Triangle Tiler:
(A) I already noted that if we are given a fractal geometry of a item
that is the same throughout as it becomes larger and larger, and if
the equilateral-triangle unit is the maximum tiler of this fractal
geometry, implies the proof.

(B) Let me offer an alternative proof scheme. The maximum tiler is
going to be the maximum tiler of various diameters of circles. So here
we focus on just
one object shape but of various diameters. It is an easy proof that
the hexagon composed of equilateral-triangles is the maximum tiler of
any given diameter circle. Unit squares or unit circles or unit
rectangles never come close to matching that 6 unit equilateral-
triangles tile a circle and all other unit objects will be less than
6.


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
From: Ostap S. B. M. Bender Jr. on 14 Mar 2010 21:56
On Mar 13, 3:44 pm, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> I sort of made the waters muddy when I first blurted out this
> conjecture yesterday by using "any shape in 2D". Muddy waters because,
> obviously if that shape is a unit square itself that the unit
> equilateral triangle is not going to tile it. So what I meant to say
> was that given a choice of a unit tiler, because a large number of
> shaped 2D objects will be given to you to tile, the question is what
> "unit tiler in 2D" is the best choice to make? Do we pick a unit
> square? or a unit area circle?
> or a unit right-triangle? This conjecture believes that the best
> choice to make for a unit tiler is the equilateral triangle. For the
> reason that two triangles can be situated to form a parallelogram and
> thus imitating the tiling ability of a square. And if the object is
> roundish, the equilateral triangle is best capable of tiling a rounded
> object since its long pointed end acts as spokes on a wheel, so that a
> circle is better tiled by equilateral triangles than is square tiling.
>
> So let me try to write out the Conjecture as clear and
> logical as possible:
>
> Equilateral Triangle Tiler Conjecture: In 2D, given a large number of
> 2D objects to tile by unit tilers, the unit tiler that achieves the
> maximum tiling is the unit equilateral triangle.
>

Wouldn't the answer depend on the nature of these "2D objects"?

>
> 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.
>
> And let me touch on the topic of how and why I think this conjecture
> is the equivalent of Least Action Principle in Physics. I believe they
> are equivalent because the residue left over in the packing by
> equilateral triangles is considered to be the residue of
> maximum energy applied to the overall system. So that
> if I can pack the object with maximum unit areas by the equilateral
> triangles, then the residue portion is the reversal of what the "least
> path in physics" is. So the equivalency is the idea that the reverse
> of one is the other.
>
> Possible proofs: of this conjecture of Equilateral Triangle Tiler:
> (A) I already noted that if we are given a fractal geometry of a item
> that is the same throughout as it becomes larger and larger, and if
> the equilateral-triangle unit is the maximum tiler of this fractal
> geometry, implies the proof.
>
> (B) Let me offer an alternative proof scheme. The maximum tiler is
> going to be the maximum tiler of various diameters of circles. So here
> we focus on just
> one object shape but of various diameters. It is an easy proof that
> the hexagon composed of equilateral-triangles is the maximum tiler of
> any given diameter circle. Unit squares or unit circles or unit
> rectangles never come close to matching that 6 unit equilateral-
> triangles tile a circle and all other unit objects will be less than
> 6.
>
> Archimedes Plutoniumhttp://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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